Adaptive Multiscale Schemes for Conservation Laws

ut

o

u x,

1 n

u o(x )
(1.6 )

11 T

o

an

d

L f (u ) n i sp dx dt i

i= l

holds for all test fun ctions ip E Co=' ( Rd+l ) where n den otes the outer unit n ormal to aD. In case of an I V? the right hand side vanishes .

Note , that equation (1.6) correspo nds in a nat ural way to the integral equatio ns, so-called balan ce equatio ns , deriv ed in cont inuum mechanics. In t his setting t he conse rvation laws (1.1) can be viewed as local balan ce equations provided local regularity ass um ptions hold . Obviou sly, weak solutions are not necessarily smooth . Hence, discontinui t ies are now admiss ible. However , t he disconti nui ti es are not arbitrary but have to sat isfy t he well- known Ran kin e-Hugoniot conditions which are present ed in t he following t heo rem for general mul ti dimension al sys tems of conse rvation laws. Theorem 2. (Rankin e- Hug on iot con ditions) Let u E L I::c([O , T] x D, Rm) be a pi ecewise smooth weak solut ion of (1.1) and I" := {( t , x) : g(t, x) = O} a discon tinuity surfa ce which is im plici tly given by th e fu n ct ion g : [0, T] x D --+ R. Th e norm al vector of I" can be repre se n ted by n *(t , x ) = n " = (no , n df

= (Otg, oXlg, · ·· , oXdgf .

(1.7)

Th en f or th e lim iting value s u ±(t , x ) := limJ-+o± u (t , x + J n (t , x )) the relati on d

- L n i( fi( u +) i= l

holds.

fi(u _ )) = no(u + - u., )

(1.8)

1.1 Conservation Laws

5

Sketch of proof: (See also [Liu76).) Consider a neighborhood Dc (0, T) x n of a point (t, x) E I" which is split into two parts by the discontinuity surface . In each of the two sub domains the solution u is smooth according to the assumptions. Therefore the partial derivatives in the integral equation (1.6) can be shifted on each subdomain from the test function ip to u and fi , respectively, by means of integration by parts. Due to the evolution equation (1.1) only the boundary integrals corresponding to the discontinuity surface do not vanish. Then the assertion immediately follows for meas(D) -+ 0 and incorporating (1.7). 0 Unfortunately, the Rankine-Hugoniot conditions do not ensure uniqueness of the IVP, i.e. , (1.8) is only a necessary condition. As is verified in several textbooks , e.g. [LeV92, Kro97], it is possible to construct different weak solutions of the IVP. In order to characterize a " physically meaningful" solution additional constraints have to be imposed. In the literature, there exists a variety of so-called entropy concepts motivated by the entropy in thermodynamics by which a unique solution can be characterized in some cases. For scalar equations Kruzhkov's concept seems to be the most general concept and shall be presented here . Definition 2. (Kruzhkov's entropy condition [Kru70j) A weak solution is called an entropy solution if and only if the inequality

1l 11 T

o

o

lu - kl-;::l

0

ao

o
i=1

o x;

+ (1.9)

T

o

~

k) ~ IIi(u) - kl~ dxdt

d

sign(k)

L Ifib(u(t,x))) i=1

fi(k)1 niif dx dt

~0

holds for all nonnegative test functions if E CO'(R+ X Rd ) and all k E R. Here n denotes the outer unit normal to on and 'Y : BV((O, T) x n) -+ £ 00 ( {O} x n) x £ 00((0, T) x on) the trace map according to [BBB73, BLN79]. In case of an IVP the boundary integral vanishes.

Note, that in the scalar case the known entropy concepts are equivalent . In particular, the Kruzhkov entropy criterion coincides with the entropy pair concept, because the cone of convex functions is spanned by [z - yl. By means of (1.9) Kruzhkov proved that the entropy solution of the IVP is unique [Kru70). The existence was proved by Kuznetsov [Kuz67). For the IBVP existence and uniqueness were also established [BLN79, Kr(97). For syst ems of conservation laws there exist by now several entropy concepts but they are not as mature as in the scalar case. For a discussion on this subject the reader is referred to Dafermos' book [DafOO) where several concepts are discussed in the context of continuum mechanics.

6

1 Model Problem and It s Discretiz ation

1.2 Finite Volume Methods In general a closed form analytical repr esentation of t he solution of t he IVP and t he IBVP is not available. In order t o det ermine an approximat e solut ion several st rategies hav e been investi gated. A naiv e discretiz ation is based on finite differences. However , in the pr esence of discontinuities t hey produce oscillations which cau se a breakdown of the computation. These schemes can be stabilized by adding artificial viscosity which lead s to a significant loss in acc uracy. In particular , t he sha pe of discontinuities is smeared . Nevertheless finit e difference schem es are frequently applied in industry, e.g. aviat ion indu stry, since t heir impl ement ation is easy and they are very efficient du e to t heir simple structure which is suppor ted by modern compute r hardware, in par ticular vector architectures. Important representatives for finit e difference schem es ar e the Lax -Friedrichs scheme [Lax54] which becam e a helpful tool in proving exist ence and uniqueness of conservation laws and the Lax-Wendroff schem e [LW60] as a typical higher order discretization. Another strategy is based on finite elem ent m ethods. On ce again , standard Galerkin discr etizations usually applied to ellipt ic problems are not stable when applied to conservation laws. These schemes also have t o be stabilized by additional viscosity , see for instanc e streaml ine-diffusion methods [Joh87]. In recent yea rs a new approac h has been derived based on discontinuous, piecewise pol ynomial fun ctions, so-called discontinuous Gal erkin discretizations [CKS99]. This approac h can be viewed as a combination of finit e element methods and finite volume methods. A natural choice for t he discretization of conservation laws are finit e volum e methods. Here Godunov's pioneering work [God59] has to be mentioned as t he proto typ e of modern FVS. The principle idea is t o evolve an approximate solution represented by piecewise constant dat a in t ime by solving a so-called R iema nn problem at t he int erfaces where t he approximate solution jumps. Since t hen Godunov-typ e schemes have been studied exte nsively. Tw o main st reams of resear ch shall be mentioned. On e is concerne d with the solution of the Riem ann pr oblem and its approximati on , see Toro 's book [Tor97] for a sur vey. Another subject is the development of high-order accurate schemes based on reconstru ction techniqu es, see e.g. [HEOC87] . Since t he pr esent work is concerne d with the acceleration of FVS the structure of Godunov-type schemes will be bri efly motivated and the core ingredients are out lined. In the context of FVM the natural discretization is by cell averages

u'J := litJ1 I

l

tn

i;

1

+

u (t , x) dx dt ,

r

!Vi I := }Vj 1 dx

where t he computational domain n is parti tioned int o cells Vk , i.e., n = The cells Vk are always assumed to be closed. They are ty pically int ervals in ID , t riangles and quadran gles in 2D, tetra hedra and hexah edr a in 3D, and t he t ime int erval [0, T ] is decomposed by a te mporal mesh

UkVk·

1.2 Fi nite Volume Methods

7

o=

to < ... < t NT = T . Here t he te mporal parti tion is assumed to be un iform , i.e., t n + 1 - t-. = T for all n = 0, ... ,NT - 1. T his is no constrai nt for t he construction of t he adaptive scheme but sim plifies t he notation, because spatial mesh adaptation is only considered here. For these averages t he evolution equation

(1.10) is deri ved from (1.2) . According to the neighb orin g cells Vi of Vk t he bo undary k ,1 , i.e. ,

aVk ca n be decomp osed into different par t s r with

n ,I :=

aVk n aVi .

In case of a bounded domain n a layer of ghos t cells outside t he domain is introduced which is attached to the boundar y an. Then equation (1.10) can be rewri t t en as

:=!

Irk,11

r.,

1 dx.

(1.11)

Since in the one-dimensional case, t he edges r k ,1 coincide wit h a single grid point of t he mesh , t he value of In ,d is by definition put to 1. Here the average flu x across t he int erface is defined by .

f k,I :=

1

1

r: iTf

-:;-I T.k ,1I i ft

n

r k,l

fn k,I(X) (U (t ,x) ) dx dt .

(1.12)

Not e, t hat the outer norm als llk,1 and lll,k corres ponding to t he cells Vk a nd Vi, respect ively, differ only in sign. Therefore t he average fluxes satisfy t he conservation propert y fk,1 = -fl~k'

Substit uting in (1.11) t he averages ti} and t he average fluxes fk,1 by some ap prox imat ion v r and Fr ,I' respectively, leads to t he general repr esentation of a FVS T '""' IT I Fn v kn+ l -_ v kn - IVkl L.. k, 1 k, I ' I

Of course, t he key ingredient is an appropriate computation of t he numerical fluxes Fr I' To obtain a robust and high- order accurate scheme t hree basic req uirements ar ise, nam ely, consistency, conservati on and accuracy , whi ch have to be taken into acco unt in t he construction of t he numerical fluxes - v~ == v ~ Fr ,1 = Ir~ , d I r k., fn k,,(X) (v) dx, - Fr ,1 = - F~k '

8

1 Model Problem and It s Discreti zation

- Fr ,1 =

st. + O(h

q

)

+ O(rP ) ,

where h denot es t he lar gest diam et er of t he cells involved in t he discretization stencil of t he num erical flux Fr I' An appropriate const ruction is present ed in t he following. It consists of f~ur basic steps: Step 1: Quadrature of Spatial Boundary Integral. The integral over t he boundary par t r k ,1 in (1.12) is approximated by a quadrature (d= 2) or a cubature (d= 3) formula of order NQ with PQ quadrature points x~ ,1 and . hts Wik,l , i.e., . weig

In th e one-dimensional case (d = 1) this step is void since single point.

rk ,1

reduces to a

Step 2: Flux approximation. The fluxes in normal dir ection have to be evalua ted at th e quadrature points where the solut ion u is not yet known . Therefore the fluxes are substit uted by approximations of t he form G(v , w , n) t hat are assumed t o be consiste nt in t he sense t hat G(v , v , n ) = fn (v) . An appropriate flux approximation is t he Godunov flux int roduced in [God59]. This approxima t ion is based on t he solutio n of t he one-dimensional Riemann problem o u(t,x) ot

+

o fn(u(t,x)) ox = 0,

U(O, X)={u L ,x< O' . UR

,x> 0,

(1.13)

The solut ion of t his pr oblem is known to be self-similar, i.e., it is constant along rays x / t = canst, and can be repr esented as u (t , x) = U (UL , UR , x / t) . For det ails on t he existence and t he const ruction of t he Riemann solut ion see [GeI59] (scalar) and [Liu75] (system) . Th en th e Godunov flux is det ermined by U(Vk , VI , 0) where t he initi al valu es ar e th e averages on both sides of the interface n,l ' Obviously, t his leads to a first ord er approxima t ion

since t he exact averages U k and UI are only first order approximations of t he point valu e u (t , x~ ,I ). If t he solut ion u would be known, t hen no erro r is int roduced when evaluating t he num erical fluxes at V = w = u (t ,x~ ,I) and n = n k ,1 (x~ ,I) according to t he consiste ncy of t he flux approximation. Solving such local Riemann problems accounts for a substant ial par t of t he overa ll computational effort . A less expensive approximation is base d on lineari zing t he Riemann pr oblem in the sense t hat t he flux f n is replaced by

1.2 Finite Volume Methods

9

the linear approximation fn(u) := A(u, n ) u with respect t o an int erm ediat e state u . This was first introduced by Roe [Roe81]. For mor e det ails on approximat e Riemann solvers see [Tor97]. Step 3: R econstruction. The low accuracy of the Godunov and Roe flux approximation can be remedied by means of reconst ruction techniques int ro duced by Hart en et . al [HEOC87] . The basic idea is to construct an approximation t o a scalar function w : [l -+ R based on its averages {wkh such t hat the corres ponding reconstructi on fu nction R N R (' , w) has a desired discretization error, satisfies t he conservation property and is essenti ally nonoscillat ory (E NO)

- R NR(X,W) = w(x ) + O(h NR),

w sufficient ly smooth,

- IJkl JVkR NR(X, w) dx = Wk, - TV(R NR(., w)) :S TV(w) + O(h C» ,

a> 1.

It should be emphas ized th at t he reconst ruc t ion funct ion is not necessarily

smoot h on the whole computat ional domain [l but may jump, for inst an ce, at t he int erfaces of t he cells Vk . In pr act ice, this does in fact happ en, since for each cell a local reconstruction function R ( ' , w) is det ermined by mean s of the averages in t he local neighbo rhood . In view of th e ENO property t he reconstruction ste ncil has to be chosen judiciously employing dat a depend ent possibly un symmet ric stencils. For the ID case an efficient strategy was developed in [HEOC87] by successively increasing t he reconstructio n st encil. For t he multidimensional case this turns out to be not feasibl e. Here a fixed stencil is in general chosen where th e reconstruction conditions are appropriately weight ed when solving a least- squ ares pr oblem (see e.g. [HC91, Abg91]). For more det ails on reconstruction t echniques, see also [Son95]. The reconstruction concept has originally been developed for scalar fun ct ions. It can easily be exte nded to syste ms by applying t he reconstruct ion component wise or vect orwise. More sophisticated techniques are based on t he reconstructi on oft he cha racteristic vari ables ui; := (li (U, n) , u) , i = 1, . . . , m , where li(U, n) denot es the left eigenvect or of t he J acobian A(u , n) . Now t he approxima t ion ord er of t he flux approximation can be increased . To t his end, t he averages are replaced by point valu es det ermined by th e loca l reconstruction funct ions Rr:R(. , u(t , .)), i.e.,

Rr:

G(Rr:R(x7,l, u(t, ')) , R{"R(x7,1, u (t , ')), n k,l) = f n k • , (x 7' 1) (u(t ,x7,l))

+ O(h NR).

Note , t hat for a first order reconstruct ion the point values coincide with the averages according t o t he conservat ion property of t he reconstruction. Step 4: Quadrature of time integral. Fin ally the time int egration has to be resolved. One possibility is t o apply a Runge-Kutt a scheme to t he evolutio n equation (1.2), incorp orating t he approximations due to th e precedin g

10

1 Model Probl em and It s Discreti zati on

steps. T his requires averages on some int erm ediate t ime levels which all have to be stored in t he comp utation of one t ime step. In order to avoid t his addit ional amount of storage one can apply a different st rategy. For t his purpose, the time int egration is approximated by means of a quadrature formul a , e.g. by a Gau ss quadrature rul e, where t he fluxes have to be approx imated at t he Gaussian points t n ::; tf < ... < t Na ::; tn+l . Since the solut ion u is not known at t ime level t~ , an approximation of u (t~ , x~ ,l ) is determined where u is expanded in a Taylor series at t he point (tn, xd . Here Xk denotes the cent roid of t he cell Vk . Then th e t ime derivatives are successively substit uted by spatial derivatives exploiting t he evolut ion equation (1.1). At t ime level t n t he averages are known. Hence t he spatial derivatives can be approximate d by t he reconstructi on function due t o t he known averages . Obviously, the computation of t he num erical fluxes is very expensive. Therefore one is int erest ed in reducing th e cost. Since upwind t echniques are only essent ial close to discontinuities, cheaper finit e difference approximations are sufficient in th e smooth part of th e solution. This requires a tool by which the data representing t he flow field can be analyzed. In t his regard Hart en introduced a sophist icated st rategy based on multiscale decompositio ns [Har9 5].

2 Multiscale Setting

The core ingredient of t he adaptive FV S is t he multiscale setting by which t he data at han d, here cell averages, can be analyzed . In t he lit erature, at least two settings are known which can be app lied for t his purpose. One framework was int roduced by A. Har t en [Har9 3a, Har93b, Har 96] and extended lat er by [ADH98, ADH99]. Here t he construction of a multi scale analysis is based on discret e dat a only and employs reconstruction and pr edicti on techniques. In view of stability invest igations, funct ion spaces have to be introdu ced in order to benefit from functi onal analytic arguments . Therefore we pr efer a different meth odology by W . Dahmen [Dah94 , Dah9 5, Dah96] and collaborators [CDP96] which is based on biorthogonal wavelet s and stable complet ions . Alt hough t he set t ings are different, t he resulting t ra nsformations can be rewrit ten in te rms of t he ot her setting, see [GM99b]. In the appendix of t his book , we pr ovide a short summary on wavelet t heory for readers who are not familiar with wavelets at all, see Sect. B. In t he following we present a self-contained multiscale setting employed in t he construc t ion of ada ptive multiscale FVS . In Section 2.1 we introdu ce a hierarchy of nest ed grids. By means of t he box funct ion and t he box wavelet we will motivat e t hat t he cell averages relative to t he nest ed grids are naturally relat ed to biorthogonal wavelets, see Section 2.2. We will explain in Sect ion 2.5 how to modi fy t he box wavelets such t hat t he new wavelets have imp roved cancellation pro pert ies. For th is purpose we pr esent a systematic fashion how to const ruc t the modified box wavelets. This is related to stable comp let ions, see Sect ion 2.4. Finally, we det ermine a mult iscale decomp ositio n of the cell averages relat ed t o a finest resolut ion level that will be utilized in Chapter 3 t o construct t he adaptive grid.

2.1 Hierarchy of Meshes As has been motiv at ed in th e pre vious cha pte r FVS are naturally relat ed to cell averages of t he solut ion. In ord er t o det ect singularities of the solut ion by mean s of t he array of averages, we consider t he difference of averages corresponding to different resolut ion levels. For t his purp ose we introdu ce a hierarchy of nest ed grids.

12

2 Multiscale Setting

DEEm j =O

j = l

j = 2

Fig. 2 .1. Sequ en ce of nest ed grids

Definition 3. (Nest ed grid hierarchy) A sequence of grids 9j := {Vj,k} kEfj ' j = 0, . . . , L , is called a nest ed grid hierar chy if the follow ing con diti ons hold

- n=

Uk Efj

Vj,k with

(partition)

IVj,k n Vj ,k,1 = 0 for k - Vj,k =

Ur EM j ,k Vj +l ,r ,

f:.

k' , k ,k' E I j , kE

t;

(refine me nt)

Note, that th e cells are always assumed to be closed.

Here t he coarsest grid is indicat ed by 0 and th e finest grid by L . Furthermore t he ind ex sets M j,k corre spond to t he new cells on level j + 1 resulting from t he refinement of t he cell Vj,k which is always assumed to be closed. A simple example is shown in Fig. 2.1 where a coarse grid is successively refined with increasing refinement level. From t he conditions of t he nested grid hierar chy we imm ediately conclude t hat t he following pr operties hold for t he refinement sets Mj ,k: - Mj ,k n M j,k' = -

U kEfj

M j,k =

0 for k f:. k' , k , k' E I j ,

(redunda ncy- free) (gap-free)

I j+ l.

For t he num erical experiments perform ed in Chap t er 7 only st ructured grids are considered. However , t he fram ework pr esent ed here can also be applied to unstructured grid s and irr egular grid refinements . For reasons of simplicity and stability only uniform refinements are considered here, i.e.,

# Mj ,k = M; = canst. Relative to the grids 9j we introdu ce the so- called box fun ction _

epj,k

()._ X

,-

1

V-i .kIIXv

" k

( ) _ { IVj,k\ - l ,

x -

0

,

x E Vj ,k X d V. l" J,k

(2.1)

defined as t he £I - scaled cha racteristic function with respect to Vj,k, i.e., Ilepj,kll£l(Q) = 1. Obviously, t he functions corr espond ing to t he same discret ization level are linearl y ind epend ent . The nest edn ess of the grids as well as t he lineari ty of int egration imply t he two-scale relation

2.2 Mot ivat ion

13

{;j ,k

~ Fig. 2.2 . Box function and box wavelet

r{Jj ,k

=

L r E M j ,k

IVJ+l,rl IV.J ,k I 'Pj+ l, r>

(2.2)

i.e., the coarse grid box function can be represent ed as a linear combination of the corresponding fine grid box functions. Consequ entl y, th e box functions can successively be computed st arting with the finest level. The reason for introducing the box functions is motivated by the fact that t he averages of a scalar, int egrabl e function u E £1(D) can be interpreted as an inn er product U j ,k

(2.3)

= (u, r{Jj,k ) n

with the box function where t he inner product is defined by

(u, v)n :=

l

u vdx.

Obviously, th e average s of two discretizat ion levels are related by U j ,k

=

L r E Mj ,k

IVJ+l,rl IV.J,k I U j +l ,r' A

(2.4)

The goal is to transform th ese dat a into a different form at of cell average s corres ponding t o a sequence of resolution levels. This will be motivated by a simple univari at e example.

2.2 Motivation We now consider the uni t interval D = [0,1] where th e grid hierar chy is determined by a uniform dyadi c partition of [0,1], i.e., VJ,k = 2- j[k , k+l]' k E I j := {O, . .. , 2j - I} . Not e, t hat the refinement sets are Mj ,k = {2k, 2k + I} . Then the £l - scaled box fun ction has the form

14

2 Mult iscale Setting {(,' k

'r

s,

= 2jX[0,1](2 j. -k) .

(2.5)

In the sequ el, we will explain how to decompos e th e cell averages into averages of a coa rser partition and details. To this end , we consid er in analogy to (2.2) the two-scale relation (2.6) by which a coarse- scale box function is reexpressed in t erms of fine-scale box functions . We now introduce the box wavelet -

'l/Jj ,k :=

1

'2 ( rpj+l ,2k -

rpj+l ,2k+d ·

(2.7)

Not e, th at the L 2 - scaled counterpart coincides with the Haar wavelet, see [Haa10] and Sect . B .2.1. Then we can write any fine-scale box function by means of the box function rpj ,k and th e box wavelet ;} j ,k . (2.8)

These relations are motivated by the illustrations in Fig. 2.2. In an alogy to (2.6) the cell averages satisfy A

Uj ,k

=

A A ) '12 (Uj+l ,2k + Uj+l ,2k+l .

On the other hand, (2.8) means

wher e the details are defined by dj ,k :=

(u, ;} j,k ) [O,l ] =

~ (Uj+l ,2k -

Uj+l ,2k+d ·

(2.9)

This relation shows how to reexpress fine-scale averages from coars e- sca le ones and det ails. In the sequ el, it will be convenient to rewrite the two-scale relations in matrix-vector form . To t his end , we introduce th e vectors (Pj := (rpj ,khEI; and .pj := ( ;}j,k hEI; ' Later we will use this notation also in the sense of a collect ion of functions . Then (2.6) and (2.7) read (2.10) where the columns of th e so-called mask matrices Mj,o and Mj,l contain the filter coefficients ~ , ~ and ~, - ~ of rpj ,k and ;}j,k, respec tively, i.e.,

In ana logy, t he two-scale relation (2.8) becomes (2.11) where the mask matrices are determined by Gj,i = 2 ML, i = 0,1. Note, that the relations (2.10) and (2.11) realize a change of basis , becaus e the composed matrices (2.12) are inverse, i.e.,

Gj =

Mjl . Hence, t he single blocks fulfill (2.13)

We now introduce a dual syst em by th e functions j j r i: , k := 2- ,ii r s,· k = X[0,1](2

in

.

-k) ,

'l/Jj ,k := T j ,(fj ,k

or in vector form Pj := 2- J Pj and tP'j := 2- j ~j. These are the L oonormalized counterparts of the box function and the box wavelet , resp ectiv ely. Obviously, the duals also satisfy two- scale relations of the form (2.10) and (2.11) with matrices M j ,i and Gj,i , i = 0,1. They are related to (2.12) by - T

M j ,i = Gj,i

From t his we infer that

,pj

and

- T

Gj ,i = Mj,i '

U ~j and Pj U IJtj ar e biorthogonal, i.e.,

(Pj , ,pj)[O,l] = (lJtj , ~j) [O,ll = I, (Pj , ~j) [O,l]) = (lJtj , ,pj) [O,l j = 0

(2.14)

where we use the notation (8 ,p):= ((B ,cp))OEl3 ,
16

2 Multis cale Setting

is naturally relat ed to t he array of cell averages. To this end , we consider a fun cti on u E £ 1([0,1]) . Then th e projection of u ont o piecewise constant s with respect t o t he refinement level j is det ermined by Uj :=

L (u ,
(2.15)

kE f j

This function can be int erpret ed as an approximate solut ion of the conservation law on a fixed t ime level corresponding to t he discretization level j . Equivalent ly, t he function Uj can be repr esent ed by Uj =

L

(u,
1,k )[O,1] 'Pj - 1,k

k Efj- l

=:

+

L

(u, t,bj-1,k ) [O, l j 'l/J j- 1,k

k Efj- l

P]-l ii j - 1 + pJ-1 d j - 1 ,

(2.16)

becaus e the two-scale relations realize a change of basis and , in par ti cular , the syste ms Pj U Pj and j U Pj are biorthogonal. This repr esent ation motivat es that t he det ails can be int erpret ed as th e updat e when pro gressing to a higher resolution level. We now have t o explain why t he represent ation (2.16) is preferabl e t o (2.15) for our purposes. To this end, we verify that the det ails may become small when the und erlying funct ion is smoot h. Fir st of all, we conclude from (2.7) and (2.1) th at (1, t,bj,k )[O,l j = O. Since the box wavelets are £l -normalized, i.e., II t,bj,kll£l([O,l]) = 1, we deduce Idj,kl ::; ~~~ I(u - c, t,bj,k)[O,ljl ::; ~~~ Ilu - cIILOO(Vj,k) (2.17) ::; C Tj Ilu'IIL OO (Vj ,k)'

From t his est imate we infer th at t he decay of the det ails is proportional t o 2- j pr ovided th e function u is differenti able and has a mod erat e derivative. Hence, th e det ails may become small in smoot h regions of u whereas it gives a significant cont ribution in (2.16) if th e gra dient of u is lar ge. This motivat es to neglect all sufficient ly sma ll det ails in order to compress the original dat a such t hat we cont rol t he loss of accuracy. Since t he compression rat es depend on t he decay 2- j it will be convenient to improve this decay. For this purpose not only constants have t o be canceled by the wavelet t,bj,k but also all polynomials p of a fixed higher degree, i.e., (p, t,bj,k )[O,l] = O. This can be achieved by mean s of higher order biorthogonal syst ems instead of piecewise const ants. In case of t he uni t int erval we refer to [CDF92 , DKU99] . By means of t ensor products this can be carr ied over to Car t esian grids. However , for genera l discreti zations, e.g., cur vilinear grids and trian gulat ions, we have to employ a different st ra tegy . Here we will use a technique introduced in [CDP 96] which is called the change of stable completio n. In th e special case it will be

2.3 Box Wavelet

17

applied here it coincides with t he liftin g schemes in [Swe98]. For t his purpose we will first genera lize t he const ruction of t he box wavelets on an arbit rary nest ed grid hierar chy. The resultin g wavelets will serve as a starting point for t he construction of modified box wavelets in order to achieve higher order vanishing moments.

2.3 Box Wavelet The goa l is to genera lize t he concept of box wavelets on a hierar chy of nest ed grids not necessaril y based on Cartesian grids. For this purpose, we pr oceed in two steps . First of all, we introdu ce the box wavelet on a Car tesian grid hierar chy usin g t ensor products of t he uni variate counte rparts. In a second step , we const ruct box wavelets on a general nest ed grid hierarchy such that some char acteristic properties for t ensor products are pr eserved. 2.3.1 Box Wavelet on a Cartesian Grid Hierarchy

The extension t o higher dimensions is st raightforward provided t hat th e grid hierar chy is det ermined by Car t esian grids. To this end , we consider th e uni t box n = [0, l ]d and int rodu ce a nest ed grid hierar chy by means of a uniform dyadi c par tition of [O , I ]d, i.e., YJ,k = TI~=1 2- j [ki , k, + 1], k E I j := TI ~= 1 {O, . . . , 2j - I} , see Fig. 2.1. The corr esp onding refinement set is charact erized by Mj ,k = {2k + e ; e E E} wit h E := TI~= 1 {O , l }d. In this shift-invar iant case , t he mult ivariate box function and t he box wavelets can be constructed by mean s of tensor pr odu cts. For this pur pose, we introduce the convention .(iJj,k,O := IjJj,k and .(iJj,k,1 = .(iJj,k where IjJj,k and .(iJj,k denot e t he uni variate counte rparts according to Sect. 2.2. Then t he mul tivari at e box fun ctions IjJj,k == .(iJj ,k ,O and box wavelets .(iJj ,k ,e , e E E* := E\ {O} are defined as produ cts of t he univari at e box function IjJj ,k and t he univari at e box wavelet .(iJj,k, respectively, correspon ding to t he cell YJ,k, i.e.,

-

'l/Jj ,k, e( X ) :=

rr i=l

'l/Jj,ki,ei (Xi) ,

e E E.

(2.18)

These functions ar e shown in Fig. 2.3 where (Xl , X2) corresponds to (x, y). In analogy to th e t wo-scale relations (2.6) and (2.7) the box function and t he box wavelets can be rewrit t en in te rms of fine- scale functions. To this end, we first obser ve t hat a cell YJ,k can be decomposed by YJ,k = U e EE YJ+ I ,2k+e. For any position x E YJ+I ,2k+ i we then obtain

where we employ (2.5) and (2.18). Fin ally, we conclude t he two-scale relation

18

2 Mu lti scale Set ting Box function

Boxwavelet of type (0,1)

o.a O.G

0.4 0.2

0 1.5 1.5

-0.5

-0.5

Boxwavelet oftype(1,1)

Box wavelet01type (1,0)

·0.5

-0.5

F ig. 2. 3 . Box fun ction and box wavelet s on [0,1]2

;j;j ,k,e =

L 2-

d

(_1)i .e 1f?Hl,2k+i '

(2.19)

iEE

As has been proven in [Got 98], p. 77, an inverse two- scale relation 1f?Hl ,2k+ i

=L

(- l)i.e ;j;j,k,e,

i EE

eEE

holds in ana logy to (2.8). Hence {1f?Hl ,2k+e}eE E and {;j;j ,k,e}eEE , respectiv ely, ar e syste ms of linearly independent functions that span the same space . By t he construction of the multivari ate box functions we fur th ermore conclud e t hat t he multivariate box wavelets are ortho gonal to constant functions, i.e., (1, ;j;j,k,e) n = 0, e E E*. Finally, a bior tho gonal system is determined by t he £ oo- scaled counterparts .'f'J, t. . k ,e .2- j d .7. . e E E, .o/J ,k ,e, accor ding to the univariat e case.

2.3 Box Wavelet

19

2.3.2 Box Wavelet on an Arbitrary Nested Grid Hierarchy

The goal is to genera lize th e concept of box wavelets on a hierar chy of nest ed grids not necessarily based on Car t esian grids. The const ruction principles are (i) t he system $j U .pj is anot her basis for t he space spanned by t he basis $j +l , (ii) the box wavelets are orthogonal to constant functions, i.e., (1, .pj )n = 0 and (iii) t he basis $j u.pj and its Loo-scaled counte rpart Pj UtPj are biorthogonal, i.e., (2.14) hold s. For t he const ruc t ion we now pro ceed analogously to the Cartesian case as has been out lined in th e pr evious sect ion. To t his end, we consider t he box wavelet as a linear combination of t he finescale box fun ctions epj+l ,r, r E Mj ,k, related to t he refinement of t he cell Vi,k , d . (2.19), ._ 'l/Jj ,k,e' -

L ( rEMj,k

IVi+l ,r I

IV.k I

1/ 2

)

j,k - . ae,r
e E E * := E\{O}

(2.20)

J,

with E := {O,. , . , M; - I} and appropriate par amet ers a~' ~ which have yet to be det ermined. Here t he weights (IVi+1 ,rl/IVi ,kI)1/2 are 'mot ivate d by t he requirement th at the box wavelet s and their Loo- scaled counte rparts are biorth ogonal, i.e., t he identi ti es (2.14) hold ", In order to distinguish t hese fun ctions from t heir modifications t o be introdu ced in t he subsequent sect ion we use the not ation 1f;j,k,e inst ead of 1f;j,k,e. Then we define .a Jo' , k .-

(

IVi+1,rl/I Vi,kl)

rEMj ,k

aj,k ,- (aj,k) e .e,r rEMj ,k' e E E*

,

(2.21)

and the vect ors of local basis funct ions

Here we use th e convent ion 1f;j,k,O matrix-vect or form as

epj,k' Thus we can rewrit e (2.20) in

(2.22) where t he square mat rices (not e:

#M

j,k = M; =

Ai.u .(aj,k) >: e,r eEE,rEMj ,k'

# E)

are defined by (2.23)

In order to realize a change of basis, i.e., we obtain two-scale relations of t he form (2.8), we obviously need that th e matrix Aj ,k is invert ible, i.e. , t he inverse denoted by 1

Choos ing t he ansatz ;j;j,k,e

= 2::rEM j,k de',~ c,Oj+l,r

t he par amet ers de',~ are sub-

ject to t he conditions (1) cV..l1 , e E E*, i.e., {I, ;j;j,k,e )n = 0 and (2) {'ljJj ,k,e, ;j;j ,k,e/ )n = oe,e' Cj,k,e . These conditions hold for de',~ = a~',~ a{',~ .

20

2 Multiscale Setting - I . - (bj,k) Aj,k'r,e rEMj, k,eEE

exists. In this case we obtain (2.24)

However , we are not only int erest ed in performing a change of basis but th e resulting syst em of box functions and box wavelet s and their LCXl- scaled counte rparts defined by (2.25)

are biorthogonal. For t his purpose we prove th e following corollary. Corollary 1. (B iorthogonality of box wavelets) Assume th at th e vectors a~,k , e E E, form an orthonormal system, i.e., (a~,k) Ta~;k = r5 e ,e" Th en the systems ~j,k and'Itj,k .- ('l/Jj,k,e)eE E are biorthogonal, i.e., ('It j,k, ~j, k)!t = I. Proof. In ord er t o prove th e assert ion we consider first ( nl. 'f/j,k,e, .i'f/ j,k,e') !t

=

IVj,k I '" L..-

'" L..-

j,k ae,rao,r' j,k j,k a j,k () ao,r e , ,r' tpj+l,r , tpj+l ,r' o

rEMj ,k r'EMj ,k

for any k E I j , e, e' E E and j E No. By definition of the box function (2.1) we infer for r E M j,k and r' E M j,k l

_ _ ) s ( tpj+l,r, tpj+l ,r' o = Ur,r'

1

_, ( j,k)- 2 1 Ur,r' aO,r -IVI'

IV1 +I ,r I -

1,k

Note, that t he refinement sets are redundan cy- free. This impli es

for r -j. r' according to t he conditions of a nest ed grid hierar chy, see Definition 3. In addition, we use (2.21). Thus, we conclude by the assumpt ion th at nl' , nt, ) _ ( o/J,k,e, If/j ,k ,e' f2 -

'" ~

aj,k aj,k _ (a j,k )T a j,k _ s e ,T e' ,T e e , - U e ,e/

rEMj ,k

which proves th e assert ion. D ' k Note , that the vector a~' can be exte nded to an ort hogonal syst em by mean s of t he Gr am-Schmidt orthogonalization process. Fin ally, t he orthogonal vecto rs have to be normalized. According to t he univari at e case, we now int roduce t he det ails corre sponding t o th e box wavelet e E E*.

(2.26)

2.3 Box Wavelet

21

By the definition (2.20) of the box wavelets we then infer the two-scale relation i. _ "" (Wi+l,rl)I/2 j,k ~ . (2.27) dJ ,k ,e L.J IV k I a e,r UJ+I,r, e E E*. r EM j ,k

J,

On the other hand, the two-scale relation (2.4) for the cell averages can similarly be written as (2.28)

In matrix-vector form th e equations (2.27) and (2.28) then read cl j ,k

= Aj,k Aj,k Uj+l ,k '

Here the vectors are defined by UJ+I,k := (Uj+l ,r)rEMj ,k and clj,k . (dj ,k ,e) eEE where we use the convention dj ,k,O := Uj ,k ' Since the matrix Aj,k is assumed to be at least invertible in order to realize a change of basis , we can locally reexpress the fine-scale averages by the coars e-scale ones and the details , i.e., ~ A-I A-I dUJ+I ,k

=

j ,k

j ,k

j ,k·

So far, we have only considered a local change of basis. This has been possible, because the support of the box wavelets .f!;j ,k, e , e E E, is completely covered by th e support of the box function rpj,k . According to the univariate case we write the global change of basis in terms of the vectors ~ j and 4"j,e := (.f!;j,k ,eh Elj ' e E E . In particular, we obtain (2.29)

and (2.30)

Here the ent ries of the mask matrices are determined by fh,J .,e _

r ,k -

rEM j ,k } ,elsewhere'

{ aje,T ,k ajO,T' ,k

0

gJ,. e k, r

= { bj,kjaj,k r ,e O,r ' r

0

EM} j,k

, elsewhere

(2.31)

for r E IJ+I, k E I j , e E E, according to (2.22) and (2.24) . Note , that bt:~ = a{:~ provided that the matrix A j ,k is orthogonal. The sparsity pattern of th e mask matrices is only presented for the two-dimensional case

**** Gj ,e

****

=

****

****

22

2 Mult iscale Setting

where we assume that each cell is decomp osed int o four subcells, see Fig. 2.1. The pattern of M j ,e coincides wit h the pat tern of t he t ra nspose of G j ,e ' In particular, t here is exactly one entry in each row and colum n due to t he nestedness of the grid hierarchy. We now gather all mask matrices corresponding to t he box wavelet s of type e E E* in one single block, i.e.,

T hen we can define t he matrices

as in t he univari ate case, see (2.12). Note, that t he change of basis (2.29) and (2.30) impli es that Mj and Gj are inverse. Similar two- scale relations hold for t he L'Xl-normalized counte rparts defined by (2.25) for th e vect ors (2.32) wit h t he diagonal mat rix Vj := diag ((llJJ,k lhE IJ . Then we infer from t he loca l biorthogonality pr operty, see Corollary (1), that t he systems ;jj U UeEE* ljI j ,e and j U U eEE* 'It j ,e are biorthogonal provided that t he matrices A j ,k are orthogo na l. Hence, the projection of any function u E £ 1(D) onto piecewise constants with respect to t he refinement level j can be represe nte d by

Finally, we have to verify that the box wavelets have one vani shin g moment , i.e., t hey are ort hogona l to constant functions. To this end we not e t hat ~ aj,k a j ,k (1 in'+1 ) rv = ( aj,k)T a j ,k .L....t O,r e,r 'r J .r t ss 0 e rEMj .k

holds for all e E E * , k E I j, j E No. Again , we need t hat t he vect ors a~, k , e E E , are at least ort hogonal. According to th e univari at e case we then infer t hat t he det ails decay like 11JJ,kI which becomes sma ller with increasing refinement level j .

2.4 Change of Stable Completion In order to improve t he compression rates we need wavelets wit h bet t er can cellation pro perties in the sense that higher order polynomial moments vanish. In [CDP 96] a systematic ansatz has been pr oposed for t his task. To t his

2.4 Change of St abl e Completi on

23

end, we first not e t hat th e ~onstruction of t he matrix M j ,l is only one way t o complement t he matrix Mj,o t o an invertible matrix Mj . There is a cont inuum ?f completi ons of Mj,o . We call a complet ion (uniformly) sta ble if th e matrix Mj and it s inverse Gj have uniforml y bounded operator norms with respect t o a suitable vector norm. For t he matrices Mj and Gj correspo nding to t he completion by mean s of t he box wavelet it can be pr oven that t hey are stable in any lP-norm, see Sect . B.1.2 for p = 2. Not e, t hat any completion Mj,l of Mj,o characterizes a particular complement of t he spaces of piecewise constants relat ed t o t he refinement levels j and j + 1. In particular , t he correspond ing wavelet basis Pj is stable in the sense th at

holds for constants c, C ind epend ent of j. In the sequel, we will const ruc t anot her stable complet ion Mj,l of Mj,o such that th e corre sponding wavelets have higher ord er vani shin g moments. The start ing point is any composed matrix

where t he matrices Lj,e E R N jx N j , N, := # I j , are uniformly bounded with respect to l l . Then we not e t hat t he mat rices

- - (I L.)

MJ· ·= MJ· .

are inverse, i.e., single blocks

Gj

0 IJ

= Mjl. From t he composed matrices we det ermine t he

and, equivalent ly, Mj,o = Mj,o, (2.33) Gj,o = Gj,o -

L

Lj,e Gj,e,

eEE*

where we employ the block st ructure of Lj, Mj,l and Gj,l with respect t o the different wavelet ty pes e E E *. Since t he matrices Lj,e, e E E* , are assumed to be uniformly bounded, t he matrix Mj,l is st ill a stable complet ion of Mj,o. In terms of t he new complet ion, the basis of modified box wavelets can now be represent ed in terms of th e old wavelet basis Pj, i.e., (2.34)

24

2 Multiscale Setting

Not e, t hat t he basis
2.5 Box Wavelet with Higher Vanishing Moments The change of stable complet ion according to Sect . 2.4 is now applied to the generalized box wavelets pr esent ed in Sect . 2.3. For this purpose, we first introduce the definition of high er order vanishing moments and outline an algorit hm for det ermining the modified box wavelets. A univariate example is pres ented as an illustration. We conclude this section with a result on compression rates.

2.5 .1 Definition and Construction We will now det ermine the free parameters introduced by the matrices Lj ,e , e E E* , such t hat th e modified box wavelets have high er order vanishing moments. To this end, we introduce th e following notion.

Definition 4. (Vanishing Mom ents) Th e wav elet basis has M vanishing moments if

(P, ,(f;j,k,e)Q = 0 holds for all polynomials pEP M -1 of degree less than M and e E E *,

k E I j , j E No. In ord er to ensure an efficient two-scale transformation between the bases of box functions
2.5 Box Wavelet with Higher Vanishing Moment s

25

Algorithm 1. (R ealization of higher order van ishing m oments) 1. Choos e the order of vanishing moments M and det ermin e th e number of required condit ions Me := (M +dd-1) = dim PM-1 ;

Lj,k

2. choose the ste nci l modified, i. e.,

7fi j ,k ,e

C Ij

of box fun ctions by which th e box wavelet is

,

= 7fij ,k ,e +

L IN

r{Jj,l,

e E E*,

lE£. j,k suc h that M e S 3. choose a basis

# £ j ,k :s: M£.

and M£. in depen dent of i . k and e;

for P M-1 with

{ w d iEPM_l

4. det ermin e th e free param eters l{:~ ,

PM-1 :=

{I , ... , Me};

I E £ j ,k' such that

(2.35) The system (2.35) results in a linear system of equat ions Al = b for t he unknowns I = (l{'DlE£. J,~ k . Here we omit the dependence on i , k, e. The matrix A and t he right ha nd side b are det ermin ed by 1

Introducing th e moment s with respect to the basis

{Wd iEPM_l '

i.e.,

Vjl, , i:=(Wi ,XV)S? ' i, l

t hese inner products can be rewritten as _ (Wi ,
I

=

) (Wi, ;r, 'f/j ,k ,e S? =

-IV IVj ,l ,i , J ,l

aj, k a j, k " O ,r e,T L...J I Vj +1 ,r ,i . r EMj.k J+1 ,r

IV

Then th e linear syst em (2.35) can be reexpressed as

v I

" " L...J

tee )s,k

I t/, ~ IV I J,l ,t

aj,k

.u: = -

"" L...J

l ,k

J ,l

rEM ]·, k

~ j ,k t/,J+1 ,r ,t. · aO' r

Concerning t he choice of an appropriate basis for t he monomials

P M -1

we might choose

d

W j(X )

=

xi

:=

II X;l. 1=1

However , it might be mor e useful to apply a local set of basis functions, e.g., I

Cj ,k

r

= IVJ,k I }Vj.k xdx,

(2.36)

26

2 Multiscale Setting

where Cj ,k denotes t he cent er of mass and i E Ng, a::; L~= 1 ik ::; M - 1. Since the solution of t he linear system is not uniquely det ermined , t he underdetermined system (2.35) is solved by mean s of singular valu e decomposition , cr. [GV96]. In pract ice, th is solver pr oves to be very reliable bu t expensive. According t o (2.33) the ent ries of t he mask matrices relat ed to t he modified box wavelet s are determined by - j ,O _ • j ,O m r,k - mr,k '

- j ,e _ • i .e m r,k - mr,k

+ l j ,e

• j,O I,k m r,I '

e E E*, (2.37)

-j,O _ «i» '""" lj ,e «i .« gk,r - gk,r - ~ eE E ' k,1 gl,r '

- j ,e _ «i,e gk ,r - gk,r '

e E E *,

for r E I j+l, k E I j . Here the ind ex l E I j is uniquely det ermined by rEM j,l . Mor eover we use the convent ion that = a if rf. £j,k' In more detail, the modified coefficient s read

If::

mj, e = r,k

{(a~,,~)2 a

(bl ,k

l

(a~',lr) - 1 a~:~ + If::) , r E M j ,l , l , elsewhere,

E L j ,k U

{k} , (2.38)

_ .0 J,

gk ,T

j ,1 ) ( aO ,r

_

-

{

1

(bI,k bjr,O,1 -

'" LJ eEE'

btr:eI [jk,1 ,e) , r

E M j ,l , l E I j ,

k E £j,1U {l} ,

a

, elsewhere,

for r E I j+ l ' k E I j , e E E* . . 2 .5.2 A Univariate Example

We now apply Algorithm 1 to t he univari ate box wavelet "jJj ,k on t he uni t interval , cr. Sect. 2.2, where we consider a uniform dyad ic par ti t ion of n = [0,1 ], i.e., Vj ,k = 2- j [k, k + 1], k E I j = {a,... , 2j -I } and Mj ,k = {2k , 2k + I}. For simplicity, we now assume t hat the order of vani shin g moments is odd , i.e., M = 2 s + 1 = Me for some s E No. In particular , t he dimension of PM- 1 coincides with the order of vanishing mom ents, i.e., M = Me, that only holds in the one-dimensional case. Since there is only one wavelet type, we will omit the ind ex e for the moment. The univariate modified box wavelet "jJj, k is now det ermined by t he ste ncil L j ,k = {k - s

+ p., .. . , k + s + f.L}

(2.39)

of box fun cti ons
2.5 Box Wavelet with High er Vani shing Mom ents

•

o

o

o

•

o

o

Fig. 2.4. Illu stration of t he set s

L j ,k

for s

27

•

o

=1

contained in [0,1], i.e. , f-l E {-8, . .. , 8}. For 8 = 1 the three different st encils are illustrat ed in Fig. 2.4. Here. indi cat es the support of the box wavelet ;j;j ,k and 0 indi cat es the exte nded support of th e modified box wavelet ;j;j, k. In par ticular , t he st encils from the left t o the right corre spond t o f-l = 1,0 , -1 , respectiv ely. We now det ermine the coefficients ii» , l E L j ,k ' For this purpose, t he param et ers 8 E No and f-l E {-8, . . . , 8} are arbit ra rily but fixed. Not e that j has t o be chosen such t hat 2j ~ M = 2 8 + 1 otherwise t he support of t he modified box wavelet ;j;j,k is not contained in [0,1] . According to (2.36) we put Wi (X) := (x - 2- j(k + 0.5)) i. Then the syst em (2.35) reads S+ JL

L

2i+l ((l + 0.5)i+l - (l - 0.5)i+ 1 ) ll+k,k = 1 - (_l) i

1= -s + JL

for i = 0, . . . , M - 1 = 28. Obviously, th e solut ion depend s only on t he positi on k and the par am eters 8 and u, respect ively, bu t is ind epend ent of t he level j due t o the uniform dyadic grid hierar chy. In particular , th e syste m of linear equat ions is regular. For 8 = 1 and 8 = 2 the solut ion is summa rized in Tabl es 2.1 and 2.2. The corresponding mask matrix L == Lj,l for t he case 8 = 1 is det ermined by the coefficients in Tabl e 2.1 and t he st encil (2.39) and read s

Table 2.1. Coefficients for s

/1 =-1 /1=0 /1=1 1

1

3

=1

Table 2.2. Coefficients for s

/1= -2 /1= -1 /1=0 /1=1 /1=2 7

- 8"

128

1

- 2

0

1

2

19

- 64"

3 128 9

64"

- 64"

- 64"

3

1

1

11

13

0

3"2

8"

8"

- 8"

-

3

7 128 15

- 8"

8"

=2

128 11

-

13

65 - 1 28 61

64" 11 -16

16 61 -64"

- 3"2

64"

64"

- 64"

64"

65 12 8

7 128

3 128

3 128

7 - 1 28

15

11

-

9

19

28

2 Mult isca le Setting

-3- 1 4 0- 1 -1 1 0 1 Lj ,e

=

1

8"

-1 0- 1 1 1 0-4 1 3 Finally, in Fig. 2.5 we pr esent t he box wavelet 1(;j,k and t he modified box wavelets 1(;j,k for s = 1 and J.l = - 1, 0, 1.

tj;j ,k

f.l = - 1

f.l = 0

f.l = 1

Fig. 2.5. Box wavelet tj;j, k and modified b ox wavelets tj;j,k for s

= 1 and f.l = -1 , 0,1

2.6 Multiscale Tr an sformation

29

2.5.3 A Remark on Compression Rates Since t he construction of modified box wavelet s is motivated by higher compr ession rat es, we pr esent a st andard result for t he decay of the det ails . It can be proven similarly t o (2.17) where we employ regularity of t he function u and t he cancellation property of t he modified box wavelet s. Corollary 2. (R egularity ~ Compression) Assum e that the support of the mod ified box wavelets ,¢j,k,e is un iformly bounded, i. e., "'j,k,e := supp ,¢j,k,e and diam "'j,k,e '" 2- j , and the modified box wavelets are L P-norm alized, i. e., II ,¢j,k,eII LP(st) '" 1 where 1 ::; p ::; 00 . For a fu nction u E W q,M("'j,k) , lip + 11q = 1, the wavelet coefficie nts can be estim ated by

pro vided that th e wavelet ,¢j,k,e has M vanishing moments. Here c constant indepen dent of j E No, k E I j , e E E * and u .

>0

is a

For our purposes we need p = 1 and q = 00. Obviously, t he size of Idj,k,e I decreases with increasing level where t he decay is the st ronger the high er t he number of vani shin g mom ent s is. However , the higher the number of van ishing moments the mor e par am eters are needed for their realization , i.e., # £ j ,k increases. Therefore the realization of t he change of basis is more expensive. On t he other hand , the compression rat es may be higher . Therefore the number of vanishing mom ents should be no t chosen t oo lar ge. Finally, we emphasize that the resulting complet ion M j ,l of t he initial stable complet ion Mj,o is stable in the sense specified in Sect . 2.4. However , it has not been verified so far that t here exists a biorthogonal syste m for an arbit rary hier ar chy of nest ed grids.

2.6 Multiscale Transformation As already outlined in Section 2.2 the ultimate goal is to transform an array of cell averages into a different format in order to compress data . This is achieved by mean s of a change of basis. For t his purpose we introduce t he vect ors Uj := (uj ,khElj and d j := (dj,k,ehElj ,eEE*. Here the averages and the details are det ermined by functionals of a fun ct ion u E L 1 (D) , i.e., Uj,k := (U ,(Pj,k) st and dj,k,e := (u , ,¢j,k,e)st. According to (2.29), (2.30) and (2.34) we obtain t he two- scale relations (2.40)

30

2 Multiscale Set ting

UL

----

UL- l

-,

d L- l

Ul

Uo

dl

do

-, -, -,

Fig. 2.6 . Pyramid scheme of mul tisc ale tran sformation

Appl ying these relations it eratively, see Fig. 2.6, th e array UL of cell averages on level L can be decomposed into a sequence of coars e-scale cell averages Uo and details d j , j = 0, . . . , L - 1. It will be convenient to introduce the not ation dT )T d (L ) .- (d T d T (2.41) .-

- 1'

0"

'"

L-1

,

for the multiscale coefficients. Since th e bases {$j }j EN o and {$j U@j}jENo ar e assumed to be uniformly stable in th e sense of Section 2.4, t here is a bijective linear opera t or ML : R N L -+ R N L by which the transformation between the single-scale coefficients UL and th e multiscale coefficients d (L ) can be carried out , i.e., d (L) = M LUL and UL = M 1d (L). L The operator ML is called t he multiscale transformation and M L1 the inverse multiscale transformation. From th e two- scale relations (2.40) we deduce t he following matrix repr esentation of th e multiscale transformat ions -1

ML =

-

GL - 1 "

-

· Go

wit h t he ma trices

Provided t hat t here is an biorthogonal system L -1

,!, (L ) :=

Po U

Uv,

j=O

t o t he multiscale basis

L-1

~ (L) := $0 U

U .pj ,

j =O

t hen t he multi scale t ransformation realizes th e chan ge of basis, i.e.,

(2.42)

2.6 Multiscale Transformation

31

L-1

UL

:=

pI UL = P6 Uo + L

Iff! d j.

(2.43)

j=O

Here UL is the pro jection of any function U E L 1 (D) onto t he space spanned by the basis PL . As alrea dy discussed in Section 2.4 th e exist ence of the biorthogonal syste m is related t o th e uniform st ability of the L 2normalized counte rparts of th e mod ified box wavelets , see [Dah94] . In the sequel, th e basis functions Pj and Iffj are always referr ed to as primal scaling junctions and primal wavelets, respectively. Analogously, t he basis functions $j and rFj are called th e dual scaling junction s and dual wavelet, respectiv ely. This te rminology is oft en used to express t hat the fun ctions under considerat ion ar e expanded in th e primal bases while bounded linear functionals are repr esented in the du al basis. In th e pr esent conte xt we are int erest ed in multiscale transformat ions of arrays of cell averages, i.e. , of junction als of th e unknown solut ion. This motivates our not ation for the box functions and box wavelet s introduced above. Accordingly we will actually explicit ly const ruct always only t he uniformly st abl e bases {$ j }jENo and {rFj}jENo t hat are du al in the above sense to pro cess t he cell average s. The corresponding primal bases will ente r only t he analysis but will never be needed explicit ly for algori thmic realizations. Finally, we would like t o conclude with some rem arks concern ing the efficiency of the multiscale tran sform ation. In ord er t o keep th e transform ation efficient it is not reasonable t o comput e th e matrix ML given by (2.42). Instead t he t ra nsformation should be performed by applying successively the two-scale tran sform ation (2.40) as illustrat ed in Fig. 2.6. The best we can hop e for is an effort proportional to the dim ension of the finest space, i.e. , O( N d. For this purpose, the mask matrices Mj and Gj should be unijorm ly banded, i.e., in each row and column there are only a uniformly bounded number of non-vani shin g ent ries. This t ask is closely conn ected to the const ruction of basis functions which are locally finit e, i.e., t he number of functions of level j t hat do not vani sh in xED is uniformly bounded. If we now assume t hat t he dim ension of the spaces increases at least by a constant factor, i.e., N j +l ::::: a N j, a > 1, then t he multiscale transformation can be carr ied out in O(Nd operations. Note, t hat it is st ill pr ohibi t ed to compute ML becau se products of sparse matrices are in general not spar se.

3 Locally Refined Spaces

In general, a uniform refinement of the discretization is not adequate, since this results in a huge number of discretization points also in regions wher e the solution is smooth and a coarser grid would be sufficient for appropriately resolving the solution. Instead, the grid should be adapted to the problem at hand to produce an economic representation of the solution. By means of local grid refinements the complexity of the problem, i.e., the number of discretization points or unknowns, can be reduced to the number of" significant" points. The meaning of significance will be clarified later. We will see that locally adapted grids correspond in a natural way to locally refined spaces spanned by box functions or wavelet functions, i.e., $L,€ C $j and

uf=o

c Uf:~l .pj,

respectively. Here the refinement strategy will be bas ed on the multiscale decomposition corresponding to the modified box wavelets. By means of a truncated sequence of details corresponding to the index set DL,€ Jj , := {j} x t, x E* , we will determine an adapted grid on which the evolution step will be performed. This is outlined in Sect. 3.1. To facilitate an efficient change of bases between the local spaces $ L,€ and .pL ,€ the notion of grading is introduced. We will investigate in Sect . 3.2 how the grading of DL ,€ affects the grading of the adapted grid. The local multiscale transformation and its inverse are presented in Sect . 3.3. We emphasize that the number of operations is proportional to #D L ,€ instead of NL. The feasibility of these transformations is verified in Sect. 3.4. Here it turns out that the grading parameter has to be chosen sufficiently high. In view of the evolution step for the averages that has to be performed afterwards, it is helpful if the adaptive grid is locally uniform. Again, the grading plays an important role, see Sect. 3.5. Finally, we outline the algorithms by which the local transformations are performed and recall the core ingredients of the setting for curvilinear grids which are involved in the numerical experiments presented in Chapt . 7. We would like to mention that locally refined spaces constructed by means of thresholded multiscale sequences are subject of current research arising in different fields, e.g. composit e functions [DSXOO] and multigrid [DMS02]. It turns out that the setting has always to be adapted to the problem at hand. .pL,€

CUf:Ol

r,

34

3 Locally Refined Spaces

- -

'--

-

I--

-

f--

Fig. 3.1. Locally refined grid with han ging nod es

3.1 Adaptive Grid and Significant Details We are now concerne d with t he construction of locally refined spaces . First of all, we introduce t he notion of an ada pt ive grid by mean s of the nest ed grid hier ar chy.

D efinition 5. (Ad aptive Grid) Let 9 be a nested grid hierarchy. Then an adaptive grid is determined by an {j} x Ij , if the two properties index set 9L C

uf=o

-

[l =

U (j, k )E ~h Vj,k,

-I Vj,kn Vj' ,k,I=O,

(j,k ) :f;U', k' ), (j,k ),(j',k') EgL

hold. The grid corresponding to t he set 9L can be int erpret ed as a locally refined grid wit h hanging nod es, i.e. , cell verti ces of neighb orin g cells do not necessarily coincide, see Fig. 3.1. In slight abuse of not ati on we will refer to the index set 9L as t he adaptive grid. An ada ptive grid is det ermined by means of t he multiscale sequence d (L) =

( l'iT ' O

T, d oT , . . . , d T) L -1

where t he det ails dj,k,e corres pon d to t he mo dified box wavelet ,¢j,k,e' This sequence is compresse d by ap plying har d t hresholding , i.e., all det ails dj,k,e smaller t han a prescribe d t hreshold value Ej are discarded . T he truncated sequence is determined by t he ind ex set

V L,E :={(j, k , e) ; Idj,k,el > Ej , e E E* , kEIj , j E{ O, . . . , L - 1}}, (3.1) where E = (EO, . .. , EL )T denot es a sequence of t olerances. Again, in slight abuse of not ati on we will refer to this ind ex set as set of significant details. The correspond ing det ails (dj,k,e)(j,k,e)EDL,. is called the sequence of significant

3.1 Ad aptive Grid and Significan t Det ails

35

details. In analogy, we int rod uce th e sequence of local averages (Uj ,k) (j ,k)E9L,e corres ponding to t he adaptive grid. The adaptive grid det ermined by t he ind ex set 9 L ,e can now be const ructed by mean s of t he ind ex set V L ,e where we apply the following refinement crite rion .

D efinition 6 . (R efin em ent criterion) Let 9 be a nes ted grid hierarchy and V L ,e a set of significan t detai ls. Th en a cell Vj ,k , k E I j , is refine d if and only if there is e E E * such that (j, k , e) E V L ,e '

For lat er use we introduce t he local ind ex sets

he := {k

he := {(k,e ) ;

; (j , k) E 9 L ,d C I j ,

(j , k , e) E V L ,d C I j

x E *,

which correspond to t he indi ces of 9 L ,e and V L ,e on one level j . The ada ptive grid 9 L ,e is now constructed by the following refinement algorit hm. Algorithm 2 . (Refinement Algorithm) 1. Initialize It := 10 ;

2. for j = 0 to L-1 do 1. Initialize It +1 :=

0,

I j- :=

0;

2. A pply th e refinement criterion, i.e ., for (k,e) E J j ,e do Ii := Ii U {k}; Tt l := It +! U Mj, k;

3. Discard the refine d cells, i.e.,

h e :=

nv:

Here t he sets It C t, and I j- C t, can be interpret ed as collect ions of ind ices indicating cells on level j which might be refined on th e next finer level when applying t he refinement criterion and t hose which have been refined , respectively. Wh enever a cell Vj,k is refined, t hen t he indi ces of t he new cells Mj ,k C I H 1 have to be added to It 1 and the index k of t he refined cell has to be removed from It. The resulting grid is an adaptive grid in t he sense of Definition 5. The grid refinement is illust rat ed schematically in Fig. 3.2 where t he refined cells are sha ded. Finally, we introdu ce t he locally refined spaces
lJIL ,e :=


In analogy to (2.43), t he pr ojecti on UL can now be represente d in terms of both bases in the form UL =

L (j,k)E9L,e

Uj,k

vs»

=

L kElo

UO ,k 'PO,k

+

L (j,k,e)ETh,e

dj,k,e 'l/Jj,k,e'

(3.2)

36

3 Locally Refined Spaces

Fig. 3 .2 . Grid refinement

Note, t hat the cardinality of DL ,e and (h,e , respectively, might be mu ch smaller t han the number of cells of the finest grid , i.e.,

which , of cour se, dep ends on the regularity of u . In order t o swit ch between the single-scale and the wavelet repr esentation of UL according to (3.2) we will apply a local multiscale transform ation analogously t o (2.40). This transform ation is to be realized with an opt imal complexity, i.e. , the number of op er ations should stay proportional to #DL ,e'

3.2 Grading In order to realize the local multiscale t ransforma t ion in on e sweep through the refinement levels and in view of its feasibility we have to inflate the set of significant det ails D L ,e by a grading procedure. For t his purpose, we first agg lomerate all wavelet indi ces corr esponding to a cell Vj,k, k E I j , i.e.,

Tj,k := {(k ,e) ; e

E E *}

and introduce t he st encil of neighbors Nj ,k C I j corr esponding to Vj,k by the recur sive definition

(3.3)

Nf,k := {r

E

Ij : :3 s

E

NJ,,,l s.t. Vj,r n Vj,s =P 0} , i

= 1 , . . . , q.

Not e, t hat all cells are assumed to be closed. Then Nj~ k is referred to as the n eighborhood of degree q E No. This set is the uni on of neighboring cells corres ponding t o Vj,k which are at tached to each other by at least one point . Hence t he support UIE N jq,k Vj,l is a simply connecte d dom ain with no holes pr ovid ed t his hold s also for t he computational dom ain fl . Obviously, t he following relations hold

3.2 Gr ading

2

2

2

2

2

2

1

1

1

2

2

1

0

1

2

2

1

1

1

2

2

2

2

2

2

Fig. 3 .3 . Illu stration of the set s Nj~k for q

1E

N jq,k {::}

kE

N q j ,l

an d

37

= 0,1 ,2 and d = 2

N q N q+ l j ,k C j ,k'

(3.4)

In Fig. 3.3 the sets Nj~k are illustrated for two-dimension al grids composed of qu ad rilat erals and triangl es, resp ectively. Here the cell V} ,k is indicated by O. The neighborhood of degree q is det ermined by all cells lab eled with the number s q , q - 1, . .. , O. By the sets M j ,k C Ij+l we know the indices of the new cells generate d by refining t he cell V}, k. Sometimes it is helpfu l to know the invers e relation , i.e. , we are interest ed in t he ind ex k of the coarse cell corresponding to V} +! ,r , r E M j, k . This information is provided by t he op erator lr j +! : I j +! -+ I j , j = 0, .. . , L - 1, defined by lrj+l (r)

= k

with

rE

M j ,k '

(3.5)

Note, that t his fun ction is well-defined, since the nest ed grid hierarchy according to Definition 3 is assumed to be ga p- and redundancy-free. Hence, for any r E I j +! there is exact ly one ind ex k E I j su ch t ha t r E M j ,k. For instan ce, in the one- dimensiona l case t he refinement set corresponding to t he cell V} ,k is given by M j ,k = {2 k , 2k+ I} according to the uniform dyadi c grid refinem ent , cr. Sects . 2.2 and 2.5.2. Hence, lrj+l (r) = lr/2 J = k for rEM j,k . By means of t he above settings we can now introduce the notion of a graded t ree . Definition 7. (G raded Tree) Th e set of significant deta ils VL ,e is called a graded tree of degr ee q, if for any j E {I , .. . , L } th e relati on

(k ,e) E hold s.

J j ,e

~ Tj-1 ,r C Jj-1 ,e ,

Vr E

N I - l ,1rj(k )

(3.6)

38

3 Locally Refined Spaces

111I I I I 111111I I I I I I I I I I I

j=4 j=3

1 0 1

j=2 1

1

0

j=l 1

1

0

j= O

I III

q=O

I I I I I I I

q=l

Fig. 3 .4. Grad ed trees of degree q q = 0 (middle), q = 1 (bottom)

= 0,1

(t op) and corresponding adaptive grids

The grading of the truncated details results in a t ree st ructure of the details. Generally speaking, a graded t ree mean s t hat for any significant detail on level j t here are significant det ails in t he neighborhood but on t he next lower level j - 1. T hus, gra dedness simplifies t he local multi scale tran sform ati on, since t here are no isolated details on higher levels. In addition, locally switch ing between finer and coarser levels becomes easier, because only two levels are involved if the gra ding par ameter q is chosen sufficient ly large. In pa rticular, t he level difference between two neighboring cells is at most 1. Hence, we have to look for all neighb ors only on the same level, t he next coarser level or t he next finer level, i.e., we avoid recursive searching in t he t ree . We emphasize that for q = 0 t he details st ill correspond to a tree but not necessaril y a graded tree. In par ticular , a gra ded tree of degree q is also a graded t ree of degree q' < q. An example is presented in Fig. 3.4 for the one- dimensiona l case . Here t he coarsest uniform grid is composed of six intervals which are successively refined by dyadic grid refinement. We assume t hat t here is one significant det ail on refinement level 3 determ ined by (j , k) = (3,23). For t he univariate setting, d . Sects . 2.2 and 2.5.2, Definition 7 implies for k E h e NI-l ,lk/2J == {max (O, lk/2J - q), . .. ,min( lk / 2J

+ q, N j

-

I )} C J j-1,e

3.2 Gr ading

39

J k-l

k+l

k

•

•

•

•

l ~J-2

l ~J-l

l ~J

l ~J

j -1

+1

Fig. 3.5 . Neste dness of neighb orh oods of degree q

=1

where we omit t he ind ex e for the wavelet ty pe (e = 1). Then t he graded t ree of degree 0 is det ermined by all int ervals with lab el "0" whereas the graded t ree of degree 1 is determined by all intervals with lab el "0" and" 1". For both cases, q = 0 and q = 1, t he corres ponding ada ptive grid according to Algori thm 2 are also pr esent ed in Fig. 3.4. Note that for q = 1 two neighboring int ervals differ at most by one refinement level. Obvi ously, this is not t rue for q = O. In t he following we investigat e t he effect of grading the significant det ails on t he adaptive grid. For thi s purpose, we first observe that the nest edn ess of the underlyin g grid hierar chy impli es the nest edn ess of the neighborhoods. This conclusion is illustrat ed in Fig. 3.5 for t he one- dimensional case . Lemma 1. (N estedn ess of neighborhoods) If th e grid hierarchy is neste d, th en th e supports of th e n eighborhoods are also neste d, i. e., (3.7) VJ,1C VJ-l,l'

U

U

I EN/,k

I ENjq_l,"j(k)

Proof: The assertion is pro ved by inducti on over q. For q = 0 we know by ini ti alization of t he neighborhood s that Nj~k = {k} and N ?-l,1rj( k) = { 1f j (k)} . Hence, t he nest edness of the grid hierar chy impli es t he assertion in this case. We now assume t hat t he assert ion hold s for some q and decompose

U IENjq,tl

U

VJ,l =

VJ,1 U

IE N jq,t l \Nj~ k

U VJ,I. I EN j\

The inducti on ass umption together with t he recur sive definition of t he neighborhoods and (3.4) imply

U VJ,1C IENjq,k

U I EN jq_ l ," j (k)

VJ-l,l

C

U I EN]~;' '' j (k)

VJ-l ,l'

40

3 Locally Refined Sp aces

By definition of

»n:

the addit iona l layer of cells is attached to the neighbor-

hood Nj~k which is also true for NT~ll,1rj (k) and N I - l,1rj( k) ' Again we conclude from t he nest edness of t he grid hierar chy and t he induction ass umption : l E

{ 1fj(l)

Nj~t \Nj\ }

Thus, we obtain (3.7) with q repl aced by q

C

Njq~ll,1rj (k)"

+ 1 which completes t he proof.

o From t his lemma we deduce the followin g properties. Corollary 3. If the grid hierarchy is ne sted, th en th e follow ing relati ons hold:

1.)

NJ k C U lE N ?

J- l,7tj ( k)

,

1fj (Nj~k) :=

M j-1 ,l;

UlENj~ k 1fj (l)

C N jQ-l,1rj (k )

Proof. 1.) The ass ertion follows by Lemma 1 and t he nest edness of the cells since V} ,l C V}-l ,l = V} ,r

U

U

l EN ].k

U

l EN jq_l ." j (k )

U

lE N jq_l ," j (k ) r E Mj _l. 1

hold s. 2.) On acc ount of property 1, the definition of 1fj and the nest edness of the grid hier ar chy we conclude

U

U

V}- l ,l

=

lENjq_ l,"j(k) rE Mj _l ,1

U

V}-l ,l .

lE N jq_ l ," j (k)

This yields the assert ion. 0 The st ructure of t he adapt ive grid can now be characte rized as follows. Lemma 2. (A - stru cture of adaptive grid) A ssume that th e tree correspon ding to th e significant details is graded of degree q ~ O. Th en th e adaptive grid constr ucted by Algo rith m 2 is A -structured, i.e ., fo r any k E I j+l ,€ th e follow ing two relati ons hold: 1.) 1fi ° oo, o 1fj+l (k) ~Ii-l ,€ ,

2.)

T i - 2,1r i _ l Ooo '01r j+,( k ) C

J i - 2 ,€ ,

i = 1, oo ., j + 1; i = 2, ... ,j

+ 1.

3.2 Gradi ng

41

Proof. For simplicity we define k; := 7ri 0 •.• 0 7rHl (k). The assertion is proven by induction over i. For i = j + 1 we conclude from the refinement criterion, see Definition 6, that there is some index (kH 1 , e) E Jj,c' Therefore Vj ,kHI is refined according to Algorithm 2. Furthermore, the assumption of a graded tree implies that Tj-1,r C Jj-l ,c for all r E NI-l ,1rj(kH!l' Since 7rj(kH 1) = k j E NJ-l ,kj C NJ-l,k j we obtain, in particular, for r = k j that Tj -1 ,kj C Jj-1,c'

We now assume that the assertion holds for some i 2:: 2. Then the gradedness yields T i- 3 "r C J j- 3 e for all r E N'~ 3 , Jr .- 2 (k't-1 ). Since 7ri-2(ki-d = t k i - 2 E N'O_ 3 k, C N~3 k. we conclude for r = k i - 2 that Ti -3 ' k-2 C .. ' 1- 2 .. , t-2 J i -3 ,c ' Furthermore, the assumption k i - 1 E I i - 2 ,c would lead to a contradiction to the induct ion assumption , since the refinement criterion implies T i - 2 ,k i _ l (j. J i - 2 ,c ' This completes the proof. D This lemma shows how the grading of the set of significant details influences the adaptive grid . In particular, the gradedness of the tree corresponding to the significant details results in a A-structure of the adaptive grid. Thus the adaptive grid can be interpreted as the leaves of the tree. An example is shown in Fig. 3.6 illustrating these statements in t he onedimensional case . Here we consider a piecewise constant function u on the unit interval [0, 1] . The multiscale analysis is performed on a uniform dyadic grid hierarchy, d . Sect . 2.2, using the modified box wavelets according to Sect. 2.5.2 with s = 2 and grading of degree q = 2. The index sets (h,c and 1JL,c are represented by plotting the position k at the cell center of an interval Vj ,k versus the level j . The function u is represented by the averages Uj ,k plotted at the cell center corresponding to the adaptive grid . t

t

2 .1 14

2

12

1.9

1. 8

~

,

~

1. 7

,,

I. ' 1.5

10

,

1.4 +

1.3 1 .2

........

. + + +-

0 0

D. '

D.'

0. 8

0

0.2

-

D.'

. ...

12

10

-.. D.'

·-·

! 0.8

..

++++++ ... .. ++

0 0

0. 2

F ig . 3.6. Averages of function u (left), distribution of (;h ,e (middle) , distribution of VL,e (right)

We ar e now able to verify that the grading of the details implies the grading of the adaptive grid, i.e., two neighboring cells of the adaptive grid differ at most by one level. Proposition 1. (Graded tree =? graded grid) Assume that the tree of significant details is graded of degree q 2:: 1. Then the corresponding adaptive grid determined by Algorithm 2 is also graded of

42

3 Locally R efined Spaces

degree q, i. e., for any k E holds:

he

and l E N j\ on e of th e following three cases

1.) l E Ij ,e or

rt Ij ,e l rt I j ,e

2.) l

and 7fj(l) E Ij-l ,e or

3.)

and Mj ,l C Ij+l ,e .

Proof. Let k E he. Then we conclude from the refinement crite rion, see Definiti on 6, that there exists (7fj(k) , e) E Jj-l,e and, hence, t he property (3.6) of a graded tree impli es T j - 2,r C J j- 2,e for all r E N J-2,1fj_l (1fj(k)) ' Furthermore, Algori thm 2 guar antees t hat M j - 2 ,r C for these indices r , We obtain now by Corollary 3 t he inclu sion

tt.,

7fj

(Nj~k)

U

C N jq-I ,1fj(k) C

M j- 2,r .

rE.N'f_2,,, j -l (" j ( k ))

From this relation we conclude that 7fj (l ) E It-l for all l E Nj~ k ' According t o Algorithm 2 there are two possibilities, eit her 7fj(l) E I j-l ,e or Vj-I ,1fj(l) has been refined , i.e., l E It . In the latter case we hav e to check whether the cell Vj ,l has been refined again but at most once. For this purpose, we assume that t he cell Vj ,l for an arbit rary l E Nj~k has been refined twice, i.e. , there is s E It +2 such that 7fj+1 (7fj+2 (s)) = l . According t o the refinement crite rion, this ca n only be t he case if (7fj+2 (s), e ) E Jj+l ,e for an e E E* . Then t he gradedness of t he det ails impli es T J· r C JJ' e for all r E N Jq,1rJ.+ l (1T.+2 ()) = NJql, ' S J Since l E Nqk we conclude by Definition 3.3 of the neighb orhoods t hat k E J, Nj~k and, hence, (k , e) E he. In view of the refinement crite rion the cell Vj ,k has t o be refined, i.e. , k f- he . This is a cont radict ion to the assumpt ion. 0 From t he condit ions 1.) , 2.) and 3.) of Proposition 1 we conclude t hat an adaptive grid is graded of degree q if for any cell Vj ,k of t he adaptive grid all cells corres ponding to the neighborhood Nj~ k of degree q is (1) its elf a cell of t he adaptive grid or eit her t he (2) father or (3) t he children are cells of the adaptive grid. This is schematically illustrated in Fig. 3.7 for q = 2. Here the cell Vj ,k and the cells of the neighborhood Nj~k are indi cat ed by 0 and . , respectively. At t he bottom of Fig. 3.7 one possible ada pt ive grid is shown t ha t is graded of degree 2. In Sect . 4.1.1 t he grad edness of the ada pti ve grid will be helpful for performing the flux computation efficient ly on a locally uniform grid . Conversely, the grading of t he grid impli es the grading of t he t ree correspo nding to the significant det ails. 1

1

~ grad ed tree) A ssume that th e adaptive grid is graded of degree p . If th e con diti on

Proposition 2. (Graded grid

(3.8)

3.2 Grading

43

j+I

•

•

•

0

•

J

j- I

F ig . 3.7. Graded adaptive grid of degree q = 2

holds for any k E Ij with Tj,k C details is graded of degree q.

Proof. Let k E I j with Tj ,k C

Tj -

1 ,r

Jj,e

and k' E Mj ,k, then the tree of the

he . Then the grading of the details implies (3.9)

C Jj-1 ,e

On the other hand we conclude from the Proposition 1 that for all k' E Mj ,k and 1 E NJ+l ,kl one of the three conditions holds (i) 1 E I j+1,e or (ii) 1 (j. Ij+1,e and 7Tj+l(l) E Ij,e or (iii) 1 (j. Ij+1,e and Mj+l,l C Ij+2,e ' The refinement criterion implies that 7Tj+1 (NJ+l,kl) C It . According to th e refinement algorithm th is holds only if 7Tj (7Tj+l

(NJ+l,k' )) C

J j - 1,e '

(3.10)

Comparing (3.9) and (3.10) the assertion follows immediately. 0 Obviously, condition (3.8) is always satisfied for q = 0, since 7Tj+1 (k') and k' E NJ+1 ,kl and, hence,

=k

Furthermore, condition (3.8) gives an upp er bound for t he grading parameter q dep ending on p .

In Fig. 3.8 the condition (3.8) is illustrated for t he one-dimensional case according to Sect . 2.5.2. Hence t he uniform dyadic grid hierarchy implies k' E Mj ,k = {2k , 2k + I} . According to (3.5) and the dyadic grid refinement, cf. Sects . 2.2 and 2.5.2, we determine the index sets of the neighborhoods NJ+l ,kl = {k' - p, . .. , k' +p} C I j+l and NI-1 ,7rj(k) = {Lk/2J - q, .. . , lk/2J + q} C Ij- 1 as well as t he index set of the projection 7Tj+1 (NJ+l,k') = {L(k' p) /2 J,... , l (k' +p)/2J} C I j . These sets are indicated by • and 0, respect ively. From Fig . 3.8 it can be motivated that the condition (3.8) holds if 0 ::; q ::; lP/4J, cf. Corollary 6 in Sect. 3.8.

44

3 Locally Refined Spaces

j+1 k' -p

I

k'

e

I

e

L ~J

-

q

k'+p

I e

e

L ~J

I e

I

L ~J

J

j-I

+q

Fig. 3.8 . Illustr ation of condition (3.8)

3.3 Local Multiscale Transformation So far we have introduced locally refined spaces whose dimension corres ponds to # D L,e and # 9L ,e, respect ively. Now we have to explain how to perform t he change of bases between t he single- scale and t he wavelet repr esent ation in (3.2) with a number of operations pr oportional t o #DL ,e' This is only possible if t he involved matrices are uniformly banded and, hence, the mask matrices are sparse , see Sect . 2.6. In th e sequel we pro ceed in analogy to the tran sformation (2.40) of th e full spaces. For this purpose we first summarize t he multiscale t ra nsformation for t he averages and det ails defined by (2.3) and (2.26). From (2.3), (2.26) and (2.29), (2.30) we first derive t he two-scale relations A

= M- jT,a U j +l ,

(3.11)

A

Uj

(3.12) where t he matrices of t he mask coefficients are det ermined by (2.31). For t he modified box wavelet s similar relations hold where th e mask mat rices M j ,e , Gj,e are replaced by M j ,e, Gj,e det ermined by (2.37) and (2.38) . According to (2.33) t hese matrices are relat ed by a modificati on matrix Lj ,e det ermined by

(L' e)k r = J"

{l{',~ , r 0

E M j,l'

, elsewhere,

l E £ j,k U { k } ,

for r E I j+l ' k E Ij , corresponding t o the coarse grid modificat ion of the box wavelet s. To obtain an efficient multiscale t ra nsformation the computation of t he modified matrices M j ,e and Gj,e according t o (2.33) should be avoided. Instead it is mor e efficient to apply th e right hand sides directly t o t he vectors of t he averages and the det ails, respectively, because this requires less

3.3 Local Multiscale Transformation

45

operations . Then the two-scale relations for the multiscale transformation, i.e., Uj and dj ,e are computed by Uj+l, read A

Uj

=

M -T

A

j ,O Uj +l ,

Here the vectors Wj,e and dj ,e denote the coarse grid correction of the box wavelets and the details of the box wavelets. Analogously, the inverse multiscale transformation, i.e., Uj+l is computed by Uj and dj ,e, reads

(3.14) In order to realize the change of basis between the single-scale representation and the wavelet representation in (3.2) with an effort that is proportional to the number of significant details # VL,e and local averages # QL,e, respectively, it is prohibited to access the full coefficient vectors and mask matrices. Instead we have to carry out the transformations (3.13) and (3.14) componentwise only for those coefficients that correspond to QL,e and VL ,e' In particular, the summation is restricted to those indices which correspond to non-vanishing entries of the mask matrices. In order to describe this local transformation the notion of the support of matrix columns and rows is helpful, i.e., A k := supp(A, k) := {r ;

¥- O} = support of kth column of A, ak,r ¥- O} = support of kth row of A.

ar,k

A k := supp(A T , k) := {r ;

These sets are only helpful when the matrix A is sparse, Le., only a few matrix entries are different from zero. In this case it is useful to know the non -trivial terms of the matrix-vector product yT = x T A that have to be computed. The support A k of a column collects all non-vanishing matrix elements that might yield a non-trivial contribution to the kth component of the matrix-vector product, i.e., Yk. Therefore it can be interpreted as the domain of dependence for Yk> i.e., the components x; which contribute to Yk. The support A k of a row collects all non-vanishing matrix entries of the kth row that might yield a non-trivial contribution to the vector y of the matrixvector product . Therefore it can be interpreted as the range of influence, i.e., the components Yr which are influenced by the component Xk . Obviously, the supports satisfy the relations and For the description of the local multiscale transformation and its analytical investigation the supports of the box wavelets

46

3 Locally R efined Spaces

- Qj,k := SUpp (Gj,e, k) , M j,k := supp(Mj,e, k) , e E E Y*e YT Y*e YT - gj,'k := SUpp (Gj,e' k) , M j :k := supp(M j,e, k), e E E

and t he supports of the modified box wavelet s

- Lj,k -

('0

~j ,k

:=

supp (Lj,e, k) , £;:~ := supp (L],e ' k) , e E E *

. _ ;;0

.-

~j , k

U re

(' e

J..-j,k'

~j , k

._ ;; e

.-

~j , k '

e E E*

o .- MY0 - M . - MY0j,k = - M j,k, M e j,k'j,k u J..-r ej,k' e E E * - M j,k 'j,k =

u £ *,e ('*,e._ ;;*,e E E* j,k' ~j ,k .- ~j ,k ' e _ M *'O .- M *'O j,k'j,k' M *,e.j,k' - M *'O j,k u c-: j,k' e E E * _

(' * ,0 . _ ;;* ,0 ~j , k .- ~j ,k

will be needed . Not e, that the supports gj,k' M j,k' gi ,': and M ;:~ are no t ind icat ed by a t ilde. In particular , t he supports of the box wavelets are det ermined by M ej,k y

;;*,e MY0 - M = ~j, k = j,k = j,k,

MY* e ;;e j:k = ~j ,k

= MY*/ k'0

eE

E k ,

E

I

j

(3.15)

where t he support M~, k coincides with the refinement set M j,k corresponding to the cell Vj,k, d . Sect . 2.1. Wi th t he aid of t his not ation we now introduce the local mul t iscale transformation consideri ng (3.13) and (3.14) component wise. For t his purpose we have to specify which dat a are acce ssed when locally performing the transformation , i.e., we onl y consider the tran sform ation for a single coefficient . Cho ose k E I j such that M~, k C I Hi ,c ' Then t he local two-s cale transformation for t he k-th component of (3.13) read s (3.16)

W [.r,« -

' " L-J

u-:«i.e » S ,T'

(3.17)

r E S j,k'

s Eq ,r

ii,J+l,s --

' 'L...t " gYj,0{j, . _ '~ " 'L...J " gyj,e T',S s-r T ,S w J,T',e , rEgo eEE* rEg~ ), S

y

d j,r,e =

) ,S

e' Z:: mYjs:rUj+i ,s,

' "

sE

U PJ+i,k' (3.18)

eEE*

(3.19)

r E Uj,k'

sE Mj ,r

dj,r,e = dj,r,e + Wj,r,e =

L

m~', ~{j,Hi , s ,

r E Uj,k'

(3.20)

sE Mj ,r

Here the ind ex sets Uj,k C I j , PJ+i ,k C Ij +l and S j,k C t, are not yet specified . The main task of t his chapter will be t o derive t hese index sets such that each ste p of the local tran sform ation can be ca rrie d out, i.e., all

3.4 Grading P ar am eter

47

data on the right hand side have alrea dy been computed. In t he proof of Lemm a 3 t hey will be identified as Ui I' * ,e j,k = l- j,k'

according to the definition s (3.27), (3.28) and (3.29). Note, t hat we compute in (3.18) a predicti on value for Uj+l ,s which is accessed in (3.20). This prediction value has to be co:uputed only if s rf.. Ij +l ,E: , or , equivalently, dj,r,e = 0 for all coefficients r E 9J,s, e E E * , i.e., t he corresponding average Uj+l ,s is not available, because th e cell VJ+l,s is not contained in the ada ptive grid. According to th e construction of th e ada ptive grid thi s may happen only if t here are no significant det ails corre sponding to th e und erlying coarse grid cell VJ,k :l VJ+l ,s , otherwise VJ ,k would have been refined . In t his case the details are zero . Therefore we introduce no err or when applying (3.18). The exact repr esent at ion would be w J' ,r ,e - dJ·,r ,e = -dJ·,r ,e instead of w J' ,r ,eThe reverse transformation is easier. For this purpose, we assume that there is an index k E I j such t hat Tj,k C Jj,E: . According to one component of t he full t ra nsformation (3.14) t he local inverse two- scale transformation t hen read s

(3.21) L...J u-« l,sUj,l, - '"

(3.22)

Wj,s,e -

1E£'i ,s

d·J,s ,e -d J,s ,e

- w )',s ,e ,

s E S ;,k'

(3.23)

Again , the index sets It +l ,k C Ij +l and S ;,k C I j are not yet determined. T hey will be derived when proving Lemma 4. There th ey are identified as

+

_

- 0

I j+l ,k - Mj ,k ,

- e

Sj,k = {k}

according to the definitions (3.32) and (3.33).

3.4 Grading Parameter In t he previous sect ion we have presented th e local tran sform ations according to (3.16) - (3.20) and (3.21) - (3.23) , respectively. However it is not clear so far t hat these transformations can be efficiently realized . In particular , t he numb er of floating poin t operations should be proportional to th e number of cells in t he adapt ive grid and t he number of significant details, respectively. Therefore it is prohibited to carry out the transformat ions on

48

3 Locally Refined Spaces

the full discretization levels which results in an effort proportional to the number of cells of the finest discretization. In order to develop an algorithm with an optimal complexity in the above sense the local transformations are implemented according to the following strategy: - We proceed levelw ise according to the pyramid scheme for the full transformation, see Fig. 2.6, i.e., all significant details and local averages belonging to the current level are computed before we proceed with the transformation on the next coarser (finer) level. In particular, local recursions should be avoid ed. - We carry out one sweep through the refinement levels. This is only feasible if the set of significant details can be interpreted as a tree. Note, that this assumption is necessary for Algorithm 2 anyway. - We compute only those components of the two-scale relations (3.13) and (3.14) , respectively, which depend on significant details or local averages on higher scales . In contrast to the transformation of the full discretization levels it is not obvious that the local two-scale relations (3.16) - (3.20) and (3.21) - (3.23) can be carried out, i.e., all data on the right hand side have already been computed and stored. In the following we will verify that the different steps of the local multiscale transformation are feasible in this sense. In a first st ep, we determine all indices corresponding to data that have to be made accessible before carrying out the different steps in the transformation. Here we distinguish between the encoding and the decoding transformation, see Lemma 3 and 4. From this we conclude the feasibility of the local transformations provided the grading parameter of the tree of significant details is chosen sufficiently large. For this purpose, we consider the following setting. Assumption 1.

(1) the support q,k of the modification matrices Lj,k corresponding to the modified box wavel et is chosen such that k E

q,k

or, equivalently, k E £;:~

(3.24)

(2) the tree of significant details 'DL,e is graded of degree q.

Then we derive sufficient conditions by which the local multiscale transformation can be verified to be feasible. Lemma 3. (Local multiscale transformation) Let Assumptions 1 be fulfilled. Let k E I j such that M~,k C Ij+l ,e . If the condition 'lrj

(Ktk) C

NI-l '1rj (k)

with Ktk =

U U £j,s eEE*

SEC;::

(3.25)

3.4 Gr ading P aramet er

holds, then th e local two-s cale transformation (3.16) out.

49

(3.20) can be carrie d

Proof. First of all, we observe t hat the ass umpt ion M J,k C Ij+ l ,c is not restricti ve. If we have already perform ed the local mul tiscale t ransformat ion for j' = L , .. . .i + 1, t hen t his assumpt ion also hold s for level j. Hence, we concl ude t hat t he coarse-scale average s Uj,r can be computed according to (3.16) for all r E Uj,k := U r EMOk M *j :r = {k} . Here we use the fact that

°

v

i .

M J,r ~ 'O =

°

{k }

(3.26)

By ass umption all fine- scale averages are available. In addit ion to this , we have to compute the det ails dj,r,e, r E Uj,k' e E E * , depending on Uj+l, s, S E MJ,k' According t o (3.20) the set Uj,k is determined by Uj,k: =

U

M*J,r,e =

rEM~,k

(3.27) where we incorporat e (3.15) and (3.26) . The det ails are compose d of two parts, nam ely, the det ails dj,r,e corresponding to t he box wavelet and t he modifi cation te rms Wj,r,e du e t o the modificati on of the box wavelet . The det ails dj,r,e, r E Uj,k' are given by (3.19) where we acces s the averages Uj+l ,s, S E Pj+l ,k' with e p J+l ,k .'=

(U rEm

O

],k

M J,r e ) \ M J, k =

UrE,C,e\{k} M J, r' O

(3.28)

] ,k

Here we apply (3.15) and Assumption 3.24. Not e, that t he averages Uj+l,s k have not necessaril y been computed yet . However , we know for any s E P~+l J , t hat all det ails in t he two-scale repr esent ation of Uj+l ,s are negligibl e and have been discarded by t he t hresholding pr ocedure. Hence, we can reconstruct t hese averages only by coarse-scale averages according to (3.18) . For t he computat ion of the det ails as well as for the reconstruction valu es, first of all the modifi catio n te rms Wj,r,e have to be det ermined for all indi ces r E SJ,k with

s:j,k '-.- o:j,k U U

';>e ::I j,s

(3.29)

SEP;"+l ,k

acco rding to (3.17) . This requires t he knowledge of t he coarse- sca le averages Uj,l , l E K t k' with

K t k :=

U( U eEE*

S EP~+ ] l ,k

OJ,s U

U £ j,s)

sESJ~ , k

50

3 Locally Refin ed Spaces

The repr esent ation of this index set becomes more simple if we take into account th e definition of the involved index sets (3.28) and (3.29) as well as the fund ament al relations for the supports of the box wavelet (3.15). Fir st of all, we rewrite t he set Sj ,k' To this end, we observ e th at

u

sE

;01*,0 J, S

= £~ ,e\{k} J,k

'<:I e E E,

'0

UrE .c~'k\ {k } M j, r i,

where we make use of relation (3.26). This implies Sj ,k = £;:~ . From th e definition of t he set K tk we now conclud e tog ether with assumption (3.24)

«: = U (£;:~\{k} sec>: U £j,s) = U U £j,s' U

e EE *

eEE* sE£ J,k '.,e

J, k

According to Algorithm 2 the cells Vj ,r , r E K t k ' ar e element s of t he adaptive grid 9L ,e or will be further refined if and only if t he det ails corre sponding to t he index set

U

(3.30)

T j-1 ,trj( r) C Jj-1 ,e

r EKt.k

are significant . On t he oth er hand , we conclud e from k' E = T j ,k C J j ,e . Then th e grading implies

I j+l ,e

that

T j ,tri+ d k f)

U

T j- 1,r C Jj-1 ,e .

rE N jq_ l."j( k)

Hence, (3.30) is sat isfied, if

U rEKt.k

T j-1 ,trj (r) C

U

Tj-1 ,r.

r ENjq_l. "j(k)

This condit ion is equivalent to (3.25). D In Fig. 3.9 t he condition (3.25) is illustrat ed for the one-dimensional case according to Sect. 2.5.2. Here we assume th at th e stencil of t he modified box wavelets is symm etric, i.e., It = 0 in (2.39). The cells corr esponding to t he index set K tk = {k - 2 s, . . . , k + 2 s} C I j and the neighborhood N/-1,trj (k) = Nj q- l ,Lk/ 2J = {lk/2J - q, .. . , lk/2J +q} C I j - 1 are indi cated by • and 0 , respectiv ely. According to (3.5) and the uniform dyadic grid refinement , d . Sects. 2.2 and 2.5.2, we obtain 7fj(Ktk) = {lk/2-sJ , . .. , lk/2+ sJ}. Obviously, condition (3.25) holds if q 2 s , d. Corollary 4 in Sect. 3.8. Not e t hat condit ion (3.25) is sufficient to perform the local multiscale t ransformation. In particular, it is a constraint for t he choice of th e gra ding

3.4 Gr ading P ar amet er

k-2s . . . k-s .. .

k

. . . k+s . .. k+2s

I• I• I• I• I• I• I• I• I• I I

I

0

L ~J

-

o

0

o

o

J

I

o

I

0

L ~J

L ~J

q

51

j-1

+q

Fig. 3 .9. Illustration of condit ion (3.25) and (3.31)

par am et er q. In order to keep t he transformation as efficient as possible we prefer q to be as small as possible. For t he local inverse multiscale transformation we obtain anot her const ra int for t he grading par am et er.

Lemma 4. (Local inverse multis cale transformation) Let Assumption s 1 be fulfill ed. Let k E I j su ch that Tj ,k C tion 1fj(£ j ,k) C

he ' If the

N jQ- l ,1r; (k )

condi-

(3.31)

holds, th en th e local inverse two- scale transfo rmation (3.21) carried out.

(3.23) can be

Proof. First of all we det ermine all fine- scale averages Uj+l ,r , r E ItH ,k ' influenced by the det ails d j ,k ,e which are assumed t o be significant . According t o (3.21) these are det ermined by

+ I·+ J 1, k :=

U -.

e

-

a

(3.32)

Q·J,'k=M J·, k '

eEE'

where we incorpora te the fund amental relations for th e support s of t he box wavelet (3.15). From (3.22) and (3.23) we conclude that t he coarse-scale averages Uj,s,

-

~

S

E Sj,k' and th e det ails d j ,s ,e ,

S;,k :=

U l EI;+l ,k

Qj,l =

-e

S

E Sj ,k are accessed where 1

U Mj:2= {k}.

(3.33)

MJ ,k

For the computat ion of t he det ails according to (3.22) we first have to perform (3.23) where we additiona lly access th e coarse-scale averages flJ' , s , S E £~J , k ' Note, t hat t he support £j,k sat isfies (3.24). According t o Algorithm 2 t he cells V j ,r, r E £j,k' are elements of t he ada pt ive grid QL ,e or will be refined further if and only if the det ails corresponding to the ind ex set

52

3 Locally Refined Spaces

UT

j - 1 ,1r; (s)

(3.34)

C Jj-1 ,e

SE£ ; .k

are significant. On the ot her hand we conclude from gra ding that

U

T j-1 ,r

C Jj-1 ,e'

C

U

Tj ,k

C J j ,e and the

rEN;q_ l .~; (k )

Hence, (3.34) is sat isfied, if

U

T j- 1,1r;( s)

s E£;.k

Tj-1 ,r'

r E N;q_l . ~; (k)

This condit ion is equivalent to (3.31). D Again, the resulting condition (3.31) is illustrat ed for t he one-dimensional case according to Sect. 2.5.2. In Fig . 3.9, the cells corresponding to the index set £j,k = {k- s, .. . , k+,s } C I j and th e neighborhoodNI_l ,1r;(k) = {lk/2Jq, . . . , lk /2J + q} C I j - 1 are indicated by • and 0, respectiv ely. According to (3.5) and the uniform dyadic grid refinement , d . Sect s. 2.2 and 2.5.2, we obtain 7fj (L:j,k) = {l(k - s)/2J , ... , l(k + s)/2J} . Obviously, condition (3.31) holds if q ~ ls/2J which is weaker t han condit ion (3.25) . From Lemma 3 and 4 we now conclude the main result of thi s section. Theorem 3 . (Lo cal transformation) Let th e A ssumptions 1 be fulfilled. Th e local multiscale transformation as well as its in vers e can be perform ed pro vided that con diti on (3.25) holds . P roof. According to Lemm a 3 and 4 the conditions (3.25) and (3.31) are sufficient for the feasibili ty of th e local transformations. Since L:)~ ,k C K\ ) , condition (3.25) is st ronger than (3.31). D

3.5 Locally Uniform Grids The feasibility of the local multiscale tr ansformations depends on th e grading of the set of significant det ails DL ,e , see Theorem 3. In Sect. 3.2 we have already investi gat ed the influence of t he graded tree on th e structure of the ada pt ive grid. Lat er , the evolut ion step of th e avera ges will be perform ed on th e adapt ive grid. This requires the computation of th e num erical fluxes at the interfaces of neighboring cells depending on a local ste ncil. For this purpose, we have to provide the averages correspo ndin g to this st encil. In case of a st ructure d flow solver as it is applied in the num erical experiment s in Chapter 7, t he adaptive grid should be locally uniform. This is t o ensure t hat the reconstruction of averages on on e local level does not involve all refinement levels bu t at most two . For t his purpose, we have to specify what we mean by a locally uniform grid .

3.5 Locally Uniform Grids

53

Definition 8. (Locally un iform grid of degree p) An adapt ive grid is said to be locally uniform of degree p, if for any k E he on e of th e three condit ions holds for any 1 E NJ,k : (1) 1 E he or - 0

(2) Mj ,l C Ij +l ,e or

(3) T j -

J j - 2 ,e

2,1r j _ ,(r ) C

'tIr E U e EE * L j,1rj(l ) .

If the grid is locally uniform of degree p , t hen we can access the averages Uj,l, 1 E N J,k' by

(1) eit her directly accessing the averages corres ponding to the adaptive grid (2) or locally applying the two-scale transformation , i.e.,

(3) or locally applying the inverse two- scale transformation, Le.,

L

Uj,l =

g~7l ,O Uj-l ,r +

rE9J _l ,1

L L

g~7l , e dj-l ,r,e,

eEE* rE 9j_ l, 1

d-j-l ,k,e = d j-l ,k,e - Wj-l ,k,e,

Wj- l, k,e =

lj-l m,k ,e Uj-l,m '

"L-t

A

mE £j_l,k

Again , if t he t ree of significant details is sufficiently gra ded, the ada ptive grid is locally uniform. Theorem 4 . (Grad ed grid => locally uniform grid) A ssume that th e Assumptions 1 are fulfill ed. Additionally, th e adapti ve grid is assum ed to be graded of degree pEN . Wh enever k E I j su ch that th ere is k' E M~,k n IHl ,e, th en the adaptive grid is locally uniform of degree p , if the grading param eter q satisfies Jr ' J

(J+) C Nq J,k

J-l ,1rj (k)

. with J+ := J,k

U

lENP

i + l ,k'

U

eEE*

L~J,1r j+,(l)

(3.35)

Proof. Let k E I j such th at k' E M~,k n IH l ,e. Then we conclude from the definition of a graded grid that for all 1 E NJ+l ,kl one of the following condit ions hold s: (1) 1 E IHl ,e or (2) 1 rt. I j+l ,e and M~,l E I H 2,e or (3) 1 rt. IHl ,e and Jrj+l (1) E he . In case (1) th ere is nothing to do, since Uj+l ,l is already available in t he adaptive grid . If 1 rt. I j+l ,e the avera ge UHl ,l has to be computed by either coars ening or refining t he grid . Fir st we consider case (2). According to t he local two- scale transform ation we det ermine Uj+l ,l by

54

3 Locally Refined Spaces

Finally, we have to check case (3). Here we have to apply the local inverse two- scale t ran sform ation in ord er to determine Uj+1,l, i.e., UjH ,1 =

L

g~:~ Uj,r +

rEgJ,l

L L eEE*

d-j ,k',e == d j, k' ,e

-

Wj, k' ,e,

g~:~ dj,r,e,

rEg;, l 'L..t " [j,e m,k' mECj,k'

Wj ,k' ,e =

A

Uj,m

o

To carry t hese computation out , we obviously need t he coarse-scale averages Uj,m for all indices m belonging to 9J,l u

U U L:j,r = M;:? U U U eEE *

rEg;,l

eEE *

L:j,r =

rEM ;:?

U L:j ,1THl(l) ' eEE*

Here we incorporat e the Assumptions 1. According to t he refinement crite rion t hese dat a are available if and only if \:I r E

U

L:j,1THl(l) '

(3.36)

eEE*

On t he ot her han d, t he refinement criterion and t he choice of k and k' , respectively, which are correlated by k = 1fj+1(k' ) imply th at Tj,k C h e. Hence we conclude from t he grading of t he t ree that

r.:«;

(3.37)

C J j - 1 ,e

In order to ensure (3.36) it suffices that

U

U

IENJ+l,kl rEU e E E *

Cj '''Hd l )

T j- 1,1Tj(r) C

U

T j- 1,r

rEN,"_l,"j(k)

holds. This condition results in a lower bound for t he gra ding par ameter q. It is fulfilled whenever (3.35) holds. D In Fig. 3.10, t he condition (3.35) is illustrated for t he one- dimensiona l case according to Sect . 2.5.2. Here we assume t hat t he ste ncil of t he modified box wavelets is symmet ric, i.e., J.L = 0 in (2.39). In par ticular , we conclude from t he uniform dya dic grid hierar chy t hat k' E M ~, k = Mj,k = {2k , 2k+ I} holds. According to (3.5) and the dyadic grid refinement , cf. Sect s. 2.2 and 2.5.2, we determine t he index sets of t he neighborhoods NIH,kl = {k' p, . . . , k' + p} C I jH and NJ-1,1Tj(k) = {lk/2J - q, . .. , lk/2J + q} C I j - 1 as well as t he ind ex set = {k - 8 - lP/2J,... , k + 8 + lP/2 J} C I j . These sets are indi cat ed by • and 0 , respectively. From Fig. 3.10 it can be motivated t ha t the condition (3.35) holds if q 2: r(rp/21 + 8) /21 , cf. Corollary 5 in Sect . 3.8.

Jtk

3.6 Algori thms: En coding, Thresh olding, Gr ading, Decoding

55

j+1 k'-p . .. k' ... k'-tp I e I e k-s - LI J

L ~J

-

q

I

k

k+s

L~J

J

+LIJ L~J

j-1 +q

Fig. 3.10. Illustrati on of condit ion (3.35)

3.6 Algorithms: Encoding, Thresholding, Grading, Decoding In t he previous subsections we have analyzed to some extent t he feasibility of t he local t ransformat ions (3.16) - (3.20) and (3.21) - (3.23), respectively. In t he following we will pr esent t he algorit hms by which t he local multiscale t ra nsformation and its inverse are realized in our computatio ns. Note, that we comment on each step of an algorithm after havin g summarized the compu t at ional steps. The st ructure of the algorit hms as well as the definiti on of the involved ind ex sets is motivat ed by the pro ofs of Lemm a 3 and 4, i.e. , t he proofs serve as a guideline for t he computation. However , in the algorit hm we pro ceed levelwise. Consequent ly, we do not carry out t he local transformation for one single ind ex bu t for all local averages and significant det ails t hat are influenced by local informat ions on t he finer (coarser) level, respectively. In par ti cular , t he local multiscale t ra nsformation (3.16) - (3.20) is carried out for all k E I j where the averages Uj+l ,r, r E M ~, k have been computed in t he pr evious step on level j + 1 or t he averages correspond to cells of t he ad apt ive grid, respectively. For t he local inverse transform ation (3.21) - (3.23) we pr oceed similarly. Here we compute th e averages Uj+l ,r , r E M ~,k ' t hat ar e influenced by significant det ails d j ,k ,e for all (k , e) E ,he: . First of all we consider th e encoding algorithm. This algorit hm need s t he following input information: (i) th e number of refinement levels L, (ii) t he local index sets he: , j = 0, . . . , L , repres enting th e ada pt ive grid and (iii) t he corresponding sequences of local averages (U j ,dkE lj" , j = 0, . . . , L . Then t he encoding algorithm provides the following out put : (i) t he local ind ex sets of det ails J j,e: , j = 0, , L - 1, (ii) th e corresponding sequences , L - 1 and (iii) t he averages (u o,kh E1o of det ails (dj ,k ,e ) (k ,e)EJ j." j = 0, of t he coarsest grid. We emphas ize t hat t he ad aptive grid corresponding to riL,e: may be det ermined by a set V L,e: t hat does not corres pond to t he local averages (Uj ,k)(j,k )E9L .• of t he input dat a. This sit uation occurs ind eed when performing an evolut ion st ep on the ada ptive grid. Thus, t he set V t..e of t he

56

3 Locally Refined Spaces

output data is not necessarily graded and it might include non -significant det ails, because no thresholding is performed.

Algorithm 3. (Local multiscale transformation) for j = L - 1 downto 0 do e .- U 1 · UjO '.- U 'o: U j'e e E E* rElj+l,e M*[.r rElj+l,EM*, j,r'

2. Uj,k := L rEMJ'k rh~:~ Uj+l ,n k E UJ 3. PJ+l := Uk E U~ M j,k\Ij+1 ,c , e E E* , Pj+1 :=

U eE E *

]

4·Sj

:=

UJ U U1 E P E+ j

1

PJ+l

9j,l' e E E *

,euJ ' .,n k E s je' e E E * 5· Wj,k,e ..- '" LJrEC j,k ljr,k _j, O ' " " «i .e k E Pj+1 ' 6· Uj+1 ,k -- '" LJrEQ io. k gr,k Uj,r - 'LJeE E* 'LJ rEQ ~ k gr,k Wj,e,r, J,

7. for e E E* do -1 - j,e ' k E Uij LJrEM ~ k mr,kUj+1 ,r, . d-j,k,e: = '" s.

2. dj ,k,e = Wj,k,e + dj ,k,e,

e E E *,

k E UJ

B.if k ~ UJ U U(l ,e)EJj,c {I} then delet e rh~:~, r E M~,k 9. for e E E* do

if k ~ Sj then delete l~:~ , r

E q ,k

The local multiscale transformation is performed levelwise starting on the finest level. For each level we pro ceed as follows. In st ep 1 we first determine t he indices of all coar se-scale averages and details which dep end on the finescale averages of the adaptive grid . The set s UJ, e E E* , can be represented by

UJ=

U

rElj+l ,E

.c;:% ,

if the relations k E q ,k and k E resp ectively, hold . Since the adaptive grid is assumed to be gap - and redundancy-free, see Definition 5, we know t hat all fine-scale averages required in the computation of the coa rse- scale aver ages in ste p 2 are available. Otherwise an err or occurs in the comput at ion. Moreover, we have to determine the matrix entries of the kth row of Mj,o if not yet available. In st ep 3, we det ermine all indices for which an average on level j + 1 will be need ed. This reconstruction value is compute d in st ep 6 and finally accessed in step 7.1. For the computation of these valu es we have to det ermine all indic es for which the coarse-scale modification terms Wj,k,e, e E E *, have to be calculat ed, see st ep 4. The modifi cation terms are then computed in st ep 5. Here we have to verify that all coarse-scale averages

3.6 Algorithms: En coding, Thresholding, Grading, Decoding

57

Uj,r have already been comput ed. According to Theorem 3 this is guaranteed provided the det ails are sufficiently graded. Note, that we introduce in step 6 an error by th e pr ediction value for Uj+l ,k, k E PHI . According to (3.21) and (3.23) t he err or is given by

L L

g~:% (dj ,r,e + Wj,r,e) =

eEE* r EQj,k

L L

g~:%dj,r,e.

eE E * r EQj,k

By t he assumpt ion on the adaptive grid we know that th e details dj,r,e are smaller than the threshold error Cj ; otherwise VJ+l,k would be part of the adapt ive grid and th e average would have been computed, i.e., k E Ij +l .e In ste p 7, we comput e the det ails dj,k,e' Since the comput at ion is not only optimized with respect t o computat ional t ime but also memory requirements, it is prohibit ed to compute and store all ent ries of the mask matrices. Instead, we have to manage those ent ries locally, i.e., only those columns ar e stored which are accessed in the local transformation process. This means th at we have to compute th e matrix entries of t he kth column of Mj,e, Gj,e and Lj,k' if not yet available before performing th e st eps 2, 5, 6 and 7, respectively. In order to avoid an inflat ion of the matrices we have to remove all ent ries of Mj,o and Lj,e, e E E* , corr esponding to th e kth column th at have not been accessed in th e tran sformation, see st ep 8 and 9. Having performed th e local multiscale transformation , the set of details VL ,c is neither graded nor thresholded. Therefore, we have t o pres ent algorithms by which the non-significant details are discarded and the tree of details is graded. First of all, we outline the thresholding algorithm. Here we do not consider a scalar function u E L 1(D, R) but a vector-valu ed function u E L 1 (D , R'") in view of applicat ions to a system of conservat ion laws. For systems it is convenient to make th e thresholding scale-invariant, i.e., the thresholding is invari ant with respect to multiplying the avera ges by a const ant factor. This becomes necessar y because for syst ems of conservat ion laws no longer a maximum-minimum principle holds as in th e scalar case. Hence, local ext rema may develop and increase in time for each conservative quantit y. If we do not introduce the scaling, then we have to choose threshold valu es Cj for each component . For t he scaling, we define the quantity Ildll oo,* for a vector d E Rm by

Ildlloo,* := Ild*lloo, d* .'= {

,uoo i < tol

di

'

~ , elsewhere '

(3.38)

UOO ,t

where tol is a machine depending tol erance to avoid division by O. Here the component s of th e vector Uoo E Rm are defined by Uoo ,i:= . max (J ,k)E9L.e

I(Uj,k)il, i

= 1, . . . , m .

Obviously, the following input data are required: (i) th e number of refinement levels L , (ii) the sequence of t hreshold valu es e, (iii) the scaling param et er tol ,

58

3 Locally Refined Spaces

(iv) t he local index sets he: , j = 0, .. . , L, repr esenting t he adaptive grid, (v) t he corresponding sequences of local averages (u j,khElj,e' j = 0, . . . , L , (vi) t he local index sets of det ails Jj ,e: , j = 0, . .. , L-1 and (vii) the corresponding sequences of det ails (dj,k,e)(k,e)E Jj,e' j = 0, . . . , L - 1. Then the t hresholding algorit hm pr ovides th e following out put: (i) the local index sets of significant details Jj ,e: , j = 0, . . . , L - 1 and (ii) th e corres ponding sequences of significant det ails (dj,k,e)(k,e)EJj,e' j = 0, . . . , L - 1. Algorithm 4. (Th resholding) for j = to £-1 do

°

1. Uoo,i := maxj=O,...,L m aXkElj,e: I(uj,k)il , i = 1, . .. , m

2. for (k , e) E Jj,e: do 1. for i = 1 to m do

if Uoo ,i < tol t hen (dj,k,e)j

:= (dj,k,e)i else (dj,k,e)i := (dj,k,e)i!u oo ,i

2. Ildj,k,e 1100,* := Ildj,k,e 11 00

3. for (k, e) E

he: do

if maxe' EE* Ildj,k,e,lloo,* < Cj t hen

he: := he:\({k } x E)

and delete dj,k,e" e' E E

In a prepr ocessing ste p we compute t he maximum values Uoo by means of the local averages (Uj,k)(j,k)E!h ,e' Then we determ ine th e maximum of the details dj,k,e according to t he scaling (3.38). If all det ails corres ponding to t he different wavelet ty pes are smaller t ha n th e th reshold value Cj t hen these coefficients are non- significant an d have to be discarded from Jj,e: and V L,e: , respecti vely, and t he corres ponding values have to be removed from t he memory. Here it is not necessarily requir ed to proceed levelwise. Furtherm ore we like t o remark t hat t he t hresholding can be alte rnatively perform ed within t he local multi scale t ransformation, see Algorithm 3, where Uoo has to be determ ined in a prepro cessing step and t he details compute d in step 7.2 are only stored if II(dj,k,e)eEE*lloo,* ~ Cj' For t he convenience of th e reader , t he local multiscale tran sform ation and t he thresholdin g are separ ated in order to emphas ize where we have to distinguish between a scalar equation and a system of conservation laws. Note, that except for the thresholding all algorithms can be applied component- wise. Before t he inverse local multi scale tran sform ation can be perform ed, t he tree of details has to be gra ded, see Theorem 3. As input dat a we need: (i) the num ber of refinement levels L , (ii) th e grad ing par ameter q, (iii) t he local index sets of significant details he:, j = 0, . .. , L - 1 and (iv) the corresponding sequences of significant details (dj,k,e)(k,e)EJj,e' j = 0, .. . , L1. Then t he graded tree of significant det ails V L,e: is determined as out put of t he following algorithm.

3.6 Algorithms: Encoding, Thresholding, Grading, Decoding

59

Algorithm 5. (Grading) for j = L - 1 downto 1 do

1. Jt-l,e :=

0

2. for (k , e) E he do Jt-l ,e:= Jt-l ,e U (NI-l ,1rj(k) x E) 3. for (k , e) E Jt-l ,e Vj-l ,e do d j-l,k,e := 0

4· Jj-l ,e := Jj-l ,e U Jt- l,e Here we have to proceed from the finest to th e coarsest level. On each level we first have to initi alize the grading set Jt-l ,e' Then we collect th e neighborhoods ~q-l ,1rj(k) C t.., of degree q for all significant details (k , e) E Jj,e and set th e corresponding details to be zero if not yet available. Finally, we have to add th e grading set to the originally det ermined det ails on th e lower level j - 1. Finally we consider the decoding algorithm. This algorit hm needs th e following input information: (i) the numb er of refinement levels L , (ii) th e local ind ex set s of significant details Jj,e , j = 0, . . . , L - 1, (iii) th e corre sponding sequences of significant det ails (dj ,k,e)(k,e)EJj,e ' j = 0, . . . , L - 1 and (iv) the average s (uo,khE1o of the coarsest grid. Note, that the set of significant det ails 'DL ,e has to be gr aded in view of Theorem 3. Then th e decoding algorithm provides the following output: (i) t he local index sets he ' j = 0, ... , L , repr esenting th e ada ptive grid and (ii) the corresponding sequences of local aver ages (u j,khElj,e' j = 0, .. . , L . Algorithm 6. (Local in verse multiscale transformation) 1 + '- 1 for j = to L-1 do

° .- °° + __

1.Ij +l

.-

2. -S j :=

. . *,e

U( e,I)EJj,e g j,l

U1E1t+l g"*j,'l°

,e Uj,r, ' k E -Sj , e E E * 3 . Wj,k,e := '" u r E.c ~ k ljr,k J,

4. dj ,k,e := dj ,k,e - Wj,k,e,

k E Sj, e E E*

' "j, O , «i ,e d 5. Uj+l ,k = '" ur EgJO, k gr,k Uj,r + '" u eE E * '" u rEg ~ k gr,k [.e .r , J.

6.Ij- := UkE1t+l

Mj:2,

he := l tV; and delete Uj,k, k E I;

7. if k rt: It +l n {k : g~:~ exists, r E 9J,d th en delete g~:~, r E 9J,k In a pr epro cessing ste p we have to initi alize th e ada pt ive grid by all cells of th e coarsest level. Then we perform the local inverse transformation in one sweep through t he refinement levels starting from th e coars est resolution. On each level we proc eed as follows. Fir st we collect in st ep 1 all t he fine-scale

60

3 Locally R efined Spaces

averages which ar e influenced by significant det ails. Then we determine all ind ices of coarse-scale averages involved in th e computation of t he modification te rm Wj ,k ,e , see step 2 and 3. Before we perform t he summation in st ep 3 we have t o check wheth er t he matrix entries of t he kth column of L j ,e are available. Otherwise we first have to det ermine t hese ent ries . In addition, we comput e th e details corresponding t o t he box wavelet according to step 4. Here we might access a non- significant det ail d j ,k ,e , i.e., k ~ J j ,E;' In t his case we set d j ,k ,e := O. Fin ally, we calculate t he fine- scale averages, see 5. Not e, that an err or occurs if t he dat a Uj, l and d j ,l ,e are not available. In this case the set of significan t det ails does not fit th e assumptions, i.e. , it is not gra ded. Moreover , we have to compute th e ma trix ent ries of t he kth column of G j ,e , e E E , if not yet available. In ord er to avoid redundanci es we remove all average s on the coarse scale which have been refined, see ste p 6. In par ti cular , we clear th e st orage. To t his end, we first det ermine all cells of level j th at have been refined and remove them from th e adapt ive grid . This proceeding is analogical to Algorithm 2. Then we know all cells of the level j and th e corresponding average s, i.e., he and (U j ,k)k Elj ,€' Fin ally, we have to remove all columns of t he matri ces Gj,o which have not been used in th e t ra nsformation step, see step 7.

3.7 Conservation Property In Chapter 4, we will employ t he local multi scale ana lysis for a finit e volum e meth od . For each t ime level, the ada ptive grid is determined by mean s of t he det ails corresponding to t he multiscale decomp osition of t he averages at hand. Th en we pr edict a new ada ptive grid for the next t ime level. This requires t he t ra nsformation of t he averages from the old to t he new grid where we access t he sequence of truncated det ails. In view of t he FVS t he question arises wheth er the truncation process violat es th e conservat ion property of t he reference FVS . However , t he following lemma shows that t he t hresholding does not affect this pr operty.

Lemma 5. (Cons ervation property) Defin e th e prediction values .

._

UJ+l ,r ' -

'"

L..J

-j ,O

gl ,r Uj ,l, r A

E I j+ l·

(3.39)

lE9J,r

Th en the truncation process is locally mass preserving, i.e.,

L

IVj+l ,rl U j+l ,r

rE M J,k

an d the global conservation propert y

= IVj,kl Uj ,k '

(3.40)

3.7 Conservation Property

L

L

IVi,kl Uj ,k =

(j ,k )Eg e

61

(3.41)

1V0,k! UO ,k

k E10

holds. Proof. First of all, we note th at

L

IVi+l,rl Uj+1 ,r

=

L

IVi,kl

r E M ~, k

A

r E M ~, k

=

IVi+1,rl Iv.J ,k I Uj + l ,r

- j ,O Uj+l ,r IVj, k I "'" ~ mr,k A

=

IVj ,k IUj ,k A

r E M~,k

where we apply t he represent ation (2.31) of th e mask coefficients m~'~ as well as the local two-scale transformation (3.16) . Together with Definition 5 of t he ad aptive grid t he global conservat ion property (3.41) follows. In ord er to verify th e local conservat ion property (3.40) we first consider t he error introduced by th e truncation , i.e.,

L

IVi+1,rl (Uj+ l ,r -

Uj+l ,r) =

L

IVi,kl

r E M ~,k

r E M~, k

=

L

IVi,kl

r E M~, k

IVi+1,rl V. I I J,k m~:~

(A

_)

Uj+l ,r - Uj+l ,r

L L

g{::dj,l ,e ,

eEE * IEg j ,"

where we incorporat e Definition (3.39) for the prediction values and the local inverse t wo-scale t ra nsforma t ion (3.21) . From the reversibility of the multiscale transform ation we conclude th at the error vani shes, since "'"

gj ,e L...t m rj ,o .k L.r

=0

rEM ~, k

for all e E E * and r E I j +1 , l E J j ,e , see also (2.13). Note , that the sets M~, k and 9j,r can be replaced by the global ind ex sets because th e remaining mask coefficients are zero according to t he definition of the supports. Fin ally, we obt ain

L

IVi+1,rl Uj+ l ,r

=

r E M~,k

L rE M~,k

IVi+l,rl Uj+ l ,r +

L

IVi+l,rl (Uj+l ,r -

Uj+l ,r)

rEM~, k

which imm ediately lead s to (3.40) when applying the local multiscale t ransforma t ion (3.16). 0

62

3 Locally Refined Spaces

From t his lemma we conclude that all mass is stored in t he averages but no mass is sto red in t he det ails. Hence the t ru ncation of t he details can not affect t he conservation property of the adaptive FVS, since we perform no threshold on the coarse-scale averages" .

3.8 Application to Curvilinear Grids The num erical experiments presented in Chapter 7 have only been carr ied out for st ructure d grids, in particular Cartesian and curv ilinea r grids. Therefore we summa rize some comp utational aspects t hat arise when app lying t he above mult iscale concept to st ructure d grids. Here we consider all t he steps that have to be performed when ap plying th e concept to a concrete applicat ion which may serve as a concrete example. Nested Grid Hierarchy. The start ing poin t is a smoot h function x : R :=

[0, I]d --+

n

which ma ps t he param et er doma in R onto the comp utationa l domai n n. The J acobian is assumed to be regular, i.e., det (8 x(~) /80 -:p 0, ~ E R . In our applications we represent t he grid funct ion by B- splines, see [BMOO]. This admits control of good local grid properties, e.g., orthogonality and smoothness of the grid, and a consistent boundary represent ation by a small number of control poi nts depending on the configuration at hand. Then a nest ed grid hiera rchy according to Sect. 2.1 is defined by means of a sequence of nest ed uniform partitions of the parameter domain. To th is end , we int rodu ce t he sets of multi- indices d

I j :=

II{O,Nj ,i -I } e Ng,

j

= O, . .. , L,

i=1

wit h Nj,i = 2 N j- 1 ,i initialized by some No ,i . Here t he prod uct denotes the Cartesian product , i.e., n~= 1 Ai := A l X ... X Ad' Then t he nest ed sequence of parameter parti tions R j := { Rj,k hE lj ' j = 0, ... , L , is determined by d

R j,k :=

II [ki h j,i' (ki + 1) h j ,i] i =1

with hj,i := I/Nj,i = h j- 1 ,i/2. Finally t he nest ed partitions of the computational domain are obtained by Vj,k := X(Rj,k ), see Fig. 3.11 for an illust ration .

1

In [CKMPO l] t he t hreshold process is also applied to the coarse-scale averages.

3.8 Application to Curvilinear Grids

63

1 x

o

1

Fig. 3.11. Tr an sformati on from param et er to comput ati on al dom ain

Obviously, the refinement relation for th e cells holds with M j,k = {2k+i ; i E E := {O,l} d},

i.e., each cell is subdivided into M; = 2d subcells. In t his case, the sequence {Qj} j=O,....i. is referr ed to as a dyadic grid hierar chy. We emphasize t ha t t he set E can be identified with the index set E = {O, ... , M; -I} by which the different types of box wavelets are indicat ed in Sect . 2.3. As a special application of t his cur vilinear setting we have already considered the Car tesian grid hierarchy, cr. Sect . 2.3.1. Construction of Box Wavelets. In order to const ruct th e box wavelets according to Sect . 2.3 we have to determine appropriate matrices Aj,k' Stability of t he multi scale transformation is ensure d when the rows of the matrix form an orthonormal syste m whereas the lin ear in dependence of th e rows already pr ovides a valid complement basis. Not e, th at in case of a cur vilinear grid the matrices Aj,k in general differ for all (j ,k) . However , t hey coincide provided t ha t t he grid refinement is unifo rm, i.e., (3.42)

cr. Sect . 2.3.1. In t he sequel, we are therefore int erest ed in a construct ion of matrices Aj ,k t hat lead s to an efficien t computation. To t his end, we assume t hat t he matrices are enumerated by th e index set of multi- ind ices, i.e., 'k

'k

Aj,k = { a~',r} eEE,rEM~,k = {a~', 2k+Je,iE E .

Then t he start ing point for t he const ruct ion is th e matrix A := { ( -l)e.i }e,iEE which is symm etric and, up to a factor, orthogonal. The inverse is given by A- 1 = 2- d A, see [Got98], Lemma 3.2, p. 77. Note, t hat A and A- 1 are ind ependent of j and k. We now define t he matrix Aj,k := A A j ,k by mean s of the coefficients j,k _ (_l) e.i (I VJ+1 '2k+d) 1/ 2 _ (_l )e'i j,k a e ,2k + i IV-) ,k I a O,2k + i '

(3.43)

64

3 Locally Refined Spaces

The inverse of t his matrix exists and is dete rmined by Aj,~ = 2- d Aj,~ A = A J-:-~ A-1 with t he coefficients , 1/ 2

bj,k.

2k+ l,e

= 2- d (_l)e-i

IVj,kl ( IV.I) J+1,2k+l

= 2- d (_l)e-i(a j ,k .)-1 O,2k+l .

(3.44)

Note, that the inverse matrix is not determined by A;'k ' since the matrix Aj ,k is in genera l not orthogonal du e to th e relation d

A j ,k := Aj ,k A;'k = A A ],kA = 2 A-1 A ],kA.

This only becomes t he identity mat rix , if t he grid refinement satisfies (3.42) , e.g. Cartesian grids . For this special configuration one obtains A ] ,k = 2- d I, cf. Sect . 2.3.1. However , in t he case of a cur vilinear grid based on a smoot h grid fun ction t he grid refinement is only locally approximate ly equidistant, in particular , for higher refinement levels j , i.e., A] ,k 2- d I. We emphasize t hat in case of a non-orthogonal matrix, t he box wavelets and their Loo-norm alized counte rpa rts do not form an biorthogonal system. Hence we can not apply Corollary 1, i.e., Mj , l is not necessarily a stable completion of Mj ,o if t he matrix Aj ,k is not orthogonal. Nevert heless, the syste m is supposed to be stable in t he sense of Sect . 2.4 due to t he above local perturbation argument . This is jus tified by the num erical result s. In spite of the missing orth ogonality prop ert y, th e resulting box wavelets form a basis, since t he matrices Aj ,k ar e invertible. Finally we present t he mask coefficients of t he corre sponding refinement equations. For the box wavelet we obtain for k , 1 E I j , e E E 'V

«i» _ 2-d IVj,kl (_l) e-i 8 gk,2I+i IV I l,k , j+1,2k+i

l-.i ,k m - (2. 31) by aj ,k d bj,k j,k an d ur,e h were we rep Iace ae,r 2k+i,e accor d'mg t 0 e ,2k+i an (3.43) and (3.44) .

Modification of Box Wavelets . In ord er to increase t he numb er of vanishing moments of t he box wavelet we apply Algorithm 1. To this end, we have to fix t he ste ncils £ j,k C I j . Here we choose only one ste ncil for all e E E. To obtain a small support we choose a local stencil that is located around the cell Vj,k where t he box wavelets ,¢j ,k,e , e E E , corres pond to, i.e., £ j ,k :=

d

d

i= l

i=1

II £ j,ki = II {lj, ki> " " lj,ki + 28} ,

°

with lj,k

=

{

8

E No ,

, O:S ki:Ss - 1 , 8 :S k i :S N j,i - 1 - 8 N - - - 1 - 28 ' }N, 1-,- - s < N J,Z --- 1 _ k 2- < _ k, -

) )1,

8

(3.45)

3.8 Application to Curvilinear Grids ,

,,, • , ,,, ,

-

-

-

-

,,- - - - -- - - ,, • ,, ,, ,, ,, ,, , ,,, ,

, ,, ,,

- I

, , , ,

,- - - - - - -- - -

,- - - - -, , , , , , , • , , - --

-

- -

'- - - - - - -- - -

,, ,, ,, ,

, ,, ,, ,,

•

, ,

,,

- -

-

•

, ,

- -,

, , ,

,

I: I:

,

,,

,

,- - - - - - --

'- - - - - - -- - -

,- - - - -, , , , , • , ,

- I

,, ,, ,, ,, ,,

65

-'

,

- I

:1 :1

,:1 ,

,

'- - - - - - -- - -

,,

--

Fig. 3.12. Illustrati on of the sets 'cj ,k for s

= 2 and d = 2

Here t he stencil width is 28 + 1 :::; N j,i in each dir ection. Insid e t he computational domain fl a cent ra l st encil is chosen , i.e., 8 cells at each side of VJ,k' Close t o t he boundar y we shift th e cent ra l stencil such t hat the support is completely contained in fl . An illustration of the set s in the bivariate case d = 2 is shown in Fig. 3.12. Here the dot s indi cat e th e positi on k and the dotted lines border t he ste ncils. For the one-dimensional case, t his set ting has already been considered in Sect . 2.5.2. In order to minimi ze t he st encil width with respect to a given number of vani shing mom ent s Me we choose 8 :=

min {r E No ; (2r

+ l)d > Me}.

(3.46)

Fin ally, we pr esent t he mask coefficients of th e modified box wavelet - j,e _ IVJ+l,2I+d (lj,e + (_l) e.i J:) IVJ,II I,k Uk ,l , m 2 I+i,k j g-k,021+' = 2- d ,

1

IVJ,II IVJ+l ,21+d

(6 _"" k,l

L."

eEE*

e E

E

,

(_l) e.i lj,e) k,I '

where we have applied (3.43) and (3.44) in (2.38). Not e t hat for a uniform Car t esian grid hierar chy, d . Sect . 2.3.1, appropriate modified box wavelets with higher vani shin g mom ents can alte rnatively be constructed according to (2.18) using t ensor products of t he univari at e modified box wavelets det ermined in Sect . 2.5.2. Supports of Box Wavelets and Their Modification. For th e repr esentat ion of t he local multiscale setting, see Sect . 3, we introduced th e supports of t he columns and rows corres ponding to th e mask matrices. In t he case of the curv ilinea r set ting t hese can explicitly be det ermined . First of all, we observe t hat all supports can be repr esented as Car t esian products, i.e., -

M ej, k --

n:

JL=l

M el-' j,kl-' ' g je,k --

rr

d JL=l

g j,kl-' el-' , e E E ,

66 -

3 Locally Refined Spaces

TId M *,e" 1'=1 j,k" '

M* ,e j ,k -

9*,e j ,k -

TId1'=1 9 j,k,, *,e" '

eE

E,

- £ j,k = TI~=l £j,k", £ j,k" := {l j,k" , . . . , lj,k" + 28}, -

TId 1'*,8 1'=1"-- j,k,,'

1'*,8 _

"--j ,k -

1'*,8._ {-:Ol -ll } "--j,k,, '- j,k,,' " . , j,k" .

-0

-1

.

Here the numbers lJ''J.L k , lJ' k , M= 1, . . . , d, are det ermmed by Lemma 6 and , p, h»; by th e stencil £ j,k according to the modi fication of t he box wavelet s. In particular , for t he box function and t he box wavelets th e supports are given by

- Mj ,k = 9; :: = {2k + i ; i E E} , k E Ij , e E E ,

- M;:~ = 9j,k = {lk/2J}, k E IH 1 , e E E

j ,

where Lk /2 J = l is th e lar gest int eger l not lar ger than Lk /2 J. Then we obtain t he supports of t he modified box wavelets

- M~,k = M~,k = {2k+i ; i E E} , k E I j

,

- Mj,k = {r E Ij +l ; Lr/2J E £ j,k} =TI~= 1{2lj,k" , ... , 2lj,k,, + 48 + 1}, kEIj , eEE*, - 9J,k = £ j,lk/2J' k E I H 1, - 9j,k = 9j,k = {lk/2J} , k E I H 1 , e E E* , * ,0 _

- *, 0 _

-0

_

.

- M j ,k - M j ,k - 9j ,k - {lk/2J} , k E I J +1 ,

- M;:~ = £ ;:Lk/ 2J ' k E I j+l , e E E *, -

9;:~ =

{r ; Lr/2J E £ ;:D =

TI~=l {2~,k" , .. . , 21~,k" + I} , k E I j ,

-9;::=9;,~=Mj,k={2k+i ; iEE} , kEIj , eEE*.

The const ruction of t he support £ j,k is mor e compli cat ed becau se t here exist t hree different ty pes of matrix st ruct ures sketched below for t he onedim ensional case (N j = 12, 10, 8, respectively, and 8 = 2).

1) 28 + 1 < N, - 1 - 28: + '" '" . . + '" '" + '" * '"

'" '" '" + '"

'" * '" '" '" + '" '"

'" '" + '" '" '" '" + '" '"

'" •'" +• +. . •'" • '" '" + '" '"

'" '" '" + '" '" '" '" '" +

2) 28 + 1 = N j

-

1 - 28:

3.8 Ap plica t ion t o Curvilinear Grids + '" * '" '" + '" '" '" '" + '" '" . . +

67

...

'"

.. '" '"

'" . . + '" . . '" '" + * '" '" '..." +* +'" '" ... . . '" '" + . .

'" '" '" '" +

3) 2s

+ 1 > N,

+'" +• •* •'" '" '" + '" '" ...... '" + [ '"

- 1 - 2s:

•... ... +'" *'" . . '" + '" '"

1

'" '" '" + . . '" '" '" '" +

Employing these different matrix st ructures t he support s of £;:~ are dete rmined by t he following lemm a . Lemma 6. (Construction of £;:~) Let N j 2: 2s + 1. Th en th e support of th e k-th colum n of Lj,k is det erm in ed by th e uppe r and lower bounds

f

- {

j, k" -

0 , 0 :::; kJL :::; 2s k J-Ls ' 2s + 1 < · - 1 ' - k JL < - N J,JL

t _{ kJL + s .k

J, "

, 0 :::; kJL :::; N j ,JL - 2s - 2 N J,JL · - 1 ' N·J,JL - 2s - 1 < · - 1 - k JL < - N J,JL

-

Proof. Since th e support £;:~ is a Car t esian product, it suffices to consider only one component . Therefore we assume d = 1 and omit th e subindex f..l for the dir ection . Then we consider the following decomp osition of the support CJ ,'~ = M 1 U M 2 U M 3 with

t, ; 0 :::;

M

1

:= {l E

M

2

:= {l E I j

M

3 :=

{l E I j

;

l :::; s - 1 , k E

«< l :::; N j

-

£j,d ,

s - 1, k E

£j,d ,

N, - s :::; l :::; N, - 1 , k E LJ,l}'

From t his we conclude

M

_ { {O, .. . , s - I} ,0 :::; k :::; 2s

0

1 -

M

2

M

= {l E I j = { {Nj

3

; -

, elsew here '

max {s , k - s } :::; l :::; min {N j

-

s - 1, k

s , . . . .N, - I} , N , - s - 2 :::; k :::; N j

0

, else w he r e

-

+ s }} , 1 .

Then we have to distinguish six cases according to t he t hree different matrix st ructures of Lj ,k'

68

3 Locally Refined Spaces

la) N j 2: 4s

M

+ 2 and

0 :::; k :::; 2s:

={O , . . . , s - I }, M 2 = { s, .. . ,k+s} , M 3=0 ,

1

Ib) N, 2: 4s

+ 2 and 2s + 1 :::; k

:::; N, - 2s - 2:

M l=0, M 2 = { k - s , . . . , k + s}, M 3=0 , l c) N, 2: 4s

+ 2 and

2s

+ 1 :::; k

:::; N, - 2s - 2:

M 1 =0, M 2 = { k - s , .. . , k + s}, M 3=0 , 2a) N j M

2b) N j M

2c) N j

M

:::; 1

2s - 2:

4s

+ 2 and

:::; N, - 2s - 1 :::; k :::; 2s:

= {O, ... , s - I} , M 2 = {s, ... , N j

:::;

1

-

= {0, .. . , s - l }, M 2 = {s, ... ,k+s} , M 3 = 0,

:::;

1

4s + 2 and 0 :::; k :::; N j

4s

+ 2 and

2s

+ 1 :::; k

:::; N j

-

-

s - I} , M

3

= {O, .. . , N j

-

I} ,

1:

=0, M 2 = { k - s, ... ,Nj-l-s} , M 3={Nj- s, .. . ,Nj-l- s} .

This verifies the assertion.

0

Neighborhoods. For th e gra ding of the det ails as well as the definition of locally uniform grids , see Definition 7 and 8, we need the neighborhoods N j\ C I j of degree q. In case of st ruc ture d grids as they ar e considered here these are determined by d

N j\ =

II {max(k

i -

q, 0), .. . , min(Nj,i - 1, k; + q)} .

i= l

Note, t ha t in case of an unbounded domain we do not have to take the maximum and minimum values into account but use instead the set of int egers {k i - q, ... , ki + q}. Grading Parameter for Modified Box Wavelets. In view of the feasibility of the local multiscale transformation and its inverse we have to verify Theorem 3. The result is motiv at ed by the illustration in Fig. 3.9 for the onedimensional case according to Sect. 2.5.2. Here we assume that the st encil of t he modified box wavelet s is symm etric, i.e., fJ, = 0 in (2.39). Corollary 4. (Verification of local transformation) In the curvilinear case the local multiscale transformation as well as its inverse are feasible, if the grading param eter sat isfies q 2: s. P roof. In ord er to verify t he assertion we apply Theorem 3. Since we consider the cur vilinear case, all supports can be int erpreted as Cartesian products. Hence, it suffices to consider the one-dimension al case in the subs equent proof.

3.8 Applicati on to Curvilinear Grids

69

In order to choose t he appropriate gra ding par ameter q we first have to determine the index set Kt k according to (3.25). For the special choice £ j,k == Lj ,k' 'Ve E E * = {I} t his set can be repr esented as -1 I j .k

U £ j,r = U {lj,i ," . ' lj,i + 28} = {I~,k "

Kt k =

rE L:.;:~

' " I},d

i=I~.k

where lj,i and ~,k ' I~,k are determin ed according to the previous subsect ions. We now determine t he lower and upp er bounds of t he index sets depending on t he level j and t he position k. To t his end, we distinguish six cases corres ponding to N , and k . 1.) N j

::;

48 + 1:

+ 28},

a) Ktk = {O,

,k

b) Kt k = {O,

, N, - I},

0 ::; k ::; N , - 28 - 2, N, - 28 - 1 ::; k ::; 28,

c) Kt k = {k -28, .. . . N , -I} ,

28+ 1 ::; k ::; N j -1 ,

2.) 48 + 2 ::; Nj : a) Ktk = {O , . . . , k b) Kt k = {k - 28,

+ 28}, 0 ::; k ::; 28, , k + 28}, 28 + 1 ::; k , N j - I} ,

c) K t k = {k - 28,

::; N , - 28 - 2,

N j - 28 - 1 ::; k ::; N j - 1.

From this we conclude IO

_

j,k [1 _

j,k -

{Ok - 28 ,0::; k ::; 28 , 28 + 1 ::; k ::; N , -

1 '

{k + 28 , 0 ::; k::; N

j - 28 - 2 N j - 1 , N j - 28 - 1 ::; k ::; N j - 1

Now, condition (3.25) can be written in t he form

1rj(Ktk ) = {l1~, k/2J , . . . , ll],k/ 2J}

c

{lk/2J - q, . . . , lk/21

+ q} = NI- l ,1rj(k)

which is satisfied if q ~ max{lk/2J - ll~,d2J , ll},k/ 2J + lk/2J} ~ 8.

o Anot her const raint for the grading par ameter q might be impo sed, if t he adaptive grid has to be locally uniform. Then we deduce from Theorem 4 t he following result for t he cur vilinear case. Once mor e the result is motivated by t he illustrati on in Fig. 3.10 for t he one-dimensional case according to Sect . 2.5.2. Here we assume that the ste ncil of th e modified box wavelets is symmet ric, i.e., f-L = 0 in (2.39).

70

3 Locally Refined Spaces

Corollary 5. (Locally uniform grid)

Let the assumptions of Theorem 4 hold. In the curvilinear case the grid is locally uniform of degree p, if the grading parameter q satisfies q 2: max(s, Wp/2l

+ s)/2l)·

(3.47)

Proof. In order to prove the assertion we have to verify condition (3.35) in Theorem 4. Again we restrict to the one-dimensional case , see Corollary 4. Then the index sets in (3.35) can be represented as N!+l ,kl = {max(O,k' - p), ... , min(N j+l -1 ,k' + p)},

7fj+l(N!+l ,kl)

= {max(O, 7fj+l(k' -

p)), . . . , min(N j - 1, 7fHl (k'

N jQ-l ,7fj(k) = {max(O,7fj(k) - q), . . . , min(Nj_ 1

-

1, 7fj(k)

+ p))} ,

+ q)},

£.j,r = {lj ,r, . .. , lj,r + 2s} , 7fj(£.j,r) = {7fj (lj,r), . . . ,7fj (lj,r) + s}, where we employ 7fj(r) = Lr/2J. Now condition (3.35) reads

U

7fj(£.j,r) C NJ-l ,7fj(k)'

rE7fj+l (N!+l ,k

l )

From this we deduce

for all r E 7fj+l (NJ+l,kl)' Since the coefficients lj,r are monotonously increasing, i.e. , lj ,r ::; lj,r l for r ::; r' ; we only have to check

7fj (lj,r_) 2: max(O, 7fj (k) - q) and 7fj(lj,r+) ::; min(Nj_ 1 - 1, 7fj(k) + q) - s with r_ := max(O, 7fj+l(k' -p)) and r + := min(Nj -1 , 7fHl (k' +p)) . We now investigate the inequality (3.48) in detail. To this end , we distinguish three cas es depending on k for the estimate to the left and the right, respectively, i.e .,

1) 0::; k ::; s ± IP/2l - 1, 2) s ± IP/2l ::; k::; N j -1- s ± IP/2l , 3) N j

-

s

± IP/2l ::; k ::; N j

-

1.

Her e the " +" and" ., » correspond to the lower (L) and upper (U) estimate in (3.48), respectively. Since k' = 2k + ik for an appropriate ik E {a, I} , this results in different bounds for k', namely

1) 0::; k' ::; 2(s ± IP/2l) - 1,

3.8 Appli cation t o Curvilinear Grids

71

2) 2(s ± IP/2l) ::; k' ::; 2(Nj - s ± IP/2l - 1, 3) 2(N j - s ± IP/2)l ::; k' ::; N jH - 1.

From t his we conclude th e upp er and lower bounds for r _ and r + 1) 0::; r Of ::; s - 1,

3) N J· - s < - Trx: , < - N J· - 1.

2) s ::; r Of ::; N , - 1 - s,

Then we obtain for the coefficient s 1) lj,T = 0,

2) lj ,T = rOf - s = I(k' =f p)/2l - s,

3) lj,T = N j - 2s - 1.

Finally, we derive the const raints for th e gra ding par ameter

Ll , V 3) q 2: l(lp/2l

2: Wp /2l L3 , V I ) q 2: s .

L2 , V2) q

+ s - 1)/2J + s)/ 2l,

= 1(IP/2l

+ s)/ 2l

- 1,

Now, condit ion (3.47) is obtained choosing the maxim al value on the right hand sides. D According to (3.47) the gra ding par ameter q is always lar ger than t he ste ncil width s . This implies th at t he local multiscale transformation and its inverse are always feasible, see Corollar y 4. In genera l, t he requirement of a locally uniform grid imp oses a st ronger const ra int on th e gra ding par amet er q t ha n t he feasibility of t he local tr ansformations . For inst an ce, in case of ste ncil width s = 1 and grid gra ding degree P = 3 the gra ding degree of th e significant det ails has to be lar ger t ha n q = 2. On t he other hand, t he gra ding of the adapt ive grid implies the gra ding of t he significant det ails, see Propositi on 2. In th e curv ilinear case we can specify condition (3.8). Corollary 6. (Graded gri d =} grad ed tree) Let th e adapt ive grid be grad ed of degree p . Th en th e tree of significa n t deta ils is graded of degree q pro vid ed that

0::; q::; lP/4J.

(3.49)

P roof. Again we consider only th e one-dimension al case. Then we have to determine t he sets which are involved in (3.8), i.e.

N jQ- l ,1r;(k) = {max (O, k" - q), . .. , min (N j _ 1 NJ+l,kl = {max(O, 2k 7fjH

+i

- p),

00"

-

1, k" + q)}

min(Nj H

-

1, 2k + i + p)}

(N JH ,k') = {max(O, ki,p" ' " min(Nj - 1, ktp}

7fj (7fjH(NJ+ l,kl )) = {max(O,k" +li,p), oo . ,min(N j_ 1 - 1, k" +ltp)}

with k' = 2k

and

+i

= 4k"

+ 2l + i, i , l

E {a, I}

72

3 Locally Refined Spaces

kt p:= k + l(i ±p)/2J ,

ltp:= l(l

+ l(i - p)/2J)/2J.

From t his we immediat ely conclude t he assertion. 0 According to Sect . 2.5.2, the resultin g condition (3.49) is illustrat ed for the one- dimensional case, d . Fig. 3.8. Here we assume t hat the ste ncil of the mod ified box wavelets is symmetric, i.e., 11, = 0 in (2.39).

4 Adaptive Finite Volume Scheme

In the previous chapters we have outlined ind epend ently th e core ingredients of a FVS and the framework of the multiscale setting. Now we are combining these two settings in order to derive an adaptive FVS . We start with a given reference FVS determined by the averages and flux balances on the finest resolution level. From this scheme we deduce evolut ion equations for the details and the averages. In order to reduce the complexity of the reference scheme these equations are only performed for the local averages corresponding to the adaptive grid 92i1 and the significant details i52i1 , respectively . These sets are determined by means of th e local multiscale transformation of the local averages corresponding to the old time level. It turns out t hat the construction of appropriate numerical fluxes on the locally refined grid and the prediction of the significant details on the new time level are the core ingredients which crucially influence the performance as well as the reliability of the adaptive scheme . By construction, the complexity of the resulting scheme is proportional to # V2 ,e whereas the complexity of the reference FVS on the finest resolution level is proportional to N L . Hence the reference FVS is significantly accelerated. In particular, the speed-up rates increase asymptotically with increasing number of refinement levels. This is verified by numerical experiments, see Sect. 7. In the sequel , we first outline the construction of th e adaptive FVS and then pr esent the main algorithms.

4.1 Construction The starting point for th e construction of the adaptive FVS is a nested grid hierarchy according to Definition 3. For the finest resolution level L we then consider an arbitrary (explicit) FVS, see Sect . 1.2, that can be written in the form n+l _ n \ Bn (4.1) v L ,k - vL ,k - AL ,k L,k ' where the flux balance is defined by

74

4 Ad aptive Finite Volum e Scheme

B2,k := L 1Ft.11 Ff:t

(4.2)

I

and T is chosen such that the CFL restriction for the explicit FVS holds for th e fin est grid . In ord er to handl e the numerical fluxes at the boundary {) [l we again introduce a layer of ghost cells for each discretization level such t hat t he hierarchy of ext ended grids is still nest ed in th e sense of Definition 3. Note, that t his does not affect the multis cale transformation but only the flux computation in the evolution st ep. In addition to th e not ation introduced in Sect . 1.2 we have to indicate the level. In the sequel, we only consider num erical fluxes that fit int o t he following setting. Assumption 2. (1) Th e numerical flu xes

Ff:t are determ ined by Ff:t := F(v2,r ; r E Ft.I) '

(4.3)

where th e ind ex set Ft.1 C h denotes an arbitrary but fixed st en cil. (2) If rtl, C {) [l , th e numerical flux es at th e boundary are computed by (4.4) where the state Vk ,1 is eithe r det erm in ed by th e prescribed boundary conditions or by th e attached flow field. (3) Th e numerical flu xes are consi stent, i.e.,

(4) Th e numerical flu xes are conservative, i.e., FL ,n _ _ FL ,n k ,l

Next we introduce the averages

-

v j,k

t .k :

(4.6)

and det ails d j,k ,e on the coar ser scales

j = 0, .. . , L - 1 by the definitions

(4.7)

d nj, k,e: =

"'"'

L...J

- j,e n mr,k Vj+ l ,r '

k E I j , e E E*

(4.8)

rE M j ,k

for an arbit rary time level t n by means of the box function and the mod ified box wavelet . For the sake of simplicity, we USe the mask coefficients

4.1 Construction

75

m~:~ instead of distinguishin g between the initial coefficients m~:~ of the box wavelets and the modific ation t erms corr esponding to the coefficients l~' ~ , Appl yin g the mul tis cale transformation (3.16) - (3.20) we obtain discrete evolut ion equat ions for the averages and th e det ails where we employ (4.7) and (4.8) as well as (4.1) n+l v j ,k

=

n \ vj ,k - "'j ,k

Bn j, k ' (4.9)

d j, + k, e n

1 -

-

d"j ,k ,e

""""' v .+ l ,r L-.... "'J

-

iii j,e Bn r ,k

j+ l ,r '

k E l j , eE E*

.

r EM j, k

Here, the flux balanc es

n Bi» >

B~\

""""' L-....

rEM~,k

ar e det ermined by

IVj ,k I I m- s» Bn IV r ,k j+ l,r

_

-

J+I ,r

""""' Bn L-.... j+l,r'

(4.10)

r E M ~,k

where we have used (2.31) and A j ,k := T/IVj ,kl · So far we have only derived evolut ion equations for the average s and details corresponding to the full grids. The adapt ive FVS is now det ermined by a significantly smaller selection of evolution equat ions corr esponding to n+ 1 . tea apt ive gnid rv t.e . i.e., h danti n+l vj ,k

= V jn,k -

\ "'j ,k

Bn

[. k»

(4.11)

Here, t he ad ap tiv e grid is det ermined by the set of significant det ails V£i1 according to the Algorithm 2. We ar e now facing two fund am ental problems, nam ely, - how to compute the local flux balances B j,k without employing the flux balan ces corre sponding to the cells of the finest grid and - how to det ermine V£i1 without evolving the averages on the globally finest level, applying the mul tiscale transformation and finally thresholding the det ails? In t he following two sub sections we will discuss these questions. 4.1.1 Strategies for Local Flux Evaluation According to the definition of the flux balan ces (4.10) the computation of B j ,k requires all flux bal an ces B2,k' corres ponding t o the cells V L ,k ' C Vj ,k of t he fin est level. However , exploit ing the conservation property of the fluxes (4.6) t he right hand side of (4.10) can be simplified . For this purpose, the boundary part r11is decomposed into smaller par ts r{ I' according to the local ind ex sets ' ,

76

4 Ad aptive Fini t e Volume Scheme

stf := {( k' ,l') ; k' ,l' E Ij' , r {,l l C r i ,t>

VJ I,l1 C VJ,d,

0::;

j ::; jf ::; L.

Then t he num erical fluxes and also the flux balances (4.10) can be rewritten as (4.12) Bj,k = !Ft.11 F{',7

L I

wit h the local num eric al fluxes

L

F{',7 =

F{,',~

(k' ,I')Est:f

=

L

Ff,'JI =

L

F(v2,T ; r

E

F f,,ll)'

(k' ,l1 ) Est: ~

(k' ,I')Es t'}

(4.13) Here, t he flux balanc e Bj,k is compute d by the num eric al fluxes Ff,'JI corresponding t o t he boundar y part r f, ,1' C ri,1 C 8 VJ ,k' All num erical fluxes corres ponding t o r f, ,I' inside the cell VJ ,k cancel each other due t o the conservatio n property (4.10). Thus, (4.12) is preferable to (4.10) with regard t o an efficient computat ion. Fur thermore, we observe that th e computation of t he num erical fluxes Ff,'JI requires t he average s V2,T' r E Ff" I' C Ii., on the fin est level which have to be pr ovided by means of t he local inverse two- scale t ra nsformation (3.21) - (3.23), see Fig. 4.1 (left) . This, in general, inflat es t he complexity by a logarithmic factor depending on the spatial dimension d. In par ti cular , for a dyadi c refinement we obtain # = (2d - 1 )L - j . In ord er t o t reat multidimensional pro blems a chea per approximation is desirabl e. If t he reference FVS is based on num erical fluxes corre sponding to a st ruc t ure d grid, one alte rn at ive is t o make t he grid locally uniform , Le., th e grid is locally refined such t hat t he fluxes can be comp ute d to dat a of the sa me refinement level, see Fig. 4.1 (middl e) . For the computat ion of t he num erical fluxes F{',7 , (j, k) E 92}1 , we have t o distinguish three cases, nam ely,

sk'f

(i) the neighb oring cell VJ,I also belongs to t he ada ptive grid, Le., (j , I) E r.n+l v i ..e » (ii) the coarser cell VJ-l ,11"j(l) belongs to t he ada pt ive grid, i.e., (j -1, 7["](1)) E 92}1 or, equivalent ly, (j - l ,7[j(I) ,e) ~V2il , e E E * , and (iii) the neighboring cell has been refined due t o significant det ails, i.e., (j, I, e) E V£i 1 for some e E E * . Here, t he gra ding of the grid implies t hat t he levels of two neighboring cells differ at most by one. In case (i) and (ii) the num erical flux is computed by (4.14) and in case (iii) by ,n F jk,1 -

'"

L.J

(k',II)Est ',{+l

Fjkl+l ,I',n =

'L.J "

(k

l,l'

)ESU+

F( Vj+l n ,T ;

r

E ,rkl 'T"j +l) ,I"

(4.15)

1

P ro ceeding in such a way, we event ua lly have to compute addit iona l averages V.f+l ,T usin g (3.21) - (3.23). In genera l, the num eric al fluxes do not

V'j,T or

4.1 Construct ion

77

only depend on two values corresponding to the two adjacent cells but on a larger st encil, e.g., high- ord er accurate FVS employing a higher order reconstruct ion . Therefore t he neighborh ood of cells to be made locally uniform is larger. To this end , we have introdu ced t he notion of a locally uniform grid of degree p , see Definition 8. Here, the degree p can be ensur ed if t he grid is graded of degree q where q is chosen sufficient ly large, see Theorem 4. This again imposes a constraint on t he grading paramet er for t he set of significant details, see Proposition 1. Another alte rnative of making t he computation more efficient is t he use of an unstructure d approac h. Here t he num erical fluxes are computed by mean s of t he averages pr ovided by t he ada ptive grid, i.e., (4.16) In t his case no local refinements are necessary, see Fig. 4.1 (right) . Although the two alte rnatives are less expensive th an th e exa ct comput ation according to (4.12) we have to be aware th at an erro r is introduced . As investi gations in [CKMP01] show, this erro r might become significant in case of a first order FVS. However , if t he reference FVS is a higher order scheme employing a high- ord er reconst ruction, t hen no significant loss in accuracy has been observed.

~-

I

' 0 -:-,0

- - I - ....' 0

0 '

• • 1• •

0 :

r- -:-0 .0 .:,.. 0 '

,, ,, 0

---- .,, ---,,

0

0

0 0

0

0

Fig. 4.1. Exact (left ), locally st ructured (middle) and un structured (right) flux eva luation , (.) numerical fluxes, (0) cell averages

4.1.2 Strategies for Prediction of D e t a il s

The core ingredient of the ada pt ive FVS is t he pr edict ion of th e set VZ~l by means of t he dat a {Vj, d (j ,k) E9 r: .€ and {dj,k,J(j,k,e)E'Dr:,€' respectively. Afirst ap proac h was int roduced by Harten in [Har9 4, Har9 5]. It is based on heuri sti c arg ume nt s employing chara ct eristic features of hyp erb olic conservation laws, see also Sect . 1:

78

4 Adaptive Finite Volume Scheme

- The speed of propagation is finite. This implies that information moves according to the locally finite characteristic speeds, i.e., the eigenvalues of the Jacobian corresponding to the normal fluxes (1.3). The time-space continuum limited by the characteristics propagating with maximal and minimal characteristic speeds form the range of influence. For systems of conservation laws this is a cone. - Discontinuities may develop although the initial data are smooth. In case of nonlinear conservation laws this is caused by the intersection of characteristics.

V2;i

by means of From these observations Harten deduced a prediction set two criteria applied to each significant detail (j, k, e) E VZ e of the old time level, namely, ' (i) details in a local neighborhood of a significant detail may also become significant within one time step,i.e.,

Ildj,k,elloo ,* 2: Cj

=>

(j,r,e) E

V2;-i , r E Nj~~ ,

e E E*,

(4.17)

and (ii) gradients may become steeper causing significant details on higher levels, i.e.,

Ildj\,ell oo ,* 2: 2M +1 Cj

=>

(j

+ 1, r, e) E V2-;g1,

r E M1,k' e E E* . (4.18)

In principle, it suffices to choose q' = 1, since the CFL restriction for explicit FVS ensures that the time step is sufficiently small such that a perturbation is propagating only from one cell to another. To guarantee a reliable adaptive scheme the prediction set V2-;g1 has to satisfy the reliability condition

tr:z .e U trr: L,e

C

n 1 Vt..e + :

(4.19)

So far, Harten's approach has not yet been verified to fulfill (4.19) . In [CKMP01] a slightly different strategy is proposed. The basic idea is to determine all details dj';k~ .e' on the new time level which are influenced by a detail dj,k ,e on the old time level. In order to represent this influence set Vj,k, e we have to determine all averages vZ ,r' r E Ej,k,e C It. , that are influenced by the detail dj,k ,e' i.e., the range of influence of dj,k,e ' Accordingly, we determine all averages V2~1, r E tj' ,k' .e' Chon which the detail dj';k~ .e' depends, i.e., the domain of dependence of dj;;k~ .e' In order to merge the index sets Ej,k,e and t j,,k'.e' which correspond to different time levels, we either have to determine the backward influence domain ir: k' , or the J , ,e forward influence domain by means of the evolution equation on level L. In the sequel we describe how to compute the sets Ej ,k,e and tj' ,k' .e'> Since we also need the range of influence and the domain of dependence for

»i;

4.1 Con struction

79

j+1 4(k-s)

4k I

I

4 (k+s)+1 e

2(k-s)

I

e

2k

I

•

I I 2(k+s)+1

J

j-1

e

k

Fig. 4 .2. Illu strati on of the range of influence Ej~k , e

the averages V'J,k we introduce the convention d'J,k ,O := V'J, k'

9;::,

Range of influence. The set E j,k ,e is det ermined by the supports e E E , according to the local invers e multiscale transformation (3.21) (3.23). From these two-scale relations it can be recursively computed U) '= {k} E J,k ,e . , E(lk+l) J , ,e

"

:=

U

(I )

r E E.

J , k, e

._ ,,(L)

L.Jj ,k ,e . -

LJ j,k ,e

91*,e , l = i ,... , L - 1, , 1'

'

9;:: 9;::

Since the supports of the modifi ed box wavelets fulfill = = M1,k' e E E*, and t he grids are nest ed , we may int erpret t he corre sponding range of influence as E j ,k ,e = {r E h ; VL,r C Vi,d, e E E* , i.e., t he partition of the cell Vi,k by cells on the finest level. This do es not hold for e = O. In t his case the support corresponding to t he ind ex set E(lk ) 0' J, , i.e. , U r EE (l) Vi,r:J Vi,k , is successively growing with increasing levell. This J .k ,O

is illustrat ed in Fig. 4.2 for the one-dimensional case according to Sect . 2.5.2. Here we ass ume that the st encil of the modifi ed box wavelets is symmetric, = {2(k- s) , . . . , 2(k+ s)+1} , i.e., J1 = 0 in (2.39). In par ticular , we obtain

9;:2

cr. Sect. 3.8. The stencils EJ~k ,o and EJ~L corr espond t o the cells indicat ed by both. or ° and 0, resp ectively. Domain of dependence. The ind ex set E j ,k ,e , e E E , can be compute d by mean s of the local multiscale t ransformat ion (3.20) and recur sively applying t he two-scale relation (3.16) f', U ) L.J

._ M e

i ,k,e '-

i ,k'

80

4 Ad ap t ive Finit e Volume Scheme 11

1 1 10101 010 18 18 18 18 10 1010\010 101010 101010 1010 10 1010 1 1 1 1 1

8(k-s) 1

0

1

I 8

0

1 8

1

4(k - s)

0

1

0

I

I

0

1

0

0

I

0

I

4k 1

1

2 (k-s)

0

1

1

0

4(k+s)+1 8

8

2k

2k+1

1

1

2 (k+s)+l

F ig . 4 .3. Illustration of t he dom ain of dependence

U

E-(1+1) j k e: = ,

-

- (1)

rEEj, k ,e

1

._

j +3

8 (k+s )+1

8k

-°

M 1+1 r' l '

= j , ... , L -

£?L 2,

(L - 1)

E j,k,e .- E j,k,e

Not e, that this set corresponds t o the support of the modified box wavelet 'l/Jj,k ,e, e E E *, according to Algorithm 1, and t he box funct ion 'Pj,k == 'l/Jj, k,O, respecti vely, i.e., supp 'l/JJo k e = U r EE- J ,k , e VL ' r - Similar t o t he set E j k e i e E E *, we may int erp ret t he dom ain of dependence as °

J

I

I

U {r E t c ; VL,r C VJ+1 ,s},

E j,k,e =

I

e E E,

s E Mj, k

i.e., t he par tition of t he cells VJ+1 ,s, s E M j,k by cells on the finest level. In analogy to the ran ge of influence, we consider again t he one-dimensional case where we employ the modified box wavelet with symmet ric support , d. Sect . 2.5 .2. According to Sect. 3.8 we det ermine M ~,k = {2k , 2k + I} and

M ],k = {2(k - s), 2(k + s) + I} . In Fig. 4.3 t he index sets by

0

(e = 0) and . or

0

EYL are ind icated

(e = 1), respectively.

B a ckwa r d (fo r ward) influence domain . By mean s of the explicit evolut ion equation (4.1) we know how t he averages of two t ime levels are correlat ed . For t he sa ke of simplicity, we assume that the discreti zation st encil which is involved in the computation of the numerical flux balan ce 82 k and the corresponding numerical fluxes F~'t , see (4.2)and (4.3) , is determi~ed by t he ind ex set Nf k indi cating t he neighborhood of degree p corre sponding t o t he cell VL,k on t he finest level, i.e., U, :Ff,1 C Nf ,k' In order t o merge t he ind ex sets E j,k,e and E j , ,k' .e' we have two alte rnat ives based on looking back or forward in t ime. One possibility is t o collect all indi ces corres ponding to averages v2,r ' r E E;; ,k'.e' on t he old time level which are needed when computing v2:~? , r E Ej' ,k ' .e' according t o t he evolution op erator , i.e.,

E-:;J, k ' .e , :=

U

Nfr' ,

4.1 Construction

I~I I

0

I

0

I

0

I

0

I

I

I

0

I

0

I

0

101

I~I

I

0

0

81

Fig. 4.4. Backward (left) and forward (right) influence dom ain

Conversely, we can det ermine t he num erical range of influence corresponding t o t he averages vl,r' r E E j ,k,e , i.e. , E tk ,e :=

{r

E

ti. ; Nf,r n E j ,k,e =l0} .

The different points of view are illustrat ed in Fig. 4.4 . Influence set . The influence set V j ,k,e of det ails dj';k1, .e' which depend on t he det ail d j ,k,e and t he coarse-scale averages v O,k = dO,k,O can be represent ed as V j ,k,e =

{(j', k' , e' ) E V L ,O

; E;; ,k' ,e'

-

= {(j' , k' , e' ) E V L ,O ; E j' ,k' ,e'

n E j ,k,e =l0} n Ej+,k,e =l0}.

To realize an efficient impl ementation we have to det ermine a priori t hose indi ces (j' , k' , e' ), which yield a cont ribut ion to V j ,k,e' We not e t hat the det ail dj,k ,e may not only cause a perturbati on in t he neighborhood of t he cell Vj,k similar to (4.17) but may also influence det ails dn'+k~ e' on higher scales. In cont rast to Harten 's st rategy, here j ' > j + 1 is J, , also admissible. Since th e addit iona l higher levels inflat e the influence set, we would like to bound th e number of higher levels t o a minimum number which st ill pr ovides t he reliability property (4.19). For t his purpose we fix some a > 1 which will depend on th e smoot hness of t he primal wavelet s, see Assump tion 4 in Section 5.3. We now assign t o each significant det ail or coarse-scale average corres ponding to (j , k , e) E V'l,e and (0, k, 0) == (0, k ), respecti vely, a unique ind ex v = v(j , k, e) such th at 2 v (j ,k,e) U

C' J

< lid'j:'.k ;« II 00 , * < 2(v(j ,k,e)+1) c '. J U

(4.20)

Since th e ind ex v(j , k, e) becomes th e sma ller th e lar ger a is, it is convenient to choose a as large as possible. Th en th e modified pr ediction set is det ermined by jjn+ l .- V n U L ,e ' L,e

u

(j ,k,e )E'f5~.e

{(j' , k' , e' ) E Vj ,k,e ; j' ::; j

+ v(j, k , e)}

(4.21)

82

4 Ad aptive Finite Volume Schem e

with D~ ,e := D'L,e U {(O , k,O) ; k E I o}. We would like to remark th at t he definition of DZ+g1 is slightly different to th at in [CKMP01], since we apply no t hreshold on th~ coarse-scale averages. This would destroy th e conservat ion property of the ada pt ive scheme, see Lemm a 5. For a scalar one- dimensional conservat ion law it is possible t o est ima te the details dj';k~ ,e' on t he new t ime level by t he det ails d j,k,e on th e old t ime level, see Sect . 5.3, from which we conclude t he reliability property (4.19). Fin ally, we remark t hat we only apply Har t en's st rategy for our computations present ed in Sect . 7, becau se it is mor e efficient and easier to impl ement . This decision is motivated by t he investigations in [CKMP01] .

4.2 Algorithms: Initial data, Prediction, Fluxes and Evolution In the sequel, we pr esent th e algorit hmic ingredient s of t he adaptive FVS . Four basic routines are needed , nam ely, (i) t he multi scale decomposition of t he initi al dat a , (ii) t he pr ediction sets D2i 1 , (iii) t he computat ion of t he local num erical fluxes and (iv) t he evolution of t he averages corresponding t o t he adaptive grid 92i1 . First of all, we consider the comp utation of th e set D~ e- In principle, we have to perform t he fu ll multiscale decompositi on accord ing t o (2.40). However , this would requir e the computation of all averages on the finest grid. In ord er t o redu ce t he comput ational complexity thi s has to be avoided. Therefore we present an algorit hm t hat pro ceeds levelwise from coarse t o fine scale according t o Algorithm 2. The basic idea is as follows: first we compu t e all averages on levell, i.e., the ada ptive grid is assumed to be th e grid of levell , and det ermine t he det ails by t he local two- scale tran sform ation (3.17) - (3.20) . If there is at least one significant det ail corre sponding to a cell V O,k t hen t his cell has actually to be refined otherwise t he ada pt ive grid is locally coarsened. Aft er having handled all exist ing cells of t his level we th en proceed to t he next high er level where we refine again t he cells of t he locally highest level and det erm ine t he det ails by the local t wo-scale tran sformation. These det ails det ermine which cells act ua lly remain . This pro cess is successively repeate d until th e a priorly fixed highest refinement level is reached . As input dat a th e algorit hm needs (i) t he number of refinement levels L, (ii) th e sequence of t hres hold values e , (iii) t he scaling par am eter t ol and (iv) t he initial function Uo. Then t he algorit hm provides t he set D~ e of significant det ails corres ponding to t he initi al dat a Uo. Note, th at t his s~t is not necessaril y gra ded. Moreover , it is not gua ra ntee d from an analytical point of view t hat this set actually contains all significant det ails . However , in many applications the initi al dat a are given by piecewise constant dat a . In t his case t he set is supposed t o be reliable.

4.2 Algorithms: Initial dat a, Predicti on , Fluxes and Evolution

83

Algorithm 7. (M ultiscale decomposition of ini ti al data) I o+ ..- I.0 for j = 0 t o £-1 do 1.

o, :=

It U U

eE E *

U kE1J+ Lj,k

2. R H 1 := U kEU; M~, k 3. Uj+l ,k := 1V;~ l . k l JV;+l,k ua(x ) dx , k E R H 1

4. Uj,k = A

j,a LJrEMo] ,k m r ' k Uj+1 ,r>

'"

-

A

lj,e 5 . d j,k,e -- '" LJ rEL ~ k r,k Uj,r + A

i.

LJ r E M ~

'"

r,

j,e Uj+1,r , k m r,k -

A

e E E * , k E 1+ J

+1,i := maxkER;+l I(Uj+l,k)il , i = 1, .. . , m 6. U~ 7. for e E E * , kElt do 7.1 for i = 1 to m do if u H 1 < tol -

OO,'l

t hen (dj,k,e)'; := (dj,k,e)i else (dj ,k,e)i := (dj, k, e)du!;};,~ 7.2 Ildj ,k,ell oo, . := Ildj,k,ell oo

7.3 if Ildj ,k,ell oo,*

> Cj t hen (k , e) E he else delete dj ,k,e

8. It+ 1 := U (k,e)EJ; ,e M j,k

F irstl y, we have to initi alize the set It of cells on level 0 for which we have to compute the det ails. In order to det ermine the significant det ails we pr oceed levelwise starting on t he coarsest level. For each level we have to perform the following st eps. First , we det ermine all indi ces of cells on t he locally highest level j which have t o be refined , see ste p 1, in view of the local t wo-scale tran sform ation. To t his end, t he set It has to be inflated , since additi onal averages Uj,k have t o be compute d in order t o perform (3.17) in step 5. Then t he indi ces of the new cells on the next finer level j + 1 and t he corres ponding averages are computed, see st ep 2 and 3. Before performing t he local two- scale decomposition (3.17) - (3.20) in st ep 5, we first have to provide the averages Uj,k not yet det ermined from the data on the finer level, see ste p 4. Similar to Algorithm 4 we perform t he thresholding and det ermine see ste p 6 and 7. By mean s of the significant det ails we deduce the cells of level j + 1 for which we have to det ermine the details in t he next it eration , see step 8. Finally we remark that all averages UO,k' k E l a, have been det ermined when performing t he loop for j = 0, since It = 10 and, hence, Uo = 10 , These data also have t o be provided by t he multiscale decomposition of t he initi al dat a.

»t:

84

4 Adaptive Finite Volume Scheme

In ord er to predict the ada pt ive grid 92i1 on th e next time level we have t o det ermine the set V'Li1 . According to Sect . 4.1.2 two alte rnatives are at hand. Here, we only outline Harten 's st rat egy which has been applied for our computat ions in Sect . 7. As input dat a we need (i) t he number of refinement levels L, (ii) the sets V2,e and :Jj,e ' j = 0, ... , L -1, respectively, of significant det ails and (iii) t he sequences of significant det ails dj,k ,e' (j, k , e) E V2,e corre sponding to t he old time level t n' Then the algorit hm provides V'L i 1 and jj~t l , j = 0, .. . , L - 1, according t o (4.17) and (4.21), respectively.

°

Algorithm 8. (Prediction due to Harten) qn+l .0 J' -- , . . . , L - 1 Vj ,e .- , for j = t o L-1 do

°

1. for (j, k, e) E .J;~e do

1.1 jj~tl

:=

.f;~tl

U {(j , r, e) ; r E

jI~~~e := jj~~~e U {(j 2. dj,k,e := 0,

3. dj+l ,k,e := 0,

+ 1,r, e)

Nj~~' e E E *}

; r E M 1,k' e E E*}

(j, k , e) E .f;~t l \:Jj~e (j

+ 1, k , e) E jj~~~e \:J1+1 ,e

Not e, t hat we ensure :Jj~e C jj~t l in step 1. Next we describ e th e algorit hms for the computation of t he num erical fluxes. Here we have to distinguish between the intern al and t he boundary fluxes, see (4.3) and (4.4) . In both cases we need t he following input dat a: (i) t he number of refinement levels L , (ii) the set Q 2i1 and i j,t 1 , j = 0, . . . , L , respectively, (iii) th e corre sponding local averages v j,k' (j, k) E Q 2i1 , corresponding to t he old time level t« and (iv) for the boundar y fluxes we have t o provide in ad dition th e prescrib ed boundary data. From t his information the numeri cal fluxes corr esponding t o th e flux balan ces Bj,k' (j, k) E Q 2i1 are det ermined . First of all, we consider th e computat ion of th e boundary fluxes. To this end we introduce the ind ex set IJ, j = 0, ... ,L, of t he ghost cells in ord er t o C 8 a, l E IJ. handle t he edges

it,

Algorithm 9. (Bounda ry fluxes) 1 for (j , k) E Q2i , 8 Vj ,k n 8 n # 0 do for 1 E

IJ, rl,l c 8 n do

j ,L do 1. -for (k' , l') E S k,l -

4.2 Algorithms: Initial dat a , Prediction , Fluxes and Evolution

85

1.1 det ermin e th e state v k ' ,I' from th e bounda ry values or th e attached flo w field

,n 2 . Fjk,1

'"

:= L.J (k' ,1' ) E St·~

F L ,n

k' ,I'

For the boundar y fluxes appropriate st ates V k' ,I' have to be det ermined from t he boundary condit ions or the at tached flow field . In t he latter case, we event ua lly need t he average s on the finest level. Then we have to apply the local inverse two- scale tran sformation (3.21) - (3.23) recur sively. According t o (4.4) t he boundar y fluxes corres pond to the fin est level in ord er to avoid inconsist ency. This is a serious dem and , in par ticular , if the boundary {J fl is cur ved. According t o Sect . 4.1.1 there are th ree alte rnatives for the computat ion of t he int ernal fluxes. Algorithm 10. (Exa ct flu x evalu ati on) for j = L downto 0 do for k E i j,t 1 do for l E t . , r il =f. 0, r i l n {J fl = 0 do 1. -for (k f , If) ~ S kj ,L do ' ,l1.1 reconstruct th e states V'l,T' r E F f,,1 " by me ans of recursively apply ing th e local inverse two -scale transformation (3.21) - (3.23), if not yet available

1.2

Ff,. ), = F(V'l,T ;

r E Ff, ,1' )

,n '" F L ,n 2 . Fjk,1 := L.J (k' , I ' ) E st·. ~ k' ,I'

Here, we pro ceed from fine to coarse. For each cell Vi,k of the ada ptive grid we det ermine the num erical fluxes at t he int erfaces to th e neighboring

it,

cells

Vi,I . According t o (4.12) t he num erical flux Fk',7 is the sum of num erical

fluxes corres ponding t o t he partition r l.l = U (k' ,1' )Est.~ rf.; ,I', see (4.13) . The computat ion of t he int ernal fluxes Ff,. )" i.e., r f.;,1' · C fl , depend s on t he averages vt. T' r E F f, I' C lt. , on the finest level. If th ese dat a are not available, i.e., (L ,r) ~ i 211 , th ese values have to be reconstructed locally. For t he boundar y fluxes appropria te states Vk ' ,I' have to be det ermined from t he boundary conditions or the attached flow field. In order to avoid t he local reconst ru ction of valu es on the full fin est level, t he second st ra tegy is based on t he idea t o make the grid locally uniform. Then we can st ill use a num erical flux based on a structured grid , see (4.14) and (4.15) . In addit ion t o t he above input dat a we need t hat the ada pt ive

86

4 Adaptive Finite Volume Scheme

grid can be made locally uniform of degree p where p is determined by the minimal degree q of neighborhoods Nj~k such that UI Fk,1 C Nj~k'

Algorithm 11. (Locally structured flux evaluation) for j = L downto 0 do for k E ij,t 1 do for l E I j ,

rt. 1- 0, r{1 n <3 a = 0 do

if (j,l) , (j -l,7fj(l)) ~ Q2;tg1 then 1. -for (k' , l') E Sj,j+1 k , ldo 1.1 reconstruct the states V.f+1 ,r' r E Fk~l~ C Ij+I' by means of the local inverse two-scale transformation (3.16) - (3.23), if not yet available j + 1,n 1.2 Fk',I'

n = F(vj+1,r;

'+1

r E F~"I')

"" Fj+1 ,n 2. Fjk,1,n := 6(k' ,l')EsL',{+l k' ,I' else 1. reconstruct the states vj,r' r E C I j , by means of the local inverse two-scale transformation (3.21) - (3.23) or local two-scale transformation (3.16), if not yet available

FL

FL)

2. F{',7 = F(vj,r ; r E In comparison to Algorithm 10 we proceed differently in case of the internal fluxes. Here we locally refine the neighboring cells at most once . This reduces the reconstruction costs which become significant for higher dimensional problems. The third strategy is based on an unstructured flux evaluation, see (4.16) , where we only access data of the adaptive grid. In principle, this grid need not be graded in view of the flux evaluation. Algorithm 12. (Unstructured flux evaluation) for j = L downto 0 do , for k E i n +1 do lor l Ej~ , lkl 1- 0, rid n <3 n = 0 do 1. determine the stencil Fk ,1 C Q2;tg1

2. F{',7 = F(vj"k' ; (j' ,k') E Fk,l) Now the algorithm for the evolution step according to (4.11) can be described. Here we need (i) the number of refinement levels L, (ii) the set QTigl and ij,t 1 , j = 0, . . . ,L, resp ectively, corresponding to the adaptive grid, (iii) the corresponding local averages vj,k' (j, k) E Q2;tg1 and (iv) the numerical fluxes corresponding to the flux balances Bj,k' (j, k) E Q2;tg1 . Then the local

4.2 Algorithms: Initial data, Prediction, Fluxes and Evolution

averages follows.

vj,t 1 , (j, k) E 9£11 on the new time level

tn+l

87

are determined as

Algorithm 13. (Evolution) for j = L downto 0 do n 1 + do -for k E i L,c-

1. Bj,k :=

I:llrl,ll F{',7

2 . vj,k n+l --

vj ,k -

n

>. j,k

Bn j ,k

Finally, we summarize the algorithm for the adaptive FVS. The input data are (i) the number of refinement levels L, (ii) the sequence of threshold values e, (iii) the scaling parameter tol, (iv) the grading parameter q, (v) the temporal step size T and (vi) the initial function uo . Then the algorithm provides the local averages v:'[, (j, k) E 9f,~, corresponding to the time

T =

TNT .

Algorithm 14. (Adaptive finite volume scheme) for n = 0 to NT - 1 do 1. perform the local multiscale transformation according to Algorithm 3 (n > 0) or determine the multiscale decomposition of the initial function Uo by Algorithm 7 (n = 0); 2. threshold the sequence of details by Algorithm significant details V2,c; 3. predict the set of significant details rithm 8 respectively;

4. grade the set of significant details

V£i1

4,

i. e., determine the set of

on the next time level by Algo-

,

V211

by Algorithm 5;

5. perform the local inverse multiscale transformation by Algorithm 6; 6. determine the adaptive grid

9£11

by Algorithm 2;

7. compute the internal numerical fluxes by Algorithms 10, 11 and 12, respectively, and the boundary fluxes by Algorithm 9; 8. perform the evolution step by Algorithm

4.

Note, that the local (inverse) multiscale transformation is only feasible if the tree of details is sufficiently graded. Moreover, the flux evaluation according to Algorithm 11 requires that the adaptive grid can be made locally uniform. To this end, an additional constraint on the grading parameter q has to be taken into account. And finally, we remark that the CFL restriction for explicit FVS has to hold for the finest grid. This is still a bottleneck in the performance of the adaptive scheme, see Sect. 7 for details.

5 Error Analysis

The objective of the proposed adaptive scheme is to reduce for a given FVS computational cost and memory requirements while preserving the accuracy of the reference scheme. In order to quantify t his we introduce the averages u2 of the exact solution, the averages v2 determined by the FVS and the averages v2 of the adaptive scheme . Note , that all sequences correspond to the finest discretization level indicated by L . In particular, the sequence VL is not generated by the adaptive scheme on the full uniform highest level L but these data are det ermined by the adaptive array {VJ,k}(j,k)EQ'L ,E and could be retrieved from them by means of the local inverse multiscale transformation (3.21) - (3.23) where the non-significant details dj ,k,e in (3.23) are put to zero . An ideal strategy would be to prescribe an error tolerance tol. Then the number of refinement levels L should be determined during the computation such that the error meets the tolerance, i.e.,

IIu2 - v211 ::; tol II . II denotes an appropriate norm to

for possibly small L. Here be specified below . For this purpose we need an error estimator which is not available for the adaptive scheme at hand. In the sequel, we therefore proceed differently. To this end we split the error into two parts corresponding to the discretization error TE := u2 - v2 of the reference FVS and the perturbation error e L := v'L -

-t.

i.e.,

II u2 - v211 ::; IITEII + IIe211 ::; tol.

(5.1)

We now assume that there is an a priori error estimate of the discretization error, i.e., TE '" h L where lit. := maxkEh diam(IVL,kI) denotes the spatial step size and a the order of convergence. Then we ideally would determine the number of refinement levels L such that h L '" tol. In order to preserve the accuracy of the reference FVS we now may admit a perturbation error which is proportional to the discretization error, i.e., lIe211 '" IITEII. From this we conclude L = L(tol, a) and E = c(L). (5.2) For scalar conservation laws error estimates have been derived, see [CCL94, Noe96]. Therefore it remains to verify that the perturbation error can be

90

5 Error Analysis

cont rolled. Note, that in each time step we introduce an error du e to the threshold procedure. Obviously, this err or accumulat es in each st ep , i.e., the best we ca n hop e for is an est imat e of the form

IIe211 ::; Cri e. However, the threshold error may be amplified in addit ion by the evolut ion step. In order to control the cumulative perturbation error we have to prove that the const ant C is independent of L, n , T and e. For this purpose we will conside r the following issues in more det ail, namely, L t he uniform bounded ness of the perturbation error, ii. the stability of the threshold procedure and Ill . the reli ability of t he pr ediction procedure.

5.1 Perturbation Error In the sequel, we investigate t he difference between the averages v2 of t he reference FVS and the averages v2 of t he adapt ive scheme. The error is measured in t he weighted [l -metric

IluLI11,L

:=

L

IVL ,kll uL ,kl ,

k Eh

whi ch is equivalent to the Ll-norm of a piecewise constant function . This metric is usu ally applied in the err or analysis of FVS. According to Harten 's ana lysis in [Har95] we will split t he error into its different contributions corresponding to thresholding and pr ediction as well as the local flux approximation. To this end , we introduce several op erators t hat simplify the representation of the err or analysis: - The threshold op er ator Tv : RNL -+ RNL is determined by means of the multiscale decomposition d(L). It puts all non -significant details to zero , i.e., d j ,k ,e := 0, (j , k , e) (j. V , and ret ains all significant details . Here V C J (L ) denotes an arbit rary subs et of all details. - The nonlinear approximation op erator A v : R N L -+ R NL is defined by A v := Mr;lTvML ' First t he sequ ence of averages is decomposed into its multiscale decomposition. Then the thresholding is performed by means of the set V of significant details and, finally , the modifi ed averages corresponding t o t he sequence of thresholded det ails ar e determined where we apply the inverse multiscale transformation. - The evolution op erator EL : RNL -+ RNL of the reference FVS is det ermined by (4.1) . It evolves the averages from one t ime level t o the next one , i.e., v2+l = EL v2 .

5.1 Perturbation Error

91

- The evolut ion operator E9 : R N L -+ R NL of th e ada ptive FVS corresponds to t he ada pt ive grid 9 = 9('0) and the set of significant det ails '0 , respectively. This operator is det ermined by t he evolut ion equ ations (4.11). To t hese equations we apply t he local inverse multiscale t ransforma t ion (3.21)-(3.23) where we put dj ,k ,e to zero in (3.23) , i.e., th e local average s on higher scales are reconst ructed only from avera ges on coa rser scales corresponding t o the adaptive grid. Equivalently, th e evolut ion operator can be derived by pu t ting djt~:= 0, (j ,k , e) (j. '0, in (4.9) and t hen applying t he inverse multiscale transform ation M'L 1 to th e sequence of evolut ion equations for th e multiscale decomp osition. By mean s of t hese operators the ada pt ive FVS mapp ed onto t he highest resoluti on level can be represent ed by

Here we assume t hat th e set '0 2-;g1 is always gra ded. Otherwise we have t o ap ply an additional gra ding operator A as is done in [CKMP01] . Since t he gra ding does not affect the approxima t ion error, we incorporat e this st ep into the construct ion of t he prediction set s. Not e, t hat for c = 0 no threshold err or is introduced , i.e., e2 = O. Therefore t he ada pt ive scheme may be int erpret ed as a perturbation of the reference FVS. Now th e following t heorem provid es an estima te for t he perturbation error. Theorem 5. (Unifo rm boundedness of pertu rbation error) Let the follo wing assumptions hold:

(A 1) the approximation error is un iformly bounded, i. e.,

IluL - AV (A 2)

C

uLlll,L ::; C1 e;

the local flu x approximation is accumcy preserving, i.e.,

IIELvL (A 3)

L

EgL.< vz.lh .z

::; C2 e;

th e reference FVS is stable, i.e .,

IIELUL-

ELv LIIt,L ::; (1 + C 3T) IluL

- vz.lh,z. ,

(A4) th e prediction is reliable in th e sense of (4.19) ;

(A S) th e initi al data are consiste nt, i.e .,

Ilvi - 'V11h,L ::; C 4 c.

Th en the perturbation error is bounded by (5.3) fo r n T

::;

T where C is independent of L, n ,

T

and c.

92

5 Error Analysis

Note, t hat t he te mporal discreti zation 7 is assumed to be uniform. This is not a severe constraint bu t simplifies t he represent ation of th e ana lysis. P roof. In a first step we split t he perturbation error int o it s different cont ribu t ions cor res ponding to t he stability of t he reference FVS (A3) , t he approximation of t he reference numerical fluxes (A2) and t he t hreshold error (AI) , i.e. ,

wit h

Then we conclude from th e assumpt ions

By recursion we obtain further n- l IleL lll, L :S [ l e~ l h. d l

+ C 3 7 ) n + e (C l + C2 ) 2:)1 + C 3 7 ) i . i=O

We t hen conclude wit h C := max (C l Ile Llh, L :S e C

+ C2 , C4 )

(1 + C 7) n + l 3

C3 7

1 -

_ e C 3 (n + l) r - 1 C ,------:S e C -----" 37

in case of C 3 =j::. 0 and

IIeLnill ,L:Se C(n +7 l) 7 if t he FVS is ll - contractiv e, i.e., C 3 = O. Since the maxim al number of t ime steps is bounded by n :S T /7 for a bounded t ime int erval [0, T], T < 00, the asse rtion follows. 0 From this theorem we imm ediat ely conclude t hat t he accuracy of the reference FVS is pr eserved pr ovided that E is chosen sufficiently small. Corollary 7. (Choice of threshold parameter) Let diam (VL,d ,...., 2- L for all k E h . If th e dis creti zat ion error of th e referen ce F VS is bounded by IluL - v Llh,L :S C T

o L

,

a> 0,

where uL denotes th e average of th e exact soluti on , th en th e accuracy is preserved by th e adaptive sch em e provided that e ,...., 2 - (1+0 ) L.

5.2 Stability of Approximation

93

Proof. Since for an explicit FVS t he t ime ste p T is restrict ed by t he CFL condition, i.e., T is proportional to t he maximal diam et er of th e cells, here 2- L , t he assertion follows by Theorem 5. 0 The usefulness of t he above theorem crucially depends on t he verification of t he assumptions (AI) - (AS). First of all, we remark t hat for scalar conservat ion laws on D = Rd t here are severa l first order accurate schemes which are iI- cont ractiv e, so-called monot one schemes, see e.g. [CM80]. Fur th ermore, we noti ce t hat (A2) is satisfied if t he local flux evaluation is performed according to (4.13) . In t his case no error is introduced , i.e., C 2 = O. In case of (4.14), (4.15) and (4.16) , respectively, we introduce an erro r, bu t so far t his err or has not been verified to be bounded by € . Concerning t he consiste nt discreti zation of the initi al dat a a natural choice is given by t he approximat ion opera t or , i.e., '1 := A v o v1. In this case, (AS ) holds pr ovided the c.e approxima t ion error is uniformly bounded in the sense of (AI). We emphasize t hat the application of th e approximat ion operator in general requires O(Nd opera t ions. In pr acti ce, a cheap er strategy is pr eferabl e, see Section 4.2. In the following sections we now address to some exte nt t he st ability of the approximation operat or and t he reliabil ity of t he pr edictiv e set of significant details.

5.2 Stability of Approximation In order to verify t he stability of t he approximation erro r it is convenient to investigat e th e convergence of t he so-called su bdivisi on algorithms. For t his purpose, we rewrit e t he inverse local multis cale t ra nsformation (3.21) (3.23) in full matrix- vect or representation

Recursively applying t his two-scale relat ion yields L- l

UL = G{;,D UD

+L

L

G1,eclj,e,

(5.4)

j=D eEE '

where th e matrix G1,e E RNL X n, is defined by (5.5)

represents t he subdivisi on procedure. Next we introdu ce t he kth column of G1,e which can be regar ded as discrete basis vect ors ,T.L ._ ,T.L C ~ j,k ,e . - ~ j ,e j,k

(5.6)

94

5 Error Analysis

by means of the Dirac vector Cj,k := (Ok ,r)rEf;, see [Dah94]. Then we can rewrite (5.4) as

UL =

L

L-l

'lJ6',k,oUO ,k +

kEfo

L L L j=O

eE E *

'lJ1,k,e dj,k ,e.

(5.7)

»ei,

Here we only consider a scalar problem. In case of a system of conservation laws we can apply the scalar investigations componentwise. Consequently, the approximat ion error can be written in the form

L

UL - Av UL =

'lJ1,k ,e dj ,k,e.

(5.8)

(j,k ,e)f/.V

In order to control the threshold error we need the subdivision scheme to converge at least in the ll- met ric, i.e., the piecewise constant function 'l/J1,k,e defined by (5.9) converges to a function 'l/Jj,k,e in t», It can be proved that the convergence of the subdivision scheme corresponds to the existence of a biorthogonal wavelet system. For general surveys on subdivision algorithms we refer to [Dyn92 , CDM91] and to [Dau92, CobOO] for their relations to wavelets. So far, results are only available in case of uniform refinements on structured grids, see e.g. [CDF92, DKU99, CDKPOO] . In case of curvilinear grids or arbitrary grids no stability results are available yet . In the sequel, we employ some standard results from the theory of subdivision schemes which shall be summarized for the convenience of the reader. For this purpose we first introduce the notion of quasi-uniform grids. Definition 9. (Quasi-uniform Grid) L et 9 be a nested grid hiera rchy . Then the grid on level j is called quasiuniform if there are constants a < c ::; C such that cT j < diam(Vj,k) ::; CTj and cTjd::; IVj,kl ::; C2- jd for all positions k E I j and levels j = 0, . . . , L ,.

Then we obtain for the primal wavelets the following results.

Proposition 3. Assume that the piecewise constant functions 'l/J1,k,e defined by 'l/J1,k,e(X) := ('lJ1,k, e)r ,

E VL,r, r E

X

h

(5.9)

conv erge uniformly in L towards a function 'l/Jj,k,e E L OO(f!) in the sup -norm. Then th e limit functions (primal waveletsy satisfy the following prope rties: 1.) Any funct ion u E L OO(f!) can be un iquely expanded in a series of the primal wavelet basis, i. e.,

u= 1

The functions

I:

kEfo

1jJj,k,O

(u,
I: I: I:

j EN

= CPj ,k

and

¢j ,k ,O

kEf; eEE*

=
(u , ;}j,k,e)£2 'l/Jj,k,e ;

will also be referred to as wavelets.

5.2 Stability of Approximation

95

2.) the primal wavelets satisfy the duality relation 1, ,, ( .o/}

k,e , .7. If/]" , k' .e ,) L2 - 8(. 1, k,e ) , (J" , k' ,e ') ,'

3.) the components of the discrete basis vectors coincide with the averages of the function ~j, k ,e , i.e., 'li1.k,e = ((~j,k, e, (h ,r))rEh;

4.) the functions ~j,k , e are uniformly bounded in the sup-norm, i.e., there exists a constant C > 0 independent of i , k and e such that II ~j,k ,eIIL =

< C;

5.) if the grid is quasi-uniform and the mask matrices Gj,e are uniformly banded, then the functions ~j, k ,e are compactly supported and, in particular, [supp ~j , k,e I :::; C 2- jd .

A proof of the existence of the wavelet expansion can be found in [Dau92, CohOO], for instance. From the existence we conclude the biorthogonality property where we consider 1.) for u = ~j , k ,e ' Then the duality relation and applying (5.7) to u = ~j, k,e implies that the coefficients of th e discrete basis vectors coincide with the averages of the primal wavelets . Furthermore the uniform boundedness of the primal wavelets in the sup-norm is a consequence of the uniform convergence of the subdivision scheme. Finally, in [CDM91] it is proved that the primal wavelets are compactly supported. In case of the box wavelets, see Section 2.3, the subdivision scheme has a simple representation. Therefore it is presented here for the convenience of the reader. Example 4. (Box Wavelet) We first observe that -T G j,e Cj,k =

«i.e Cj+l,r _'"' (j ,k/ ao,r j,k) Cj+l ,Tl 6 gk,r 6 ae,r rEM1,k rEM1,k '"'

(5.10)

where we make use of (2.31). According to the definition of the subdivision scheme (5.5) we then deduce 'li1.k,o

=

L

GY+l,O Cj H ,r

rEM1,k

=

L

CL,r, rEEj,k.O

(5.11)

where Ej,k,O denotes the support of 'li1.k,o, see also Section 4.1.2. By means of (5.5), (5.6) and (5.10) we further conclude

(G- Tj,e CJ,,k) -_ j,k,e -_ GL j+l ,O

,T.L '¥

'"'

6

(

rEM1.k

j,k/ ao,r j,k) ae,r

,T.L '¥

j+l,r,O'

(5,12)

96

5 Error Analysis

Then the components of th e discret e basis functions are det ermined by '" j,kl j,k lEE 'lt L ) = w rEM ~, k ae,r aO ,r , j,k,O ( ],k,e 1 { o , elsewhe re '

where we use (5.11) in (5.12) and use th e definition of th e Dirac vectors. Thus we obt ain by th e Holder inequ ality

11'lt1,k,ell;",:::;

L

L

(a~:~) 2

rE M~,k

L

(a~',~)- 2 = Ila~,kll ~

r E M~, k

(IYi ,kl/IYi+l ,rl) .

r E M~, k

Since the vect ors a~,k , e E E , are assumed t o be orthonormal , th e right hand side can be est ima te d by a constant independ ent of j, k and e provided that t he grids are quasi-uniform. Consequentl y, th e discret e wavelets 'lt1,k,e are uniformly bounded in the sup-norm. We now introduce th e piecewise constant functions 7/J1,k,e according to (5.9) which can here be repr esent ed by 7/JJ,k,e L =

'~ " (aj,klaj,k) XV. e,r O,r r EM ~,k

J +1, 1'

.

We notice that t he functions 7/J1,k ,e' e E E , are ind epend ent of L , i.e. , the subdivision scheme converges strongly t owards 7/Jj,k,e := 7/J1,k,e' Finally, we observe t hat the limit functions are uniformly bounded with resp ect t o the sup- norm, becau se the relation 117/J1,k,eIIL'" = 11'lt1,k,eI1 1'" holds . From t he above observations we now conclude th e following st abili ty result . Theorem 6 . (Stability of approximation operator) Let Cj = 2 (j- L )d C and C > O. If th e computational doma in is bound ed, i. e. Inl < 00, and th e grids are quasi-unifo rm , then the approximation operator is stable in th e sense of

provided that th e subdivision scheme converges strongly in L '" : P roof. According to (5.8) t he approxima tion err or can be est imated by

IIUL -

A D uLIh,L :::;

L (j,k,e)(J.D

11'lt1,k,eI11,L Idj,k,el:::;

L

11'lt1,k,eI11,LCj '

(j,k,e)(J.D

Here we employ Definition 3.1 of t he set V and not the scaled significant det ails according t o (3.38) , becau se a maximum-minimum prin ciple holds for scalar conservat ion laws, see e.g. [CM80].

503 Reliability of Predicti on

97

The norm of t he discret e basis vectors can be repr esent ed as

11'lJ'1,k,elh,L =

L

IVL ,rl l('lJ'1,k,e)rl

rEEf.k o •

where E f.k,e chis t he support of 'lJ'1,k,e' Next we conclude from Propositi on 3 and

I

U r EE L J,k , e

VL,rl = Isupp 1/Jj ,k ,eI :::; C Tjd

Combining t hese two est ima tes we finally obt ain

IIUL -

A'D

uL111 ,L :::;

C

L

Tjd Cj o

(j ,k ,e)fl'D

Since th e grid is assumed t o be quasi-uniform, we conclude that N j and, hence, the number of non- significant det ails is bounded by

L (j ,k ,e)fl'D

1:::;

L (j ,k ,e)fl'DL. o

""

2j d

L-1 1=

L L L 1 = N L - N o :::; C 2

Ld

.

j=O eE E * kE lj

This proves t he assertion . 0 We remark th at the above threshold values coincide with Har t en 's choice in [Har95]. A similar result ha s been obtained in [Got 98] where t he uniform stability of t he multi scale t ransformation relat ed t o the L2-normalized counterparts of the modified box wavelets is employed. This is dir ectl y related to the existe nce of biorth ogonal wavelets. In t his setting we can apply well-known stability results corres ponding to Cartesian grids. Finally, we point out t hat in t he univari at e dyadic case t he L 2-n orm alized counterparts of t he modified box wavelets according t o Algorithm 1 coincide with t he biorthogonal wavelets corresponding to th e du al pair of B-splines of ord er 1 and du al generators of order 1 in [CDF92] ([2 = R) and [DKU99] ([2 = [a, b]) . In this case the stability of t he mul tiscale transformation is guaranteed . By means of tensor products these can be exte nded t o th e multivariat e dyadic case. However, the resulting wavelets do not coincide with those constructed in Algorithm 1.

5.3 Reliability of Prediction The cruc ial assumption to be satisfied in Theorem 5 concerns the reliability of t he prediction set vt j}. For thi s purpose we have to verify t hat th e est imate (5.13)

98

5 Error Analysis

holds for (j' , k' , e' ) t/. i52i1 where C is a constant independ ent of i', k' and e' . Not e, that it suffices to bound th e detail by a const ant times the threshold value becaus e oth erwise we replace e by elC . As we infer from the evolution equa t ion for th e det ails (4.9) the main t ask is to est ima t e the details of th e flux balances. In collaboration with A. Cohen et al. we have been able to prov e that th e estimate (5.13) hold s provided th at th e pr ediction set is det ermined by th e modified approach (4.21) . The main problem arises from the nonlinearity of the flux balances. The objective is to est ima te the flux bal an ces by the det ails bu t th ey do not depend dire ctly on th e det ail coefficients. To t his end we int erpret the det ails as finit e differences. Then the finit e differences of t he flux balan ces can be estimated by finit e differen ces of t he averages where we make use of th e chain rul e. By means of an inverse est ima te the finit e differences of the averages can be est ima te d by details . For this purpose, we have t o confine the discussion to a scalar nonlinear conservat ion law in one space dim ension and a uniform discretiz ation , see [CKMP01]. The proof consists of three basic st eps: - For any smoot h function u the det ails of the corre sponding wavelet exp ansion can be est imat ed by

This est ima te can be shown provided that u E CSU\k) , th e size of th e sup port E j,k of the modified box wavelet is of ord er O(2- j) and the wavelet s are L1-normalized , see [CKMPOl] . A discret e counte rpart is proved in Proposition 4 where we est ima t e th e det ails by finit e differences. - The derivatives of any compo sit e function go u of smooth functions g and u can be repr esent ed by

OR (g(u(x)) = oxR

~ g(i)(u(x))

6

i= l

jE{ l ... . ,R -i+ l} i kE{O ,i } i k lit + ki j i =R

successively applying the chain rule. This motivates the result of Proposit ion 5 where we derive a discret e counte rpart by which the finit e differences of the flux balances are est imated by th e finit e differences of the averages. - Motivated by t he standa rd invers e est ima te

with 11" f := L~o ("{) (_l) i f( ·+ih) we derive a discrete inverse est ima te, see P roposition 6, from which we conclude th at th e finite differences can be estimated by the det ails.

5.3 Reliability of Prediction

99

In the sequel, the proof is outlined for the present setting. To this end , we confine the discussion to univariate dyadic grids, i.e., Vj ,k

= [Tj k,

z! (k + 1)], k E t,

:= Z,

where, in particular, the computational domain is assumed to be the real line, i.e., n = R. Since the assumption of shift-invariance simplifies the multiscale setting significantly we rewrite the basic relations according to Section 3.8. See also Sect. B.1.1 for details on shift-invariant spaces. We start with the two-scale relation for the averages 2 L - i_1

Uj ,k

= T

1

(Uj+1 ,2k

+ Uj+l ,2k+d =

j L 2-

L

UL ,2L - ik+ i '

(5.14)

i=O

where we use (3.16) and M~,k = {2k , 2k+ 1} as well as IVj,k I = 2- j . Moreover, we observe that the construction of the modified box wavelets according to Algorithm 1 with £.j,k = {k - 2s, .. . , k + 2s} , see (3.45) , leads to coefficients l~ k which are independent of the level j and th e position k. Note , that no J, boundary adaptation is necessary because n = R. Then the two-scale relation (3.17) - (3.20) for the details reads 4s+1 dj ,k

=

L

m - Uj+1 ,2(k-s)+r,

(5.15)

r=O

where we have used that MJ,k = {2(k - s), ... , 2(k - s) + 1}. Since there is only one wavelet type in the univariate case, i.e., E* = {1}, we omit the index e for the details. In the following we heavily exploit the definition of a finite difference of order N which in the current setting reads N

.6.~ UL ,k := L( _l)i (~) UL,k+iK

(5.16)

,=0

with its stencil S(N, K, k) := {k + iK : i = 0, . . . N} C h depending on the order N > 0, the step size KEN and the local position k E h. First of all, we verify that the details can be written as a linear combination of finit e differences of order M.

Lemma 7. (Characterization of details by finite differences) Let k E Z and j E {O, .. . , L - 1}. Then there are coefficients ttu , 0, . . . , M := 4s + 1 - M , independent of k and j such that M

dj ,k

=

L mi .6.j"1 Uj+l,2(k-s) +1 1=0

provided that the modified box wavelet has M vanishing moments.

(5.17)

100

5 Error An alysis

P roof. First of all we remark t hat th e det ails vanish if the coefficient s Ul ,r represent t he averages of a polynomial of degree M - 1. In par ticular, this hold s for the polynomials Pi( X) := I:~=o ai,mXm, i = 0, .. . , M - 1, where t he coefficients a , = (ai,m)m=O ,...,i are det ermined by aT := e, Bi 1. Here ei := (bi,m) m=O ,...,i denotes t he Dirac vector and Bi = (b~ ,r) m , r=o , ...,i t he lower trian gular matrix defined by t he elements

bi .= m,r .

_ 1_

rn-l-I

(m+1) 2-(j +l )m r

{

°

°< , '

-

r

<m -

elsew he re

Not e, t hat th e inverse Bi 1 exists because all diagonal elements are positive. For t hese polynomials the averages are determined by

with l,

= (lm)m=O ,...,i. According t o (5.15) we obt ain for k = s 4s+1 dj,s(Pi) =

L

4s+1 m r Pi,j+1,r =

r=O

L

m..r" = 0,

r=O

i.e., th e vector of t he mask coefficients m = (mr )r=O ,...,4s+l is orthogonal to the vectors r, = (r i )r=O,...,4s+1, i = 0, . . . , M - 1. Since th e vectors r , ar e linearly ind ependent, we conclude that m is orthogonal t o t he linear space R M := span{ro , . . . , r M-d . Next we verify t hat t he vectors h i = (h 1,i)I=O ,...,4s+1- M,i=O, ...,4s+l defined by t he element s h1 ,i := {(_l) i-l

(/'!.I)

, I ::; i

° ,

::; I + M

el sewher e

form a basis for the compl ement space UM of R4 s + 2 , i.e., R4 s + 2 = R M EB UM. Obviously, these vectors are linearl y independ ent . Therefor e it remains to prove t hat th e vect ors hi , I = 0, . . . , 4s + 1 - M , are orthogonal to t he vectors r i, i = 0, ... , M - 1. For t his purpose we consider the inn er product 4s+1

L

p=O

h1 ,ppi

I+M

=L

p=l

(_l)p-l

()

~I

P

pi

=L i

; r:: AM ,r

( .)

r=O

with AM ,r := I)-l)P (M)p r . p=o p

By induction over M we pr ove that AM ,r vani shes for r = 0, ... , M - 1. In t his conte xt , we employ th e addition th eorem for binomials.

5.3 Reliabili ty of Predicti on

101

Since m is orthogona l to R M an d UM = span{h o, . .. , h 4 s + I - M } is ort hogonal to RM we conclude t hat m E UM, i.e., there exist coefficients tni , l = 0, .. . , 4s + I- lvI , such t hat 4s+I- M

L

m=

ml hl.

1= 0

Using t his relat ion in (5.15) proves th e assert ion. 0 From Lemm a 7 we now deduce t he result of t he first st ep . Proposition 4. (Estimate of details by finite differences) Let k E Z, j E {O, . .. , L - I } and I< := 2L - j - l . Furth erm ore, let M denote the nu m ber of vanishing m om ents of the m odified box wavelet. Th en the detail d j ,k can be estim ated by (5.18) where th e cons tant C is in dependent of k and j . Proof. First of all, we deduce from (5.16) and (5.14) 2L

AM

.w I

A

.

_

U j ,l -

-

i _ 1

L

2 j- L

Li~-i UL ,2 L - il +i'

i =O

By means of (5.15) and (5.17) we then conclude 2L - i -

M j

. -- '~ " -ml 2 d j,k

L I - +

1= 0

1_1

L

Li~ -i -l

U L,2 L - i -

1(2(k -s)+ I)+i

i= O

and, consequent ly, Id j ,k l ::;: C

sup { I Li~ UL ,K(2 (k -

s )+ I)+ i l :

i = 0, . .. , I< - 1, l = 0, . .. , M }.

Here t he const ant C is det ermined by t he sum C := L;~o ImL\ . According to t he support E j ,k of t he modified box wavelets, see Section 4.1.2, only th ose finite differences are involved in t he representation of t he detail d j,k whose st encil S (M , I<, r ) lies fully inside the support Ej ,k. 0 In th e second st ep we apply t he finit e differences to the flux balances. The object ive is t o est ima te t hese differences by t hose of the averages. Since t he flux ba lances are in general det ermined by a nonlin ear function t his can not be done in a straight forwar d manner . However , it is possible to est imat e t he differences of t he flux balan ces by lower order differences of t he averages . To t his end, we rewrit e t he num erical fluxes Ff,t and t he flux balan ces IEk' see (4.13) and (4.12), for t he univari at e case t hat is considered here. In ag~eement wit h Secti on 4.1.2 we assume t hat t he st encil of t he flux balan ces is det erm ined by t he set Nf.k ' Consequently, t he num erical fluxes read

5 Error An alysis

102

L,n _ F ( n F k,k-l V L ,k -

and thus

n

p "'" VL,k +p -l

,n B Ln ,k '.-- FLk,k+ l

-

)

L,n _ F ( n F k,k+ l V L, k -

,

FL ,n

_ B ( n

k,k - l -

VL ,k-p"

n

n

p+ l ' . .. ,VL ,k +p

" , VL ,k+p

)

)

(5.19)

for a fun ction B : R2 P+l -+ R. In the following, the flux balan ce fun cti on B is ass umed t o be regul ar . Assumption 3. Let D C R2P+l a bounded dom ain of admiss ible sta t es. Th en th e flu x balan ce fu n ct ion is assumed to be regular in th e follo wing sense: 1.) B is piecewise sm ooth, i.e., th ere are open subs ets D , CD , i . K R unili D = U;=o D ;, such that BEG (D;);

= 0, .. . , K,

2.) B is Lipschit z- continuous on D ; 3.) th e derivatives of B can be exten ded continuously to th e boundary aD; such th at

Ok B

sup

v EDi ako vo .. . a k2PV2p

for k

= l:;~o k i ,

(v) < Gk -

k E {O, ... , R } .

Next we conclude t hat the adapt ive FVS is uniformly bounded in the supnorm provided that t he reference FVS does not increase the sup- norm too mu ch . In particular , t he bound only depends on T and IluollLoo. Lemma 8. (Boundedn ess of adapti ve FVS in sup-norm) A ssum e that th e follo wing con diti ons hold : (A 6) the subdivis ion scheme conve rges un iformly in th e sup-norm; (A 7) th e local fl ux evaluati on is based on the exact strategy, see (4.12) and (4·13); (A 8)

th e refere nce F VS sat isfies

IIELvLIIloo :::;

(1 + GT)

IlvLllloo ;

(A 9) th e error of th e ini tial data approximation can be esti m ated by

Ilvl- v111loo :::; G el r and Il v1- ft111l oo :::; G ef-t where ft1 denotes th e averages of th e initial data ; (AID)

th e threshold values are det erm in ed by Cj = 2 j - L C with e '" 2-( 1+ o< )L fo r some a > 0;

(All)

th e CFL con diti on holds on th e fin est resoluti on level, i. e., T '" 2- L .

Th en the approximation vI correspon ding to th e adaptive FVS is uniformly bound ed in th e sup - no rm, i.e.,

5.3 Reliability of Prediction

11"211[00 ::; C(T, uo)

103

for nT::; T,

pro vided that in each time st ep th e predict ion se t V'L ,€:, 0 ::; IJ th e reliab ility property (4.1g) .

::;

T / T, satisfies

P roof. In ord er to est imate the ext rema by t he time evolution we consider

(5.20)

e 1V- n-l c - n-111 [00+ II "cL -n-111 -"LV VL [ 00. li "gnL L t.,«

According to Section 5.2 th e first term can be estimated by

From assumption (A6) and Proposition 3 we conclude that t he supports of t he discr et e basi s vectors 'lJy'k ,e overlap only at a fixed number of positions ind ependent of i . k and e. This implies

""' L...J

dj,n -k ,e1 'lJLj, k,e

( k ,e)E Jj~e- l

where we employ that the prediction set is reliable in t he sense of (4.19). Thus t he first term can be est imate d by

The second term in (5.20) vanishes, becaus e we use th e exa ct local numerical flux evaluation. In this case, th e evolut ion operators tL and tg coincide . T he third t erm can be est imated according to (A8) which results in t he recur sive estimate

and, hence,

The initial dat a can be further estimated

104

5 Error Analysis

From assumption (AlO) and (All) we conclude that the ratio elr is small in comparison to IluollL'>O' Consequently, we can estimate the supremum of vI. by a constant only depending on T and the supremum of the initial data uo .

o So far we have not verified the reliability property (4.19) for the prediction set DZi1 . However, we may assume that (4.19) holds for all previous time steps J = 0, ... , n in order to check the reliability of the prediction set for the new time step. As we will see in the proof of Proposition 5, it will be sufficient to know that the data of the old time step are uniformly bounded by the constant C(T, uo) . Proposition 5. (Finite differences for composite functions) Let the assumptions of Lemma 8 hold and assume that the flux balance function B satisfies Assumption 3. Introducing

DN(vI., K,

i\d

I(R) := {(j,k)

:=

j

{ILl% vI.,k,1

sup

; S(N, K, k')

c Ej 'k}

j E {l ,oo. ,R}R, k E {O, .. . ,R}R,

L:

and

1

j1k1 =

R},

we obtain

D R(BI.,K,E j,k):::; C sup

{II: (Dj/(vI.,K,E;'k))k/ 1

(j,k) E I(R)} . (5.22)

Proof. We want to estimate the finite differences Ll~ B2 k by means of finite differences of possibly lower order. To this end, we first 'introduce Lagrange polynomials Pi E PR, i = 0, ... , 2p, of degree R defined by the interpolation conditions (5.23) Pi(m) = vI.,k-p+i+mK' m = 0, ... , R ,

and the composite function G : R -7 R G(x) := B(po(x), .. . ,P2p(X)) == B(P(x)) .

The definition of the finite differences implies

Ll~ B2,k =

R

l; (_l)m (~) G(m)

=: Llf G(O)

and thus we deduce from standard results for finite differences, see e.g. [SB80, DL93],

ILl~B2,kl:::;C

sup IG(R)(x)l·

(5.24)

xE[O ,R]

Furthermore the R-th derivative of the composite function can be represented by

5.3 Reliability of Prediction

105

where we employ the smoothness of t he flux balance function B according to Assumption 3. When P(x) E {)D i , we consider th e onesided continuous ext ensions of the derivatives. This representation can be derived successively applying the chain rule for differentiation. Then we can estimate the R-th derivative by sup

IC(R)(x)l:::; C sup

xE[O,R]

xE [O,R ]

{II~ Ip;:d(xWI ;

where t he constant C depends on the coefficients and the bounds sup {{) xE[O,R ]

(j,k) E I(R)} ,

(5.25)

1-1

Cj ,i ,k

and R, respectiv ely,

()k B{) (P( x)) : i E {O, .. . , 2P}k } . Pi! . .. Pik

From the Lagrangian repres entation of the polynomials Pi we conclude that there exist s a uniform bound such that sup

IPi(X)I:::;Cllv2111=,

i=0 , . .. , 2p.

x E [O,R]

According to th e assumptions and Lemm a 8 we know that the reference FVS is uniformly bounded in the sup-norm. Hence, the bounds Ck only depend on T and IluoIIL=. We now consider the Newton representation of t he int erpolation polynomials Pi , i.e., R

Pi(X) =

v -1

L:>dO, . .. , II] II (x v =O

1=0

l) =

R

v-1

v=O

1=0

L ~II. L\r Pi(O) II(x -l) ,

(5.26)

where pdQ , . .. , II] denotes the II- th divid ed difference, see e.g. [SB80, DL93]. By means of induction and using the addition theorem for binomial coefficients we notic e th at

Incorporating t his relation in (5.26) we dedu ce sup

x E [O,R ]

Ip;j)(x)! :::; C _ max .1L\{Pi(l)l , j:::; R ,

(5.27)

I-O ,...,R-J

wher e C only depends on R. Not e, that t he t erms for II = 0, ... , j - 1 in (5.26) give no contribution to t he j th derivative. Furthermore th e definition of t he finite differences and the int erpol ation conditions (5.23) yield

106

5 Error Analysis (5.28)

Combining (5.24) , (5.25), (5.27) and (5.28) we obt ain

(i, v) E I(R ,p,j)} , where I(R,p ,j) := {(i,v) ; iE {O, .. . , 2p}R, VI E {l , ... , R - j d R } . Note, t hat t he bound C only depend s on R and Ci : We now esti mate t he finit e differences on t he right hand side by

ILl}( VI: ,k-p+H vl K I ::; D j l (vI:, K , Ej,k) ' where we take into account t hat the set Ej,k is inflated by the ste ncil of the numerical fluxes specified by the par am et er p. Therefore we need E~k inst ead J, of E j,k. Finally we obtain the asse rt ion taking the supremum over all k' such t hat S (R , K , k') C Ej,k' D Next we deriv e a converse result to Proposition 3, i.e., we want to est imat e t he finit e differences LlW U L ,k by certain det ails dj,T' For this purpose we employ an inverse (Bernstein-typ e) est imate which is a standa rd t ool in approxim ation t heory. Lemma 9. (In verse estimate) Let 1::; p,p' ::; 00 such that l/p+ l/p' = 1. As sum e that the scaling junctions CPj,k and
holds, where the constant C is independent oj j . In case of h ~ 2- j t he result can immediate ly be inferred from the definition of t he finit e differences for any j E LP. For the complementary case the smoo t hness of t he approximation space s Sj has to be exploite d . For a det ailed proof see [CohOO], Ch ap t. III. Note, t hat we can apply in particular t his result to the primal wavelet s 'l/Jj,k E W j C Sj+l we are considering here. Since the real axis is discretiz ed by a uniform dyad ic grid, the convergence of the subdivision scheme, see Section 5.2, ensure s t hat t he pri mal wavelet can be repr esented by the shift s and t ranslates of a scaling fun ction. By means of the above lemma we are now able t o prove t he following inverse estimate.

5.3 Reliability of Prediction

107

Proposition 6 . (Discrete inverse estim ate) Let K = 2L - j' -1 . Assume that the subdivision schem e conve rges uniformly For in th e sup-norm and th e corresponding primal wavelets 'l/Jj ,k are in N > we obtain L- 1 · I', E-J,k nS(N , K , k') ...J. -r- rII} VJ , ILJ. NK UL,k'I <_ 0 'L" 2- rnill{ N, r}( j' - j-1) + sup{ldJ,k j=-l

or.

°

(5.30) where d- 1,k := UO ,k' 1_ 1 := 10 , E- 1,k := EO ,k,O and (j' - j -1)+ := max{j'j -1 ,0} . Proof. The vector UL can be expanded in a series of th e discrete wavelet basis according t o (5.7), i.e., L- 1 UL =

L L dj,k 'lt7,k'

j=- l kEf;

where we use the convention 'lt~l ,k == 'It{;,k,O and 'lt7,k 0, . .. , L - 1, see Section 5.2. From this relation we deduc e

'lt7,k,l ' j

L-1

ILJ.~ UL,kll :S

L L

j=- l kEf;

Idj,k LJ.~ ('lt 7, kh ll ·

Since the differences on the right hand side vanish if the supports Ej, k and S( N , K , k') do not overlap, t he summation is only carried over those (j, k) such that Ej,k n S (N, K , k') ::p 0. Fur thermore, we know that the support s of t he discrete wavelet s 'lt7,k overlap only at a fixed number of positions independ ent of j and k , see also t he proof of Lemm a 8. Obviously, this also holds for the differences LJ.~ ('lt7,kh l. Similar to (5.21) we th en obtain L- 1

ILJ.~ UL,kll :S

L

sup { l dj, k l l l LJ.~ ('lt7,kh ,lll<X>

Ej ,k n S(N, K , k')

::p 0}.

j=-l

(5.31) It remain s to estimate the finite differences of t he discrete wavelets in t he supnorm. To t his end , we first recall that t he discrete wavelet 'lt7,k coincides with t he cell averages of th e continuous wavelet 'l/Jj,k, see Proposition 3, assertion 3. In particular , we obtain for h = 2- L K

and, hence,

108

5 Error Analysis

Therefor e we can est imate t he finit e differences of the discret e wavelets by t hose for t he cont inuous wavelet s. We are now abl e to employ t he inverse est imat e (5.29) for p = 00 . In case of N :S r it can be directly applied to f = 'l/Jj ,k E S j +l ' For r < N we first obs erve by the recursive representation of t he finit e differen ces t hat

Finally, we end up with

IILl hN 'l/Jj, k IIi»

:S C

II'l/Jj ,k II£ 00 (min . {1, h2 J+l})rnin{N,T} •

Combining this with (5.31) leads to (5.30), because the pr imal wave lets are uniformly bo unded, see Propositio n 3. 0 In the last step we estimate the powers of t he low order finite differe nces in P ro positi on 5 by the t hr esho ld values Cj ' For t his purpose we exploit Proposition 6 where the finit e differences are estimated by the details. In addit ion , we incorporate the definition of t he prediction sets V2i1 , see (4.21). Not e, that in the sca lar case the norm 11· 1100,* is rep laced by the ~bsolute value ',1 in (4.20) . However , we have t o sp ecify first the choice of the parameter a in (4.20). To this end, we make the following as sumption. Assu m pt io n 4. A ssume that the primal wavelets have C T Holder sm oothne ss, i.e., 'l/Jj,k E C T, and the dual wavelets have M vanis hing moments. Th en we choose some o suc h that 1
>0

su ch that

R - 1 < r :S R ,

(5.33)

1 + {3

(5.34)

< a < 1 + R - {3.

Not e, t hat t he smoothness param et er r is bo unded by the number of vanishing mom ents M of the dual wavelets ~j, k , i.e. , r < M, and thus o < M + 1 and R:S M . Proposition 7. L et the assumptions of Lemma 8 and Proposition 6 as well as Assumption 4 hold. Let (j' , k') f. V2i1 , K := 2£-/ - 1, N > 0 and k such that S(N, K , k) c

E; ,kl'

Th en we get th e estim ate N /R ,l } j' , ILl KN -VLn ,k I <_ C c rnin{

where the threshold values are given by Cj = 2j th en the cons t ant C depends on T and uo .

L

(5.35)

c. In part icular, if N < R

5.3 R eliability of Predi ction

c

Proof. Since S(N, K, k)

109

E;; ,k" we infer from (5.30) the estimate

L-1

I

I
L\ N v n K

""' L..-t

2- m in{N,r}(j'-j-1)+

sup{ld"! I ' EJ'lnE~ ,...to}. J,l ' , J ,k r

j= - l

(5.36) Again we notice that the supremum has only to be t aken over all indices (j, I) E D'L ,e such that E j ,l n E;; ,k' i- 0, since otherwise d'],l = O. The details and t he coarse-scale averages vg,k = d~l,k corr esponding to these indices ar e bounded by

e, 2(v(j,l)+l)
= 2 j-L +(j'-j)
,

(5.37)

because (j' , k') ~ 152i1 impli es j' > j + v(j, I) according to (4.21) . We first consider N 2: R. Since R is chosen such that (5.33) holds , this implies min {N, r} = r, Then we can further estimate the finit e differen ces where we combine (5.36) and (5.37) L-1

IL\~ v 'L,kl ::; C £2-L

L

T r(j '-j-1) + 2
= C £T L (E 1

+ E2 )

j=-l

with i' - 2

E 1 :=

L

2(
2 j( r-
j= - l

" ..L.2 .-

2
Since (5.32) holds , we can estimate E 1 and E 2 by C 2j ' which proves (5.35) in this case. Next we consider N < R . Then (5.33) impli es min{N,r} = N . First of all, we conclude from (5.15) and (5.14) t hat

Not e, that t his also holds for d~l ,O infer from Lemma 8

Id JTf,l IP =

Id"! Ip J,l

1

= vg,l' Combining this

Id n

J, l

I< C 2 j-L+(j '-j)
with (5.37) we

(5.38)

provided that p > 1. We notice that C is proportional to (C(T, UO))p-1 . Now we pu t p := R/N. Then we can est imat e the pth power of the finit e differences where we use Holder 's inequality

110

5 Error An aly sis

with , ,_ 2- M( / - j - 1)+

a) .-

b,) ..- 2-,61/

2,6 1/-j- 1 IO' / p s)',

S j :=

sup{ldj,tl ;

Ej,l

nt

;; ,k '

# 0}

- j-1 IO' / p .

The second sum in the right hand side is bounded by a constant , since 2- ,6O' /(p-1) < 1. In order to est imate the first sum , we first notice that t he pth power of t he supremum S j is bounded by sP < ) -

sup{ld )n,1IP

j- L+(/- j)O' E: .' si., n i: .: 0} < )' ) ' .k' r - C2 ,

where we employ (5.38). Then we obt ain L- 1

L

L- 1

(a j)P::; CE:2- L

j= - l

L

r

R (j ' - j - 1)+ 2 ,6 lj'-j- 112}+(j'-j)O'

j=-l

with / - 2

E1

:= 2(-R+ ,6+O')/ +R- ,6

L

2 j(R- ,6-O'H) ,

j= - l L-1

"

._ 2 (0'- ,6)/ +R

.02 . -

L

2 j (,6-O'+1) .

j =j'-l

Both t erms ar e bounded by C »', since (3 is chosen such that (5.34) hold s. This proves t he assertion. 0 Now we ar e able to verify t he reliability property (4.19) for the pr ediction set (4.21) . For this purpose, we combine the results of the Propositions 4, 5 and 7. Theorem 7. (R eliability, [CKMPOlj) Con sider th e un ivariate shift-invariant setting and let the Assumptions 3, 4 as well as th e assumptions (A6) - (All) of Lemma 8 hold. Th en th e prediction set defin ed by (4.21) fulfills th e reliability property (4.19). Proof. Let (j' , k') ~ i52i1. Then we hav e to verify that the corresponding det ail on the new time l~vel is not significant , i.e., (5.13) holds . In t he first ste p we estimat e the detail by t he supremum of certain finit e differences of order M according t o Proposition 4, i.e.,

(5.39) with K := 2L - j ' - 1 and M the number of vanishing mom ents of the dual wavelets. Next we apply the evolut ion equat ion (4.1) t o t he averages V2~1

5.3 Reliability of Prediction

111

where we replace the flux balances LlFE k by BE k according to (5.19). Then the finite differences in (5.39) can be estimated by

ILlPf V1~11 :s ILlPf v2 ,rl + AL,r ILlPf BE,rl· Sinc e R

(5.40)

:s M , Proposition 7 implies ILlPf v2 ,rl :s C Cj '.

(5.41)

From the CFL constraint we furthermore deduce 1 AL ,r = 2L T :S C (maxlul:::: C(T,uo) 1!,(u)lr =: C(T,uo ,!') .

(5.42)

Combining (5.39) , (5.40), (5.41) and (5.42) we obtain

Idj,;-l/l:S

CCj'

+ C(T,uo ,!') DM(B2 ,K, Ej' ,kl)'

Here we employ the definition of DM(B2, K, Ej l,kl) introduced in Proposition 5. Hence it rem ains t o verify that the differences of the flux bal ances ar e bounded by C cj" To this end, recall the meaning of R from (5.33) and note that ILlPf BE,rl = ILl~-R Ll~ BE,rl :s C ILl~ BE,rl· Hence we infer that

DM(B2 ,K,Ej' ,k'):S C DR(B2,K,Ej',k')'

(5.43)

In order to est imate the right hand side we apply (5.22) to obtain

D R(B2 , K, Ej' ,kl)

:s C sup {II:1 (D jl (v2, K, E; ,kl))kl ;

(j, k) E I(R)} . (5.44)

For (j,k) E I(R) and r E h such that S(jI ,K,r) conclude from Proposition 7

c E; ,kl'

1

:s l :s R, we

Aj, -n I < C rnin{1,j, /R} v L ,r _ cj' . I£..1K

Thus, t he supremum can also be est imated by this bound, i.e., 1,jl /R} D JI. (-n vL , K , LJf,jl,k' ) < _ C Cjrnin{ l

and we obtain R

n K f,(DJ.I (-vL' , LJ j' ,k'

II

))kl
1=1

1=1

Since (j, k) E I(R) , we conclude from Assumption 4

~ ilk l = R = 1 Z:: R R . 1=1

Finally, the right hand side of (5.44) can be est imate d by

D R(B2 , K, Ejl ,kl ) :s C Cj" Combining this with (5.43) proves the assert ion. 0

L:~l jlkt/R

Cj l

.

6 Data Structures and Memory Management

In the context of adaptive schemes the design of data structures and an appropriate memory management have a significant influence on the performance of the computation. In the following we will discuss the design of data structures where we deduce some fundamental design criteria from algorithmic requirements, see Sect. 6.1. Here the concept of hashing turns out to be an efficient tool which fulfills these criteria. The basic ideas of hash tables, hash functions and collision strategies are summarized in Sect. 6.2, although they can be found in standard text books, see e.g. [Str97, OW96]. Nevertheless we believe it to be convenient for the reader to repeat some details, since we add some special features to the standard setting which are needed for the application at hand. Finally, in Sect. 6.3 we present the data types by which we realize the adaptive code. These are derived from the template library igpm_ t_lib developed at the Institut fiir Geometrie und Praktische Mathematik, RWTH Aachen, see [MVOO].

6.1 Algorithmic Requirements and Design Criteria In the previous chapters we developed an adaptive scheme with an optimal complexity in the sense that the number of operations is proportional to the number of unknowns, Le., # DL,e and # fh,e, respectively. We are now concerned with the design of an optimal code in the sense that the memory requirements and the CPU time are proportional to the complexity of the adaptive algorithm. Here it turns out that the choice of data structures and memory management has a significant influence on the performance of the computation. Our experience shows that the design of appropriate data structures crucially depends on the underlying adaptive algorithm, i.e., the data structures have to be adapted to the algorithmic requirements and should not be designed independently. Therefore we first have to determine these requirements. To this end, we consider the Algorithms 3 and 6 by which the local (inverse) multiscale transformations are performed, since these form the backbone of the adaptive scheme. From these we deduce the following demands:

114

6 Dat a St ru ctures and Memory Man agement

- The local transformations are performed level by level. Therefore dat a st ructures by which the local averages and the significant det ails are sto red have t o maint ain t he level inform ation. - For each level we have to collect the indic es of cells to be refined or coarsened. Therefore we need a dat a structure for ind ex sets. - For t he computation of t he local (inverse) two- scale transformat ion (3.16) - (3.20) and (3.21) - (3.23) , resp ectively, we have to det ermine t he supports of the mask matrices Mj,o , Gj,o and Lj,e. The summation is then perform ed on a column of these matrices. Obviously, we never access a single element of the ma trices but a colum n corresponding to t he non-vani shing ent ries indicat ed by the support. This dat a group has to be st ored as on e element by a dat a st ruc ture. - Whil e performing t he local t ra nsformations we have t o check whether a matrix column or a local average as well as a significant det ail exists, i.e., wheth er this information has alrea dy been computed and has been st ored. - The data provid ed by th e local transformations have to be inserted or have t o be deleted from a cur rent data st ruc t ure . - If we access det ails dj,k,e, we always consider all wavelet types indi cated by e E E *.

- The algorit hm is ind epend ent of t he spatial dimension d and t he number of conservat ion laws m. Therefore t he dat a structures have to be designed fl exibly with respect to thi s par amet ers. From t hese algorit hmic requirements we dedu ce t he design criteria for an appropr iate dat a st ru ct ure for our sparse data distribution. This data st ructure should provide - fast random data access, e.g. check whether an element already exists, - fast insertin g and deleting of elements, i.e., copying and sort ing of element s within the dat a structure should be avoided, - fa st dynamic m emory allocatio n and extens ion, since the memor y requirement is not a priori known but can only be approximate ly predicted and - support group info rmation , i.e. , connect ions of dat a corres ponding to a common level, should be maintained.

By means of these design crite ria th e templ at e libr ar y igpm_t_lib has been developed , see [MVOO]. This libr ary provides te mplate classes for - a vector with an arbit ra ry but fixed dimension , see tvector_n, - an in dex composed of a multiind ex or , additio nally, indi cated by a level and blocks of multiindi ces repr esenting t he supports, see tmultiindex, tlevelmultiindex and tmultirange ,

6.2 Hashing

115

- hash map s by which elements can be sto red in an unordered way providing fast access, see thashtab, thashtab_linked and thashtab_ linked_one.

The fund am ent al dat a st ructures used in th e impl ementation of the adaptive scheme are various hash maps which optionally support grouping of t he dat a by means of level information.

6.2 Hashing In t he context of ada pt ive schemes it is not sufficient to t ake into account only the complexity of the algorithm, i.e., th e number of floating point op erat ions , bu t also t he overhead ste mming from the und erlying dat a structures. Two main problems arising in th e implement ation of ada ptive schemes are (i) dynamic m em ory operat ions and (ii) fast data access with respect to inserting, delet ing and finding an element. In the following t hese two issues shall be bri efly discussed . Dynamic memory operations. Due to refinement and coarsening operati ons in t he algorithm, memor y space for new elements or freeing unus ed elements are frequ entl y performed and th erefore should be very fast . Since t he memory operations prov ided by t he programming language are univ ersal, t hey can be mad e much mor e efficient if we t ake into account that our dat a are of t he sa me type. In particular , thi s can be realized by allocating a sufficient ly larg e memor y block and manage t he memor y requirements by t he algorithm with a specific dat a st ruct ure . Since the overa ll memory demand can only be est imated , t he dat a st ructure should provide dyn ami c exte nsion of the memory. Fast data access. For inst ance, lists might be considered as a container for fast inserting and deleting element s. However , findin g an element requires t he search in t he whole list from it s beginning until t he positio n of the element is found. On the other hand, a fast way of findin g an element by means of an index is an array. But inserting and deleting element s in an array requi res copy operations. However, if one does not need to maint ain any ordering in the data , then ot her dat a st ruc t ures, e.g. hash maps , might be a good alte rnative. In view of an optima l memor y management and a fast dat a access we design a hash m ap th at is composed of two parts, nam ely, a vector of poin t ers, a so- called hash table, and a memor y heap , see Fig. 6.1. The hash t abl e is connecte d t o a hash fu nctio n f : I -+ P which maps a key, e.g . an ind ex, t o a position on th e hash t abl e, i.e., a number between 0 and #P - 1 where #P denotes t he length of the hash table. Not e, th at in general t he set of all possible keys is much larger than t he length of t he hash t abl e, i.e., # P « # I , Obviously, the hash function is not injective. This leads to collisions in t he hash t abl e, i.e. , different keys might be mapp ed ont o t he same position by t he hash fun ction . The corres ponding values of t hese keys are t hen linked t o

116

6 Dat a Structur es and Memory Management

t he list that start s at the position f( key). Each element in the hash t abl e is a point er to a connecte d list whose elements are stored in the heap . Here each element of t he list can be a complex dat a st ruct ure its elf. It contains at least t he key and usually addit ional dat a , th e so-called value. In genera l, th e valu e consists of t he data corres ponding to an ind ex. In addit ion, the level might be addit ionally pr ovided . In ord er to maint ain t he connection of th e valu e to its key, t he key also has to be sto red in this list .

x

List 0 List 1

Elem . of KeyT Elem . of KeyT Elem . of KeyT

I

-I

II I I I I I I

0

[,

0

[,

0

[,

ValueT oJ

List 1

ValueT .J

List 0

ValueT. J

List 0 +

I

...

(memory heap)

(hash table) Fig. 6.1. Hash map

Obviously, t he performance of th e hash map cru cially depends on t he number of collisions, since t he access to th e values by t he linked list is th e fast er t he short er t he list . To thi s end, the number of keys mapped to the same position in t he hash table should be as sma ll as possible. This is controlled by t he fill rate of t he hash table, i.e., t he prob abili ty of collisions. If # p = # I and f is injecti ve, then no collisions occur. However , t his case lead s t o a tremendous wast e of memory space in t he hash table, since t he ada pt ive scheme will only access a sma ll number of keys I C I with # I « # I . On t he ot her hand , if t he length of t he hash t abl e is to o sma ll this might cause a lot of collisions. In order to balan ce t he length of the hash table and the number of collisions, severa l st rategies have been developed for the design of an ap propriate hash fun ction, see [Str97 , OW96]. For our purpose choosing t he modulo funct ion as th e hash functi on t urne d out to be sufficient. It map s an int eger generated from a key to a position f (i) E P = {O, .. . , p - 1} in t he hash table by f( i) = i mod p.

6.2 Hashing

117

As alre ady discu ssed t he length p of t he hash table cru cially influences t he performance of t he hash map . In general, p is chosen to be t he lar gest prime number less t han # I in or der to reduce t he probability for a collision . However, in our computations it t urned out t hat the cho ice of p as a prim e number is not significant . Therefore we choose a simpler st rategy. To t his end , we guess t he number of expected elements to be stored, say nMax, and choose t he length of t he hash table as nMax * nFactor , where nFact or equals 3 in our computations. At the sa me t ime we allocate memory for only nMax elements in t he asso ciated heap . This implies that we work with a hash table filled about dFill = 1 / nFactor bu t allocate an optimal chunk of memory for our dat a . This strat egy combines t he advantages of a lar ge hash t able, i.e., small number of collisions, with t he requirement to allocate only adequate sto rage memory. So far we have outlined the idea of hashing as it can be found in st andard t ext books. However , for our applicat ions we hav e to modify t he st andard concept, since we need t o know the connect ion of the values to a certain level j E {O , . . . , L - I }. Rememb er that the local transformations are performed level by level. For t his purpose, t he hash map is exte nded by a vector where its lengt h is det ermined by t he number of levels L . The jth comp onent of t his vect or contains a pointer t hat points to the first element of level j pu t int o t he memory heap. Addi ti onally, t he value has to be int ernally extended by a po inter t hat points onto t he next element of level j. This is sketched in Fig. 6.2. Then we can access all element s of level j by travers ing t he resu lting connect ed list.

x

List 0 List 1

I·

·1

1·

1

I I

I I I

I

(hash table)

Elem . of KeyT. [,ValueT.]

List 1

Level 2

Elem . of KeyT. [, ValueT.]

List 0

I Levell

Elem . of KeyT. [, ValueT.]

List 0

t Level 2

• I I

... ....

. . . .

I

.. ..

I

............... .

I

I (m em ory heap)

I Level 0

Levell

I Level 2

(array of levels)

Fig. 6 .2. Linked hash m ap

L -_ ]

118

6 Dat a Structures and Memory Manag ement

6.3 Data Structures In the followin g we describe t he dat a types that are needed for t he implement ation of the Algorithms 3 and 6 of the local transformations. Here we restrict ourselves to t he cur vilinear case as summarized in the Sect. 3.8. The dat a ty pes are derived from t empl at e classes provided by the t emplate libr ar y igpm_t_lib, see [MVOO]. First of all we deriv e the array types typedef tvector_n< double, m > uvector typedef tvector_n< double, 2-d-1> evector typedef tvector_n< uvector, 2-d-1> dvector from the te mplate class tvector_n by which we can st ore the vect or of conservativ e qu antities v j,k E R'" , th e det ails dj,k,e E Rm and the vector of det ails corres ponding t o all wavelet types (dj,k,e)eEE• . In the cur vilinear case, t he average s and the det ails ar e enumerated by multiindices k = (k 1 , • •• , kd ) E Ngand level-multiindices (j , k). To this end , we derive the int eger arrays typedef tmultiindex< d > mi typedef tlevelmultiindex< mi > lmi from t he t emplate classes tmultiindex and tlevelmultiindex, respecti vely. For the local t ransformat ions we always access simultaneously all nonvani shin g elements of a column of t he mask matrices. Therefore it is convenient to agg lomerate t he indic es of t he corre sponding supports in one dat a st ructure. However, we do not have to st ore ind ex by ind ex, since in the curvilinear case t he supports can be repr esent ed as a mul tidimension al integer int erval k + [0, i ]d C Ng charac te rized by t he multiindex k and the number i , By this knowledge we designed t he special te mplate class tmultirange. According t o the supports of t he box function and the modified box wavelet s, see Sub secti on 3.8, we need four different typ es of mul tiran ges typedef tmultirange< 1, mi> mr1 typedef tmultirange< 2, mi> mr2 typedef tmultirange< 4*8+2, mi> mr48 typedef tmultirange< 2*8+1, mi> mr28 for t he supports M;:~ (9J,k), M~,k (9;:~) , M j,k and £ j ,k' resp ectively. Here the first argument specifies the length i and the second argument the initial index k. Analogously, we can sto re t he non -van ishing matrix elements corres ponding to t he columns of the mask matrices by multiran ges where in addit ion to the blocked multiindi ces the corre sponding matrix element s are st ored whose t yp e is specified by the third argument . Here we distinguish between t he t ypes typedef tmultirange< 1 , mi, double >mrA_GOc typedef tmultirange< 2, mi, double >mrA_MOc for t he matrix columns of Gj,o and Mj,o , respect ively. In case of t he matrices LJ·, e, e E E*, we pr oceed differentl y. By construc t ion the supports £'7 J , k are

6.3 Data Structures

119

the same for all wavelet types e E E* , only the matrix elements l~'~ dep end on the wavelet type. Moreover , in the local tran sformations we al~ays have t o acce ss all ty pes corres ponding to an ind ex pair (j , k) . Therefore it is mor e convenient to st ore all matrix elements l~:~ , e E E* , in one multirange typedef tmultirange< 2*s+1, mi, evector > mrA_L. In t his we avoid t he multiple st orin g of the ind ex k and the length i of t he int eger int erval. The mask matrices Mj ,o , Gj,o and Lj = (Lj,e)eEE* are then st ored in hash map s, see Sect. 6.2, where each element is a multiran ge representing the nonvanishing matrix element s corre sponding to one matrix column typedef thashmap_linked< lmi, mrA_MOc > liMap_MOc typedef thashmap_linked< lmi, mrA_GOc > liMap_GOc typedef thashmap_linked< lmi, mrA_L > liMap_L Here the first argument of the hash map represent s the type of the key, e.g. a level-multiindex , and the second argument t he type of t he valu e, e.g. a mul ti ran ge. Analogously t he local averages V j ,k , (j , k) E 9L,e , and t he significant det ails d j ,k ,e , (j ,k,e) E V L,e , are st ored in linked hash map s typedef thashmap_linked< lmi, uvector > liMap_u typedef thashmap_linked< lmi, dvector > liMap_d , where each element is eit her a vector of conservative quan ti ties or an array of vectors representing the det ails of all conservat ive qu antities and all wavelet ty pes . Note, t hat t he ind ex sets I j,e and :/j,e are implicitly det ermined by t he linked list s corres ponding to the different levels. The local tran sform ations are perform ed levelwise. To t his end, we need dat a structures where t emp orar y dat a corres pond ing to one level can be stored . Again we use hash map s. Since only on e level is involved , we can simplify this ty pe of hash map. Therefore the dat a ty pes typedef thashmap_linked_one<mi > iSet typedef thashmap_linked_one<mi , uvector > iMap_u typedef thashmap_linked_one<mi, dvector > iMap_w use t he te mplate class thashmap_linked_one. Note, t hat t he hash map has t o be linked , since we have t o traverse throu gh all elements. In particular , t he dat a type iSet is a hash map where only keys are st ored bu t no valu es. This can be int erp ret ed as a set of keys, e.g . mul tiindices. Finally we need a data structure for the man agement of t he ada pt ive grid and corres ponding geometric inform ati on . For inst an ce, t o each cell volume, mono mials of higher order, norm als, arc lengths etc . have t o be st ored . Which information are needed does not only depend on t he mul tiscale set t ing bu t also on the FVS. We therefore do not specify the det ails bu t collect t he geomet ric dat a in a class of its own class cell_data { ... }. Then t he adaptive grid is stored in the hash map ty pe

120

6 Data St ru ctures and Memory Management

typedef thashmap_linked liMap_grid where each element is an element of th e class cell_data. By means of t he above dat a structures we now introduce the hash maps MOc liMap_MOc liMap_GOc GOc liMap_L L for t he management of th e mask matrices Mj,o, Gj,o and Lj , j = 0, . . . , L - 1, as well as t he local averages and th e significant det ails liMap_u v_map liMap_d d_map corresponding t o t he adapt ive grid and th e set of significant det ails. The adaptive grid and related geomet ric inform ation are st ored in t he hash map liMap_grid adapgrid_map where not only t he locally finest cells are taken int o account but also all cells on coarser levels. The initi alization of t hese hash maps crucially influences t he performan ce of t he computation. We th erefore describ e t he choice of nMax , see Subsect ion 6.2, which determines t he length of the hash table and the size of the memory needed for st oring nMax elements. First of all, we consider the mask matrices. We observe t hat t he number of elements corresponds to the number of columns of t hese matrices, since each element in the hash map represents the non-vani shing matrix coefficients corresponding to one column. For the matrices Mj,o and Lj th e t ot al number of columns for all levels is det ermined by ~ 1_ qL

c: N j

=

N L- l

l-=-q

j=O

since N , = q N j H

.

In case of the matrix Gj,o we obtain the total number L 1 _ qL+l "" N , = N L - - ~ j=O

1-q

This is also an upp er bound for t he hash map adapgrid, since we st ore not only cell information corres ponding to the ada pt ive grid bu t also the information for all coarser cells. For th e avera ges and th e det ails the to t al num ber of elements is restricted t o N L for the full grid on t he finest level and accordingly all det ails are significant , i.e., NL - l (1 - qL)j(l - q). From t he t otal number of elements t hat have to be stored in the worst case of a un iform refinement over all levels, we determine t he number nMax = nTotal * rFill of pr edict ed elements where rFill denot es t he fill rat e. Not e, t hat rFill differs from t he fill rat e dFill of t he hash t abl e. In Sect . 7.1.4 we will invest igat e the influence of rFill and nFactor on th e performan ce of t he computation. Finally we would like to remark t hat the hash map s for t he mask matrices as well as t he adapti ve grid are initialized once and then t he length of t he

6.3 Dat a Structures

121

hash table as well as t he memory heap size remain un chan ged t hroughout the computat ion. Therefore t he choice of rFill is mor e significant for the mask matrices and t he adaptive grid. In particular, t he corres ponding hash maps require t he bulk of memor y. This is different for t he hash maps in which t he local averages and the det ails are st ored, since the ada pt ive grid and t he set of significant det ails, respectively, are changing from one time step t o t he next one. Here t he length of th e hash t abl e has to be adapte d in each t ime st ep and t he memor y event ually has t o be exte nded dyn amic ally. From Algorithm 2 we conclude that t he complexity of th e sets 9L,e and DL,e are relat ed by #9L,e = #DL,e , since a cell is refined as long as t here exists a significant det ail. Then we can adapt the size of the hash maps v _map and d_map t o t he actual number of elements det ermined by one of the hash maps with nMax = # DL,e for v _map and nMax = q# 9L,e for d_map , respect ively.

«:

«:

7 Numerical Experiments

In the previous chapters we have developed and investigated a new concept for the construction of an adaptive FVS where we apply local multiscale techniques to a reference scheme. For scalar conservation laws we have been able to derive an error estimate of the form (5.1). This is based on an a priori error estimate for the discretization error of the reference FVS and the stability of the perturbation error in the sense of (5.3) . In [CKMP01] parameter studies have been presented for scalar one-dimensional problems which confirm the analytical results in Chapter 4. In the sequel, we verify that the adaptive concept can also be applied to systems of conservation laws. In particular, we are interested in applications to real-world problems arising from problems in engineering. For this purpose, we present several computations for the two-dimensional Euler equations for a polytropic gas with "( = 1.4, see Example 3. These have been carried out on PC's with a 600 MHz processor (Pentium III) . The numerical investigations are divided into two parts. In the first part we are concerned with the computational complexity of the adaptive scheme with respect to computational costs and memory requirements as well as the stability of the perturbation error. To this end, we present several parameter studies for three test configurations, namely, (i) shock reflection, (ii) implosion and (iii) wave interaction. These are instationary problems performed on Cartesian grids. In the second part we report on some computations for realworld applications. Here we consider (i) the flow over a bump and (ii) the flow around a profile of an airfoil. Both configurations result in steady state solutions. Since the geometry of the computational domains is no longer that simple as before, we discretize the flow fields by block-structured curvilinear grids .

7.1 Parameter Studies The analytical setting of Theorem 5 is much too restrictive with regard to realistic problems. Therefore we are interested in the performance of the adaptive scheme applied to an extended setting not necessarily covered by the analytical investigations. Here three issues are of special interest, namely, (i) the detection of all kinds of singularities, e.g. , discontinuities and kinks,

124

7 Numerica l Experim ent s

corresponding to physically relevant effects by means of the multiscale decomposit ion, (ii) t he complexity of th e adaptive FVS with respect to computationa l costs and memory requir ement s in comparison to the reference FVS and (iii) th e st ability of the perturbation err or. These issues are investi gated by means of several par ameter st udies with respect to an increasing number of refinement levels where th e threshold value e is fixed for all computations. Not e, t hat this does not quite agree with th e ideal strat egy as outlined in t he beginning of Chapter 5. In ord er to balance the discretization err or and t he perturbation erro r , t he th reshold value c has to depend on the numb er of refinement levels L according to (5.1) and (5.2). However , for systems of conservation laws no error estim ate is available so far for the discretiz ati on error. On t he other hand , th e st ability of the perturbation error in the sense of (5.3) has only been verified for scalar problems . Therefore in the following we focus in our num erical investi gations on the stability of t he perturbation error, in par ticular , with regard to its dependence on t he highest resolution level L. Fir st of all, we summariz e t he setting of t he test configurat ions. Then we specify the discreti zation and present th e reference FVS. By mean s of par amet er st udies we investigate and discuss t he computationa l complexity of t he ada pt ive FVS as well as the stability of the perturbation err or for an increasing number of refinement levels L. We conclud e with some par amet er st udies concerning t he parameters rFill and nFactor of the und erlying dat a structures. 7.1.1 Test Configurations We consider t hree configurations where t he computationa l complexity of t he problem strongly varies due to t he singularit ies occurring in the solut ion. Shock Reflection. Thi s is a quasi-one-dimensional problem where we impose a single shock aligned with one of t he coordinate axes, see Figure 7.1. The shock is moving to t he right approaching a wall. At the wall t he shock is reflect ed and moves back to th e left. Not e, t hat th e initial dat a in front of

UL

f--.-

UR

Units St ate L St at e R 1.3 0.8

!2

kg/rn"

Ux

1300

0

uy

mls mls

0

0

p

hPa

11050

5200

Fig. 7.1. Initial configurati on for shock reflection

7.1 Param eter Studies

125

the shock corres pond to a sup ersonic state. Implosion Problem. The initial configur ation is det ermined by two st at es and U/ as well as t he radius r of a circle, see Figure 7.2. Inside t he circular domain we imp ose low pr essure and out side high pr essur e. Again the resulting flow field is quasi-onedim ension al due t o t he rotation al symmetry. With pr opagating time t hree waves develop , nam ely, (i) a rar efaction wave, (ii) a contact sur face and (iii) a shock wave. The shock wave and the contact sur face are movin g t owards t he cente r of t he circle. The rarefaction wave is movin g into the opp osite direct ion where the corres ponding rarefaction fan is expanding. Here we are int erest ed in t he inst anc e when the sho ck wave focuses in the cente r .

UE

Unit s St ate I St at e E

o

e kg/rn" U

m/s

p

Pa

T

m

E

1.251

2.502

0

0

101280 202560 0.15

Fig. 7.2 . Initial configuration for impl osion pr oblem

Wave Interaction. The initial configurat ion is det ermined by four states corres ponding t o th e four quadrants of t he coordinate syste m , see Figure 7.3. Away from th e origin of th e coordinate syste m, the solutio n exhibits a one- dimensional wave pat t ern consist ing of a rar efaction wave, a contact sur face and a shock wave. This structure is obtained by t he solut ion of a one-dimensional Riemann problem characte rized by t he const ant states U L = u j , UR = uj/ and U L = U/ t . U R = U/ , respectively. Close t o t he origin t he different one- dimensional waves int eract formin g a genuinely twodim ension al wave pat t ern.

State I St at e II II

I

I

II

e

0.125

1.000

Ux

1.000

0.000

u y 1.000

0.000

0.100

1.000

P

Fig. 7.3. Initi al configurat ion for wave interaction

126

7 Numerical Experiments

7.1.2 Discretization As the reference FVS we apply an essent ially non-oscillatory (ENO) scheme described in [Mii193] . This E NO scheme is charac te rized by a one- dimensional second ord er accurate reconstruction t echnique via primitive fun ctions, see [HEOC87] . Here the primitive variabl es are reconstruct ed by mean s of piecewise linear pol ynomials. At the cell int erfaces one-dimensional Riemann problems in normal dire ction are solved applying Roe's approxima te Riemann solver , [Roe81]. For all three configurations t he boundary condit ions are simpl e. In principle, three ty pes of boundar y condit ions occur , namely, (i) free stream condit ions at inflow boundaries, see t he left side boundar y in Figur e 7.1, (ii) slip con diti ons at a wall, see t he right side boundar y in Figure 7.1 and (iii) attached flo w fi eld whenever the flow field at t he boundar y is constant in norm al direction. The discreti zat ion of the computat ional domain is summa rized in Tabl e 7.1. Since we employ an explicit FVS, th e st ep size 7 in t ime has to satisfy a CFL condi tion for t he fin est discre tization level L , i.e., AL ,k

max {IAk(U, n)1

; U E D , IInl12=

I} :S c

< 1.

n:=1

h j,i = h j and h j,i Note, t hat for a uniform discretization IVj,k I = 2- 1 h j - 1 ,i = 2- j hO,i . For t he par amet er studies we successively increase t he number of refinement levels L . In order t o maintain the CFL number we have to redu ce 7 by the factor 0.5 when adding an addit iona l refinement level, i.e., 7 == 7L = 2- 17L_l = ... = 2-L70 . From t his we conclude AL ,k = 2 (d- l ) L 70 / h o. Obviously, t he ste p size 7 is charac te rized by the number of refinement levels L and th e ratio 70/ho of th e coarsest discretiz ation list ed in Tabl e 7.1. Table 7.1. Discreti zation pa ram eters rclh« [s/m 2] Reflection [0,1] x [0,0.4] 3.2 x 10- 4 1.28 X 10- 4 10- 6 [0,1]2 3.6125 X 10- 4 Implosion 1.5 x 10- 1 Interaction [-0.4 , 0.4]2 2.5 Case

Q[m]

T [s]

The multis cale analysis is based on t he modified box wavelets. These are const ruc te d according t o Algorithm 1 where t he degree of vani shin g mom ents is chosen as M = 3. In t he cur vilinea r case , the st encils £ j ,k are det ermined by Lj,k with s = 1 according to (3.45) and (3.46). The sequence e of threshold valu es is recursively det ermined by e i. = C, Cj = 0.5 cj+l , j = L -1 , .. . , 0, according to P roposition 3. Here th e threshold valu e is C = 0.001 (implosion,

7.1 P ar amet er Studies

127

wave interaction) and e = 0.01 (shock reflection) for all computations of the par am et er st udy. Fin ally, we have to specify t he flux evaluation. Here we use t he locally st ructure d strategy (4.15) because t he reference num erical flux is only defined on a struc ture d grid. For this purpose, t he adaptive grid has to be locally uniform of degree p = 3 according to the flux stencil. From Corollar y 5 we conclude t hat t his is gua ra ntee d if the tree of significant det ails is gra ded of degree q = 2. 7.1.3 Computational Complexity and Stability

For t he t hree t est configurations we have performed par amet er studies with respect to an increasing number of refinement levels where t he t hreshold valu e € is fixed for all comput at ions. In Figures A.l , A.2, A.5, A.6, A.9, A.I0 t he density is pres ented for th e initial data t = 0 and th e time t = T listed in Tabl e 7.1. The ad aptive grids are plotted in Figures A.3, A.4, A.7, A.8, A.ll , A.12. All these figur es reflect the comput at ional results for t he high est refinement levels list ed in t he Tables 7.2, 7.3, 7.4. We observe that the high er resolution levels are only accessed near discontinuities. This verifies t ha t t he ada ptation crite rion based on det ails is able to detect t he relevant physical effects in t he flow field . We are now int erested in t he computationa l complexity of our scheme (wit h regard to comput ational t ime and memory) for t he prese nt st udies of stability. This is document ed in t he Tabl es 7.2, 7.3, 7.4. The quanti ties list ed in the different columns are - t he number of refinement levels L , - the number N L of all cells corres ponding to the full grid on level L , - t he numb er N 9 := # 9L ,c / N L repr esenting the number of cells corres ponding to t he lar gest adaptive grid determined during t he computation relative to t he full grid on level L , - t he comput at ional ti me

CMS

used by t he ad ap tive scheme,

- th e memory size M em which corresponds to t he maximum amount of memor y that has been allocated in one time st ep, - th e speedup rates SM S := CF VS / C MS det ermined by th e ratio of the computat iona l times for the ada ptive scheme and t he reference FVS on t he full grid of level L and - the per turbation error e L := '1£ - V L E R N L det ermined by t he difference of t he averages '1£ := {vL ,d kEh and VL := {VL ,khEh pr oduced by t he ada ptive scheme and t he reference FVS on t he full grid of level L , respectively. The local averages from t he adaptive scheme are mapped onto the full grid by means of local reconstruction according to t he inverse mult iscale transformation.

128

7 Numerical Experim ents Table 7.2. Param et er st udy for shock reflection

NL

Ng

C MS

%

[min]

M em [MB]

S MS

2560 31 1.50E-1

0.9

2.1

1E-3

10240 16 6.84E-1 40960 83.03E+0 6 163840 4 1.36E+ 1

1.8 3.6

3.7

6E-4 4E - 4

L

3 4

5

7 655360 82621440

2 5.98E+1 13.04E+2

IleLlk L,.

6.8 7.3 12.6

2E-4

14.5 21.8

1E - 4

29.0 34.3

Table 7.3. P arameter st udy for implosion

NL

L

Ng

C MS

%

[min]

M em [MB]

S MS

IleLil1 ,L,.

16384 45 6.833E-1

8 1.34

1E-3

65536 28 3.667E+0 6 262144 14 1.603E+1 7 1048576 76.1l2E+1 8 4194304 42 .194E+2

20 2.10

6E-4

40 3.61

4E-4

84 7.73

4E-4

165 16.51

916777216

334 36.44

4 5

2 7.470E+2

Table 7.4. P ar am et er study for wave inter action

L 2 3

NL

Ng

C MS

%

[min]

6400 89 0.73E+0 25600 58 4.53E+0

4 102400 34 2.33E+1

M em [MB]

S MS

6 0.82 1.2E-3 16 0.98 l.lE-3 39 1.62 1.2E-3 90 2.67 1.2E-3

5 409600 19 1.12E+2 61638400 10 4.99E+ 2

186 4.79

5 1.98E+3

337 9.67

7 6553600

IleL111 ,L,.

Computational complexity. Considerin g t he three tables we make t he following observations: - The number N L increases by a factor of 2 d becau se t he grid is uniformly refined by subdividing a cell into 2d congru ent subcells. - The relative numb er Ng decreases asympt otically by a factor TL := N 9 ,L/N g,L-1' Therefore t he tot al numb er of cells in th e adapt ive grid increases by a factor qt. := # YL ,c:/# YL-l ,c: = 2d TL . In general, t he ratio

7.1 Param eter St udies

129

r L is sma ller than 0.5 and , hence, t he number of cells in the ada pt ive grid increases only by a fact or of 2 at most with each additional refinement level instead of 2d for t he uniform refinement. This results in an exponent ial redu ction of cells in comparison to t he full grid t hat corresponds to significant speed up rat es. However , we would like to emphas ize t hat t he full grid may give us t he more accurate approximation.

- T he computational t ime CMS increases with L -+ 00 becau se t he number of ti me steps is doubl ed according to t he CFL condition for the finest level. On t he ot her hand , t he numb er of cells in the ada ptive grid increases by t he factor qt.. Therefore CMS increases by t he factor 2 qt. which is small er t han t he 2 d +l for t he reference FVS on t he finest level. This results in an expo nent ial increase of t he speedup rates SMS, i.e., t he comput at ional t ime becomes significantl y sma ller for the ada ptive scheme in comparison with the computation of th e reference FVS on the full grid. - Since the number of cells in th e ada pt ive grid increases by t he factor qL , we have to allocate mor e memory space. This is reflect ed by t he quantity M em list ed in t he tables. Comp aring t hese numbers we conclude th at memor y size is increasing by a fact or proportional t o qt.- We emphasi ze th at for Ca rtesian grids th e mask coefficients do not depend on th e level and the position except for boundar y ada ptations and , t herefore, require much less memory. Addit ionally, t hese valu es can be computed in a pr eprocessing st ep. This is not taken into account here. In pr inciple, t hese observa t ions hold for all the t hree test configurations although t he num bers differ according to t he wave st ructures developin g in t he different flow fields . Here t he shock reflection gives t he highest speedup rates whereas t he wave interacti on exhibits t he lowest rates due to t he complex wave st ructures. Therefore we conclude t hat t he computational effort and t he memory requirements are prop ortional to t he number of cells in the adaptive grid. We emphas ize th at this can not be realized by Har ten 's original st rategy because t he resultin g scheme still involves t he full grid on t he finest level. This is based on the fact that locally expensive num erical fluxes are replaced by cheaper approximations, i.e., Har t en 's st rategy is based on a hyb rid flux evaluation where t he switching is perform ed by means of the multi scale analysis. However , th e adaptive strategy at hand reduc es the total number of flux evalua t ions not only th e number of expensive flux evaluat ions, i.e., we do not comput e t he chea p flux approximations in smoo t h regions at all. In combinat ion with the local multiscale t ransformation and its inverse, t he number of floatin g point opera t ions as well as the memory size of the resulting scheme is pr oportional to t he number of cells in t he ada ptive grid. Stability. The ob jecti ve of th e adaptive st rategy is to redu ce the computationa l complexity while maint ainin g t he accuracy in comparison to t he reference scheme on a uniform grid. In order to balance t he discretization error an d t he pert urbation error in (5.1) according to Corollar y 7 we need t he

130

7 Numerical Experiments

stability of the perturbation error in the sense of (5.3) . We would like to emphasize that the analytical results have only been verified for one-dimensional scalar problems, see Section 5.3. Here we consider a two-dimensional system. In addition, the numerical flux of the reference FVS does not match the assumptions of Theorem 5. Since for systems of conservation laws bounds on the global discretization error are not available we focus in the following just on the stability of the compression scheme . That is, instead of balancing discretization and perturbation error we fix the threshold value c and explore the dependence of the perturbation error on the level L, the number of time steps n and the temporal step size 7. For this purpose, the perturbation error is also listed in the Tables 7.2, 7.3, 7.4 as far as the computation could be performed on the available computers. Here the error eL is measured in the weighted II -metric where each conservative quantity is scaled by its global maximum in the computational domain, i.e.,

with et.; := (eL,k, ihEh ' Recall that the thresholding is also performed by means of the scaled details , see (3.38). We observe that the error is proportional to the threshold value c . For the shock reflection this error is even significantly smaller because the solution is piecewise constant. Hence no error is introduced by the truncation in regions where the solution is constant. To some extent, this also holds for the implosion. In case of the wave interaction the solution is no longer piecewise constant but only piecewise smooth exhibiting a genuine two-dimensional structure. We conclude that for all test cases the perturbation error does not increase with increasing number of refinement levels L. Note, that for an additional refinement level the number of time steps is doubled and the temporal step size is halved according to the CFL constraint. An additional error is introduced because the local flux evaluation is performed by (4.15) and not by (4.13) which is used in the analysis. Investigations in [CKMPOl] show that this does not affect the perturbation error provided the reference FVS is higher order accurate. This no longer holds for a first order FVS . In this case a significant error is observed. We conclude the discussion on the parameter studies with a remark on the "ideal" computation in the sense of (5.2) . Whenever the perturbation error is smaller than the discretization error, we waste performance in the sense of computational time and memory size, because the discretization error is still dominating. Conversely, we loose accuracy if the perturbation error is larger than the discretization error. Therefore the ideal computation of the parameter study corresponds to the refinement level where both the perturbation error and the discretization error are balanced. Finally, we would like to discuss the limitations of the adaptive scheme . For this purpose, we present some plots of the error, see Figures A.13, A.14 and A.15. We observe that the error accumulates near discontinuities. This

7.1 P ar amet er Studies

131

seems to be a cont ra dictio n because near discontinuities t he adapti ve scheme is expecte d to locally refine t he grid. However , erro rs introduced away from t he discontinuities are transported to a shock layer by means of th e cha racte ristics intersecting t here. On t he other hand , t he reference FVS is known to smear contact discontinuities. This leads to a smoot hing of t he solut ion which is detected by the multiscale analysis, i.e., near contact discontinuiti es t he adaptive scheme does not refine t he grid until t he finest level is reached. This confirms t hat the adapt ive scheme can, of course, not improve the soluti on of t he reference FVS but it can only provide a perturbation of this solut ion in a much mor e efficient way. The previous discussion indicates th at t he ada pt ive scheme meets t he design criteria, namely, (i) the computationa l effort and th e memory requirements are proportional to t he numb er of cells in t he ada ptive grid and (ii) t he perturbati on error is proporti onal to t he t hreshold value. Since in our par ameter st udies t he threshold value e is fixed , Le., e f: c(L), we can not dir ectly conclude on t he discretization err or . If we assume that t he discreti zation error for t he uniform grids of level L behaves like 2- a L we see from t he tables at which level the record ed perturbation err or is of compa ra ble size which gives an impression of the int err elation between t he overall accuracy and t he amou nt of computationa l work . However , t he computat ions on the full uniform grid of level L can only be realized for a small number of refinement levels L du e to t he huge amount of memory needed. 7.1.4 Hash Parameters

The efficiency of th e computations is significant ly influenced by the initi alization of t he hash maps MOe , GOe , L and v _map, d_map as well as adapgrid_map by which we store t he mask mat rices, t he sequences of local average s and significant det ails as well as t he ada ptive grid . For t his purpose, we carry out two parameter st udies concern ing th e parameters rF ill and nFaetor , see Sect ion 6.3, for the implosion configurat ion. Here one of t he two par ameters is fixed while t he ot her varies, see Table 7.5 and 7.6. We are primaril y int ereste d in t he imp act of th ese par ameters on t he computationa l time, the size of allocated memory and t he act ua l fill rates of t he above hash maps. Fir st of all, we investigate t he influence of t he par ameter nFaetor by which t he length of t he hash table is varied. The larger nFaetor is chosen the smaller becomes t he fill rate dFill of t he hash table, i.e., t he probability for a collision decreases, and visa versa. For this par ameter st udy rFill is fixed by 1, i.e., the size of th e allocated memory heap for sto ring t he keys and values is t hat of t he worst case corres ponding to the uniform finest grid. Considering Table 7.5 we not e t ha t t he memor y size increases with nFaetor because t he hash table becomes larger. At t he same time, t he act ua l fill ratio of the hash maps is reduced. However, choosing nFaetor larger t ha n 3 does not significant ly affect th e computationa l time. Conversely, if we redu ce nFaetor t hen t he size of th e hash t able becomes smaller which result s in

132

7 Nume rical Experim ent s Table 1 .5 . Implosion: par am et er st udy for nFactor wit h rFill

~

I

1

2

3

4

CPU [sec]

51

50

49

fill rat e [%] Memory [MB]

58 7.9

29 8.5

19 9.2

50 15 9.2

InFa ctor

=1

L=4

L= 5 CPU [sec] 249 238 233 235 fill rate [%] 40 20 13 10 Memory [MB] 23 25 26 31

L=6 CPU [sec] 1068 1044 1030 1032 fill rat e [%] 29 15 10 7 Mem ory [MB] 64 71 75 79

more collisions. T his leads to a significant increase of t he computationa l time because accessing data in t he has h maps consumes more time du e to collision ma nage ment. T he result s in Table 7.5 suggest to choose nFactor = 3 in view of an opt ima l computational time. T able 1 .6. Implosion: par am eter st udy for rFill wit h nFactor

L= 4 rF ill

0.12 0.25 0.5

1

2

49 77 8.0

49 19 9.3

49 10 12

0.08 0.17 0.35 0.7

1.4

CPU [sec] 50 161 fill rate [%] Memo ry [MB] 7.6

49 39 7.9

L =5 rFill

238 234 236 234 234 C PU [sec] fill rate [%] 166 78 38 19 10 Memory [MB] 19 18 19 22 31

L=6 rFill

0.04 0.08 0.15 0.3

0.6

1060 1035 1029 1036 1026 CPU [sec] 243 121 65 33 16 fill rat e [%] Memory [MB] 36 38 38 40 55

=3

7.2 Real World Appl icati on

133

Another parameter st udy was perform ed for rFill predicting t he relative size of the ada pt ive grid in comparison to N L of the uniform finest grid. This fact or influences the size of t he memory heap for sto ring the keys and values as well as the length of the hash t able. Considerin g Table 7.6 we not e t hat the size of allocate d memor y is t he larger t he higher the value for rFill. At t he same time, t he fill rate of t he hash t ables decreases, Le., the number of collisions redu ces. Thi s has a positive effect on the computationa l time. The actua l fill ra te s for the implosion problem can be deduced from Table 7.3 as 0.45, 0.28 and 0.14 for t he refinement levels L = 4, 5 and 6. Then we conclude from Table 7.6 th at t he memory size and the computationa l time are balan ced if we choose these numb ers for rFill. Since in general t hese numbers are not available before th e computation has been carried out, t he value rFill has to be predicted . Here t he par ameter st udy suggests t ha t the pr edict ed value should be at least as high as t he real value, i.e., overpredicting rFill is preferable to und erpredicting. If it is smaller t hen th e computational time increases significant ly du e to a higher collision rate. Finally, we would like to remark that t he collision rat e can be reduced if an opt imal prime numb er , i.e., a good choice for a modulo numb er, is chosen for t he length of t he hash t able. However , investi gations in the context of t he ada pt ive scheme show that the effort for determining an optimal number is much mor e expensive t ha n collisions caused by our choice for th e modulo number .

7.2 Real World Application Cha racterist ic for real world applications are m ultidim ensional configurations with complex geomet ries. These configurat ions can only be adequa tely discretized by mean s of unstructured or block- structured grids, respectiv ely. Although unstructured grids, e.g. arbitrary triangul ations, are easier to generate we prefer st ructure d grids, because th ey can be aligned with t he streamlines in t he flow field . In part icular , this is useful near surfaces in order t o resolve boundary layers. On th e other hand , many applications are steady state problems which are determin ed by t he limit of inst ati onar y computati ons. In t his conte xt implicit schemes are preferable to explicit schemes in order to use lar ger time ste ps when approaching t he steady state . In t he following we will verify that t he ada ptive concept can also be applied to an implicit FVS although no rigorous ana lysis can be prov ided so far. 7.2.1 Configurations

We consider two configurations which are relevant for the design of an airplane wing. Burnp . This configuration is considered in [RV81]. It is determined by a wall

134

7 Numerical Exp eriment s

wit h a bump where the bump is a circular arc with a secant of length l = 1 [m] an d a t hickness of h = 0.042 [m], see Figur e 7.4. We impo se a homo geneous flow field cha racte rized by th e free- stream quanti ties. Here we consider a t ra nsonic probl em cha rac terized by th e Mach numb er , th e temp erature and t he pressur e. Cha racteristic for t his configuration are t he kinks in th e sur face which influence t he solut ion in t he neighborhood of th ese singularit ies in t he geomet ry. In addition, a local supersonic domain develops caused by t he expa nsion along t he bump surface provided th e free-strea m velocity is sufficiently high . Fur ther downstream t he flow is compressed aga in . This lead s to a subsonic domain. According to physical properties t he subsonic and t he supersonic domain are sepa ra te d by a compression shock. Profile. This configurat ion is determined by a profile of an airfoil, see Figur e

Unit s

U oo

M a oo

------- ------

U oo

0.8

T oo

K

poo

Pa 101300

285

Fig. 7.4. Initi al configurat ion for circ ular arc bump

7.5. Here we choose the SFB401 cruise configuration accor ding to t he BAC 3-11/ RES/ 30/ 21 profile of Moir, see [Moi94, Pie97]. Cha racte ristic for t his problem is t he nonsymmetric shape of the profile and the two inflection points of t he lower surface where the surface cha nges from convex to concave and visa versa. Here we only consider axis-symmet ric free- stream condit ions, i.e., t he angle of attack 0: is put to zero. For sufficient ly high free- stream velocities two sup ersonic domains develop at the upp er and lower surface accompa nied by a local shock. In addition, a slip line occurs emanating from the trailing edge which is not a straight line du e to t he nonsymmetric shap e of t he profile. 7.2.2 Discretization The computations are performed by mean s of th e new flow solver QUADFLOW, see [BBMOO, BGMH+01]. Thi s solver merges t hree basic tools, namely, (i) an implicit FVS on unstructured meshes, (ii) th e local multiscale ana lysis and (iii) a grid generator. The reference implicit FVS is a high- ord er ENG scheme defined on uns tructured grids. The core ingredients of this scheme are (i) a genuinely multidimensional linear reconst ruction, (ii) t he limiter due to Venkat akrishnan [Ven95] and (iii) t he HLL Riemann solver

7.2 Real World Appli cat ion

---

Units

U oo

M a oo

135

U oo

0.76

Too

K

285

p oo

Pa

101300

Fig. 7.5. SFB401 cruis e configurat ion

modified according to [TSS94, BLG97]. For th e time discretization we apply local time st epping, i.e., for each cell we determine a local ste p size Tj ,k by a pr escribed CFL numb er CT according to

Hence the solut ion pro ceeds differently in tim e for each cell. This is only justified in case of st eady state problems. The implicit time discreti zation is resolved by a Newton- Krylov typ e method based on a matrix free GMRES and an ILU preconditioner. At th e boundaries we apply eit her slip conditions at t he solid wall, i.e., the norm al velocity component vanishes, or cha rac teristic boundary conditions for subsonic inflow and outflow boundaries in case of the bump configuration . For the profile we use a point vortex correct ion according to Thomas and Salas [TS86] at the far field boundary. The ada pt ive implicit FVS can be derived ana logously to the explicit case. However we have to replace th e explicit FVS (4.1) by an implicit FVS that can be writ ten in th e form

Here t he influence of the implicitn ess can be controlled by t he par ameter B E [0,1 ], e.g. B = 0 (explicit ), B = 1 (fully implicit) . For our computations we always use t he fully implicit scheme. Applying the local multiscale t ransform ation to these equations we derive t he local evolut ion equa tions for t he adaptive grid

in ana logy to (4.11). Since t he reference FVS is an unstructured solver , we can perform t he local flux evaluation by (4.16) instead of (4.15). Hence t he grid need not be locally uniform . In t his case it suffices to choose the gra ding par ameter q such t ha t t he local t ra nsformations are feasible. According to Corollary 4 t his is gua ra ntee d pro vided q 2: s = 1. For th e computations t he threshold value is put to E = 10- 3 . Moreover t he ada pt ive grid is updated

136

7 Numerical Ex periments

afte r N adap t ime steps inst ead of each time st ep in case of an instationary pr oblem. T he underlying grid genera to r is based on a block-structured concept where t he computational dom ain [l is decompo sed into severa l blocks, i.e., [l = Uk [lk· In Figur es 7.6 and 7.7 the decomp ositi on of the computational domain s for the above t est configurations are sketc hed.

Fig. 7.7. Cruise configura tion

Fig. 7 .6. Circular arc bump

In each block [lk we apply a st ructured grid genera t or. In view of a sparse repr esent ation of the nest ed grid hierar chy it is prohibited to st ore t he grid points. To t his end, the grid hierar chy is det ermined by a grid function which map s t he paramet er dom ain R onto the block [lk , see Section 3.8. In order to provide a fast computation of th e grid points as well as to cont rol grid properties, e.g. ort hogonality, smoot hness, etc., we repr esent the functi ons by B- splin es, i.e. , d

x(~) =

L Pk IT Nt (~i) , i

k

i =l

where Pk E Rd denot e t he cont rol points and N i4 t he B- splin e basis fun ct ions of ord er 4. In genera l, t he number of cont rol points is much smaller t han the number of grid poin t s corres ponding to t he finest resoluti on level L , see

[BMOO] . 7.2 .3 Discussion of Results

For each of t he two configurations we perform severa l computat ions with increasing number of refinement levels L and a fixed threshold value c. T he characterist ic valu es for t he und erlying discreti zati ons are summa rized in Tabl e 7.7. As initi al grid we choose t he uniform grid of levell , becau se the

7.2 Real World Application

137

Table 7.7. Discreti zation par amet ers

# bump profil e

blo cks N x ,o x N y ,o NT N adap CT 5x5 200 20 50 3 4

5x 5

250

25

50

initi al dat a are constant in the entire computat ional dom ain according to the free- stream conditions. After N a dap time st eps we adap t th e grid to t he current flow field by means of the local multiscale analysis. The st eady st ate is app roached afte r NT time st eps. In Figures A.16, A.17 and A.20, A.21 the local flow field nearby th e bump and t he profile, respect ively, as well as th e corresponding adapt ive grids are pr esent ed. Not e, th at due t o the sub sonic free- stream conditions the computational dom ain has t o be chosen sufficientl y larg e in ord er to avoid perturbations from th e far field boundary. For our computat ions we use 1.5 and 2.5 of th e bump span before and behind t he bump, respectively, and 2 bump spans in height . For th e profile we use about 20 wing spans in each dir ection. The plots of th e num erical results again verify t he reliability of th e multiscale analysis. All physic al meaningful effects have been detect ed. For both th e bump and the profile the shock is recognized. In addit ion, th e stagnation points of the bump configuration which coincide with t he kink s in th e sur face as well as the slip line in the wake of t he profile are det ect ed . Not e, t ha t in t he lat t er case there is no ju mp in the pressure field and t he normal velocity field . Since no singularity has been imp osed by t he initial data it is remarkable that t hese discontinuities evolve with pro ceeding t ime. Therefore it is import ant th at det ails on lower scales may pr edict significant det ails on higher scales within the pr ediction step , see Section 4.1.2 . Table 7.8. Paramet er study for the bump L

NL

4

19200 9.5

5

Nt;; e M S % [min]

IlrLill IlrLlloo

14 2E-9 3E-7

96800 2.5

14 2E-8 3E-7

6 387200 1.1

25 1E-8 5E-7

71548800 0.5

47 2E-8 2E-6

Again t he computationa l complexity behavior in case of st eady st at e problems is docum ent ed in Table 7.8 and 7.9. We note that t he number of cells in t he final adaptive grid is significantly smaller than that of the uniform grid on the finest resolution level L. In par ti cular , th e relative number Ng decreases

138

7 Nume rical Experiments Table 7.9 . Param et er st udy for the pro file L

NL

4

25600 28

N g eMS [rz.lh % [min]

5 102400 15 6 409600 8 71638400 4

Ilr£\1 00

30 6E-7 2E-3 51 3E-7 8E-5 92 lE-6 2E- 5 164 6E- 7 8E-6

much mor e rapidly with increasing numb er of refinement levels for the bump as in case of t he tes t configurations considered in t he pr evious section . This can be explained by t he local st ruct ure of t he singularities. For the profile t he relat ive number N9 does not decrease so rapidly with increasin g refinement level because t he slip line t raverses th e ent ire wake whereas the singularities occur ring in the bump configuration are local. In contrast to inst ati onar y problems, we have to consider t he asy mpt ot ic behavior of the solut ion which is expected to approach a steady state . An indicator is defined by t he difference rj,k := vj,t l - vj,k of the conservati ve qu anti ties on two successive time levels which frequently referr ed to as the (temporal) residual. In Table 7.8 t he maximal residualllrLlloo and the average residual llrLill defined by

IlrLlloo

:= . max

IlrLill

:= . max

. max

t=l, ... ,m (J,k )E9 L .€

'l.=l, ... ,m

L

lrJ ,k,il (I VJ,kl/lnl)l rj,k,il

(j, k) E9 L ,€

are listed . For our computations t he volume of t he computational domain is about Inl = 10 (bump) and Inl = 1271 (profile) . It t urns out that the maxim al and average residu al are redu ced by four ord ers of magnitude. during t he computations for both configurations. Thi s is verified by t he plots of t he tempora l variation of t he maxim al residu al for t he density, see Figur e A.18 and A.22. We observe tha t between two adaptation ste ps the residua l decreases almost monotonically. However afte r each grid ada ptation th e residual significantly increases by severa l ord ers of magnitude. This is caused by t he t hresholding t hat is performed wit hin t he local multiscale ana lysis. Note , t hat t he new residu al after adapting t he grid is in t he order of the t hreshold value c. Nevert heless the residu al decreases more st rongly after each ada ptation ste p . Approaching th e steady state in not only reflected in t he decrease of the residu als, but also in t he ra tio of the numb er of grid points before and after t he adaptation step . In Figur e A.19 and A.23 this ratio is presented . Here aga in we observe an asymp totic behavior. In the limit we approac h 1, i.e., t he number of grid points is maintained.

A Plots of Numerical Experiments

Fig. A.I. Shock reflect ion: density cont our s (L

= 8, T = 0 [s])

Fig. A.2 . Shock reflecti on : ada ptive grid (L = 8, T = 0 [s])

140

A Pl ot s of Nu merica l Ex periments

Fig. A.3. Shock reflectio n: densit y contour s (L

Fig. A.4. Sho ck reflection : adaptive grid (L

= 8,

= 8,

T

T

= 3.2 X 10- 4

= 3.2 X

[s])

10- 4 [s])

A Plots of Numerical Experiments

Fig. A .5 . Implosion : densit y contours (L = 9, T = 0 [s])

Fig. A.6. Implosion : adaptive grid (L

= 9, T = 0 [s])

141

142

A Plots of Numerical Experiments

Fig. A.7. Implosion: density contours (L

Fig. A.B. Implosion: adaptive grid (L

= 9,

T

= 3.6125 X

10- 4 [s])

= 9, T = 3.6125 X 10- 4

[s])

A Plots of Numerical Exp erim ents

Fig. A.9 . Wave int eracti on: density conto urs (L

Fi g. A .lO. Wave inte rac t ion: ada pt ive grid (L

= 6,

T

= 0 [s])

= 6, T = 0 [s])

143

144

A Plots of Numerical Exp erim ents

F ig . A. l1. Wav e int eracti on : density cont ours (L = 6, T = 0.15 [s])

Fig. A .12. Wave inter acti on : ada ptive grid (L

= 6, T = 0.15 [s])

A Plots of Numerical Exp erim ents

Fig. A .13 . Shoc k reflection: conto urs of erro r in density (L

= 7, T = 3.2 X 10- 4

Fig. A.14. Im plosion: contours of error in density (L = 7, T = 3.6125

X

145

[s])

10- 4 [s])

146

A Pl ot s of Numerical Experiment s

Fig. A .I5 . Wave int eraction: conto urs of error in density (L

= 5, T = 0.15 [s])

A Pl ot s of Numerical Experiment s

Fig. A.16. Bump: Mach cont ours (L

-

I

-

Fig. A .17. Bump: adaptive grid (L

= 7)

= 7)

147

148

A P lots of Numerical Experiments

0 .1 0 .01

..t+ +

++ + ++ + ++

++

0.001

+

,t++*+

+

0 .0001

+ +

+++

+:

++

+

+

trip- \.+

1e-05

t+ ~.t

~

~

~ 1j..

+++

V~+~ .;+;.t ......

1e-06 1e-07

.... +

++

o

80

40

120

160

200

n Fig. A.IS . Bump: temporal variation of maximal density residu al (L

4 0

3 .5

-..-l

.w

ro

H

3

.w 2.5 s:: 0 2

•..-l

0..

ro 1.5 H o 1

•..-l

+

0 .5 0

40

80

120

160

200

n Fig. A.19 . Bum p: te mporal variation of grid size rati o before/after ada ptation (L = 7)

= 7)

A Plots of Numerical Experiments

Fig. A.20. Profile : Mach contours (L = 7)

Fig. A .21. Profile : adaptive grid (L

= 7)

149

150

A Plots of Numerical Experim ents

0.1 0 .01 r-I

cO

0.001

;::::l

+

+ +

\

+

+

'D

•.-1

Ul Q)

p::

0.0001 1e-05 1e-06

o

50

100

150

200

250

n Fig. A.22 . Profil e: t emporal vari ation of maximal density residu al (L

0

4

•.-1

4-l

cO H

3

4-l

I=:

-.-I

0

o,

2

ro

•.-1

H

1

(9

0 0

50

100

150

200

250

n Fig. A.23 . Profile: t emporal vari ation of grid size ratio before/after ada ptat ion (L = 7)

= 7)

B The Context of Biorthogonal Wavelets

This section provides a short introduction to (biorthogonal) wavelet t heory. Here we focus on some basic aspects that are restricted to our purposes. For a general overview on wavelet theory we recomm end st andard t extbooks such as t he recent book by A. Cohen [CohOO] . We start with a genera l multiresolution analysis following th e methodology of W . Dahmen [Dah94, Dah95 , Dah96] and collaborators [CDP96]. In particular, we out line the connection between stability and biorthogonal wavelets. As an example for th e general setting we consider wavelets on t he real line according to [CDF92].

B .l General Setting In t he sequel, we consider a Hilbert space ! H = 1-l(rt) of functions endowed with an inner product (-, -)t{ and associat ed norm II . lit{ := (-, . ) ~2 . Then a sequence S = {Sj LENa of closed linear subspaces Sj C 1-l is called a m ult iresolution or multiscale sequence, if the subspaces are n est ed, Le., So C Sl C .. . C

s, C Sj+l

C .. . 1-l

and S is dense in 1-l, i.e., clos-,

(U .

l ENa

Sj) = 1-l .

The nest edness of the multiresolution sequence implies t he existe nce of the compl em ent or wavelet spaces W j such t ha t

1

Not e, that st ability is best investigated in £ 2 . Similar results ca n not b e derived in £1. However , in Sect . 2, t he box wavelets and their modification hav e to b e scaled with respect t o £1.

152

B The Context of Biorthogonal Wavelets

B.l.l Multiscale Basis

We now assume that the subspaces Sj and W j are spanned by sets of basis functions Pj := {cpj,k; k E I j},

enumerated by the index sets I j and Jj, i.e., Sj = clos-, (span{Pj}) = : S(Pj),

W j = clos-, (span{tlij}) =: S(tli j).

We refer to Pj as the basis of scaling junctions? and tlij as the wavelet basis. Here the index sets are either finite or at most countable sets . In particular, in the finite case we have N j H = N, + K j with N j := #Ij , K j := #Jj. For later use it is convenient to interpret a set of functions G j , e.g., G j = Pj or Gj = tlij, as a column vector with respect to a fixed but unspecified ordering of basis functions . Then it is possible to represent any function U E span{G j} by U

=

GJ u := L

Uk OJ,k,

k

The collection {Gj}jENo is called uniformly stable, if there exist constants c, C > 0 independent of j such that the Riesz property

(B.l) holds for all u E R#8; (see [CDP96]). In particular, if Gj is an orthonormal system, then the Riesz property holds with" = " instead of ":s" and c = C = 1. The Riesz property (B.l) with uniform constants ensures the uniqueness of the expansion of U with respect to the basis Gj . If the basis G j satisfies the Riesz property then it is called a Riesz basis for the space S (G j) . Keeping our application in mind , we confine the discussion to uniformly stable Riesz bases, Le., {Pj}jEN o and {tlij}jENo satisfy the Riesz property (B.l) . In the context of principal shift-invariant spaces Sj, i. e., if 1i is a Hilbert space over Rd and S satisfies

f

E

s,

if and only if

the notion of a multiscale sequence coincides with the concept of Mallat's multiresolution analysis [Ma189]. An example is discussed in Sect. B.2. From the nestedness of S and the uniform stability of the Riesz bases, we conclude the existence of bounded linear operators Mj,o and Mj,l such that the twoscale relations (B.2) or componentwise 2

Here the notion of scaling function is used in a generalized sense .

B.1 General Set ting . t.
""

L.J

=

153

j ,l

mr,k 'Pj+l ,r

rEIj+l

hold . T hese relati ons can be int erpret ed as a cha nge of bases realized by t he mat rices

j,o) M j,O = (mr,k rEIj+l ,kE Ij

,

M j ,l =

(

j , l) mr,k rE Ij +l ,k E J j ,

pr ovid ed t hat the inverse of t he composed matrix Mj := (M j ,o, Mj ,l) exists. Note , t hat in pr act ice a two-scale relati on for t he sca ling fun ctions P j is known. Then t he t ask is to determine Mj, l such that t he composed matrix M j is invertible. If t he P j are uniformly stable t he existe nce of the inverse Gj := Mj l t ogether wit h the uniform boundedn ess of Mj and Gj , i.e. ,

j

~ 00 .

(B.3)

is equivalent t o t he uniform stability of P j U lJIj , i.e., P j U lJIj is also a Riesz basis for S ( P j +l) = S(P j U lJIj) , cr. [CDP96] . According t o t he blocking of Mj it is convenient to repr esent t he inverse by Gj := (G J,o , GJ,l)T where t he blocks are det ermin ed by

) Gj ,O = (g rj,o ,k rE Ij+l ,kE Ij'

Gj , l = (g rj,,kl ) r EIj +l ,k E Jj

.

In particular , t hese blocks fulfill G j ,i M j ,i l

= 6i,id ,

i ,i' E {O, I} .

(B.4)

By mean s of t hese identities we dedu ce t he two-scale equation T

p JT+ 1 = p J GJ·'

°+ lJITJ G '

J· 1

(B.5)

or component wise _ 'Pj+ l,k -

""

L.J

rEIj

j,O gr ,k 'P j,r

+ "" . /, L.J g rj,l ,k 'i'i.»: rEJj

Note , t hat we can equivalent ly switc h between t he two-scale represent ati ons (B.2) and (B.5 ) where we employ t he ident iti es (B.4).

B.1.2 Stable Completion Fro m t he discussion above we conclude t hat t he task of det ermining a basis for the complement spaces is dir ectl y connected to t he task of det ermining an appr opriat e matrix Mj ,l , i.e., t he problem has been tran sferred from fun ctional analysis to linear algebra. This is summarized in t he following definition.

D efinition 10. ( Unifo rml y sta ble com pleti on) A n y m atrix Mj ,l E [RN i+ 1 , RKjJ satisf yin g th e relati ons (B.4) and (B.3) is called a stable complet ion of Mj, O'

154

B The Context of Biorthogonal Wavelet s

Although the problem of const ruc t ing a wavelet basis has now become a pure algebra ic problem it is nevertheless not easy to solve. However , the set of all stable compl etions of Mj,o can be char acterized if just one st abl e complet ion Mj ,l of Mj,o is known. Theorem 8. (Change of stable compl etion [CDP96j) L et Mj ,l be a fixed st able completi on of Mj,o and t , E [R K ; , RN ; ], Kj E [RK ; , RK ; ] uniformly bounded lin ear operato rs such that Kj is regular. Th en

(B .6) is also a stable completion of Mj,o. In particular, the inverse of Mj = (Mj ,o, Mj,l) is det erm in ed by GJ = (G j ,o , Gj ,d with

(B.7) The reverse st at ement is summarized in t he following theorem. Theorem 9. (Chara cterization of stabl e completi ons [CDP96j) L et M j ,l, Mj ,l be two stable compl etion s of Mj ,O. Th en th ere exist uniformly bounded lin ear operators i, E [R K ; , RN ; ] , K j E [R K;,RK;] with in vers e Kj 1 E [RK ; , RK ; ] su ch that (B .6) and (B.7) hold.

In te rms of t he new st abl e complet ion, t he modified wavelet basis can be repre sent ed in t erm s of the old wavelet basis tPj employing t he matrices Lj and Kj , i.e., T tJfj

T

T

•

=

T +
·T

= tJfj Kj

+

(B .8)

Not e, th at the basis
So far we have only considered t he relations between two successive levels. Now we are int erest ed in t he multisc ale case. For t his purpose, we introduce the multiscale basis L-1

U

tJf(L) :=

tJfj ,

tJf- 1 :=
j=-l

By means of t his basis an element u E SL can be exp anded with respect t o the bas es
U

=


(tJf(L ) f d (L)

=

L

j= - l

tJf[

a,

(B.9)

B.l General Setting

155

with the multiscale coefficients d (L )

--

T (d T-1' d 0"

" ,

d L-1 )T ,

(B.lO)

In the sequel the two different basis expansions are referred to as single -scale representation and wavelet or multiscale representation, respectively. If the systems {Pj}jEN a and {Pj U Yij}jEN a are uniformly stable Riesz bases , then there is a bijective linear operator ML : RNL --+ RNL which performs the change of bases between PL and Yi(L), see (B .9), i.e., d(L)

= MLUL

and

UL

= M:L1d(L).

The operator ML is called the multiscale transformation and M:L 1 the in verse multiscale transformation. In order to obtain an explicit representation of ML and its inverse we conclude from the two-scale relations (B.2) and (B .5) that the coefficients of two successive levels are related by Uj+l

= Mj (~~)

,

(~~) = Gj U j+1 '

(B.ll)

Iterative application of (B.ll) yields the pyramid schemes in Figures B.l and B.2 which are similar to the up - and down -sampling in signal processing. In matrix notation, the multiscale transformation is given by ML

=

Go '"

GL-1'

M

-1

L

= -ML-1'"

-

(B.12)

Mo

with the matrices

Analogously to the two-scale setting we are again interested in the stability of the transformation and the uniqueness of the basis expansion. These correspond to the uniform Riesz basis property (B.l) for the multiscale basis Yi(L) , L E No. Investigations in [CDP96] show that this property is equivalent to the uniform boundedness of the multiscale transformations, i.e.,

L --+

00,

where the index set J(L) := Uf:~l Jj, J- 1 := 10 , corresponds to the multiscale basis Yi(L). This condition is equivalent to the existence of a dual Riesz basis for H . Theorem 10. (Dahmen [Dah94J) If the bases {Pj }jEN a are uniformly stable, then the multiscale transformation is uniformly stable if and only if Yi := Po U {Yij LENa is a Riesz basis for H. In this case, there is another Riesz basis tP := 1>0 U {tPj }jENa of H which is biorthogonal to Yi, i. e., (Yi, tP) = I and u E H has the unique expansion u=

L (u ,
jENa kEJj

(u, ;}j,k)ll7/Jj,k .

(B .13)

156

B The Context of Biorthogonal Wavelets Mo ,o

MO, l

U2

-:

.> do

-

M l ,o

Ul

Uo

dl

Ml,l

.>

-

M L- l ,o U L-l

.>

d2

UL

.>

d L- l

M L-l , l

Fig. B.l. P yr ami d scheme of the mul t iscale tran sformati on

UL- l

-

d L- l

G L- 2 , l

GL-l,O

UL

<.

G L- l , l

-

G L - 2 ,O

U L- 2

<,

<, d L- 2

<.

-

Go,o

Ul

d1

~ GO, l

Uo

do

Fig . B.2. Pyr am id scheme of t he inverse mul t iscale tran sform ati on

We emphas ize t hat t he uniform stability of the multiscale bases {ciioutPou ... u tPL}LE N can only be expected to hold in a Hilbert space, e.g. 1i = L 2 . In the present work , however , the relevant to pology here is ultim ately determ ined by £1. In particular , we make essent ial use of t he uniform stability of t he levelwise bases { ciij hE No and {ciij u tPj h ENo in L 1 . In the sequel, t he bases Pj and lJfj are always referred to as prim al scaling functions and prim al wavele ts, respectively. Ana logously, the bases cii j and tPj are called the du al scaling functions and dual wavele ts , respectively. This terminology is ofte n used to express th at t he functions und er consideration are expa nded in t he primal bases while bounded linear fun cti onals are repr esente d in t he dual basis. In parti cular , if th e bases {ciij h ENo and {cii j UtPj} jENo are uniforml y stable and there are two-scale relations of t he form (B.2) and (B.5) for t he du al bases with appropriate matrices Mj,i' Gj,i ' i = 0,1 , then t he following relat ions hold between t he primal and dual setting, see [CDP96], (B.I4)

(Pj , tPj hi = 0 , (lJfj ,ciijhi = 0 ,

(B.I5)

Sj -LWj := spa n{tPj), Sj := spa n{ciij }-LWj.

(B.I6)

An exa mple for t he bases Pj, lJfj on t he primal side and ~j , t-Pj on t he du al side is given in Sect. B.2 within t he context of principal shift- invaria nt spaces. Note t ha t t he dua l multir esolution sequence S = {Sj} jENo is also a dense sequence of closed linear subspaces in 1i and sati sfies relations ana logously to (B.2) and (B.5).

B.2 Biorthogonal Wavelets of t he Box Funct ion

157

Fin ally, we would like to remark t hat in t he literat ure t he concept of pr ima l an d dua l wavelets is referred to as t he concept of biorthogonal wavelets. For t he const ruction of biorth ogonal wavelets in one dimension see [CDF92] (f? = R) and [DKU99] (f? = [a, b] C R).

B.2 Biorthogonal Wavelets of the Box Function In t his sect ion, we consider an example t hat concerns t he class of prin cipal shift -invariant spaces where the multi scale sequence S coincides with t he classical concept ion of a multiresolution an alysis [Dau88, Mal89, Mey90, JM 91, Chu92, CDF92, Dau92]. Here, t he Hilbert space is H = L 2 (R d ) , and t he subs paces S j are generated by mean s of one single functi on t.p E L 2(R d ) . Neste dness and stability imply that t.p is refinable, i.e., there is a sequence a = {ak h Eld E l2(Zd) denot ed as mask of t.p such t ha t t.p (x)

=L

a.e.,

ak t.p(2 x - k ),

(B.17)

k EZd

is satisfied. We star t with spaces j E Z,

(B .18)

with a countable basis consist ing of t he functions j E Z, k E Zd.

(B .19)

Here t he normalization factor 2j d / 2 is chosen such t hat

B .2.1 Haar Wavelets

As a spec ial case, we consider t he box function or B - splin e of order 1 in one spatial dimension 1, 0 < x < 1 t.p (x ) := X[O ,l) (X) = { 0, elsewhere '

(B.20)

which obviously satisfies t.p(X) = t.p(2x)

+ t.p(2x

- 1),

(B .21)

i. e., ao = a l = 1 an d ak = 0 ot herwise in (B.17). Haar [Haal 0] established already in 1910 t hat t he system j ,k E Z,

158

for

B The Context of Biorth ogonal Wavelet s

I , 0 :::; x < 1/2 'Ij; (x ) :=
(B.22)

is an orthonormal basis for L 2 (R) which is called Haar basis. Since ort honormal bases are, in par ticul ar , Riesz bases (wit h
is also an ort honormal basis for L 2( [0 , 1]) due t o t he sma ll compact support of sp and 'Ij; . Another advantage of th e Ha ar bas is is t hat th e multiscale transformations ar e very efficientl y execut able which is also cau sed by th e minimal compact support. For the int erval [0,1], t he mat rices Mj and Gj are banded with

o o

1 ( _1 )i

o o

o

1 ( _1 )i

(B.23) 1

o for i i. e.,

( _1 )i

0 0

0 0

( - 1)i

1

= 0,1. Unfort unately, t he Haar wavelet has only one vanishing moment , ! 'Ij; (x ) dx= O, R

!

xm'lj;(x) dx -=f 0, mE N.

R

B .2 .2 Biorthogonal W avel e t s on the Real Li n e

In view of a high compression of th e multiscale coefficient s, we ar e int erest ed in another function 'Ij; th at has mor e vanis hing mom ent s. In [CDF92], a mor e suitable Riesz basis is constructed. First, it is pr oved t hat for each fixed odd number N = 2m + 1 > 0, t here is a du al funct ion 0 E L 2 (R) = 1,fir0 E L 2 (R) which is refinab le, has compact support [1 (
such t hat t he sets

N, N] and satisfies

= 6k,O,

k E Z,

B.2 Biorthogonal Wavelets of t he Box Fun ction

~j

q,j = {
159

= {
are biorthogona l to each ot her , i. e.,

where

(B.24)

kE Z

-

-

has only non- vanishin g elements for k = 1 - N, ... ,N and can explicit ly be determined as coefficients of th e Laurent polynomial

t_

ak z k = (4z)-m(z

+ 1)2m+l

k=l-N

r: n)

f

(-4 z )-n(z _1) 2n. (B.25)

n=O

Addi tionally, it is shown in [CDF92] that t he functi ons N L bk
'ljJ(x)

=

¢(x)

=
-
(B.26) (B.27)

with b = {bd kEZ and bk := (-1) k+l a1_k generate the complement spaces

Wj

S('I! j ) =

clOS£2 (R)

(span {'ljJj ,k IkE Z}),

w,

S(if,j ) =

clO S£2 (R)

(span {¢ j ,k IkE Z}) ,

where v s» and 'ljJj,k are defined ana logously to
160

B The Context of Biorthogonal Wavelets

o

o '. (1_N+ l (1 _N+2

1

" . b_ N+1 b_N+2

0

0

(1_N+3 (1-N+l

1

0 0

b_ N+3 b_N+l

J2

J2 aN

b-N

aN _ 2 aN _1

o o

o o

aN

o

o A fur ther res ult of [CDF92] is that