High Accuracy Non-Centered Compact Diffe

0.3. High-order schemes

9

Some similar ideas were proposed in [44], the slight difference being due to the way of using u£*'. The idea of using the differentiatioin of governing equations to obtain higherorder schemes was also exploited in [24] when constructing fourth- and sixth-order methods for convection-diffusion equation.

Compact schemes

So far, "conventional" type of discretizations was discussed in context of constructing high-order schemes. The alternative approach is based on the so called compact approximations. Just to fix the idea, let us consider the problem of numerical differentiation, that is calculating the approximate values of f = du/dx

(0.10)

for known values of u in the grid points x, = jh, j = 0,1,..., J. fi - f(*i)

Usually, we define

«

where o j , k — —q, —q + 1, are some constant coefficients. In contrast to Eq. 0.11, compact approximation of Eq. 0.10 has the form 0-lfj-i

here cti and

+0 fi O

+ I = C..,Uj_| + C. Uj + 0

+

(0.12)

(j = —1,0,1) are some constants.

If three-point stencil is considered, then Eq. 0.12 provides higher order of approximation than Eq. 0.11 with q = 1, r = 1 due to two additional coefficients on its left-hand side. Using them, one can annihilate the truncation error for Eq. 0.12 up to 0(h ) thus obtaining the fourth-order approximation. 3

Of course, some additional computational efforts are needed to get fj from Eq. 0.12 while solving the system with tridiagonal matrix. However, if Eq. 0.12 is used as a part of some difference scheme then operation count may be approximately the same as in the case of usual implicit schemes. In general, some advantages of using Eq. 0.12 are as follows:

Introduction

II)

(i) high-order approximation is usually accompanied by relatively small numerical coefficient in the truncation error thus providing significant accuracy of the discretization; (ii) small number of grid points in the stencil of a scheme facilitates the formulation of numerical boundary conditions and treatment of near-boundary points. (iii) using Eq, 0.12 as a part of implicit schemes, the inversion of implicit operator can be reduced in many cases to the solving of tridiagonal systems. Henceforth, we will consider as compact approximations those high-order formulas for the discretization of derivatives in which three-point stencil is used. Compact approximations will be referred to as centered and non-centered ones. We will call them centered if they are formed by self-adjoint and skew-symmetric three-point operators. For the particular case Eq. 0.12, this implies 0_i =0i,

a., = -Ot,

that is symmetry and antisymmetry with respect to the mid-point x = Xj. The centering also means that the coefficients in Eq. 0.12 or other compact formulas are the same for each interior grid points, thus being independent of the differential equations which are discretized. On the contrary, non-centered compact formulas use essentially asymmetric operators which can provide positive approximations for convection-type terms. To achieve this, the coefficients in Eq. 0.12 are in general dependent on these terms and may be different for different grid points in the same way as the usual upwind schemes. Although centered compact differencing seems to be known for a long time (see, e.g. [75], (26]), the first publications which describe their implementation in difference schemes appeared somewhat later than the first paper with the description of non-centered compact schemes [107]. Several authors proposed fourth-order symmetric compact schemes for convection-diffusion equations almost simultaneously ([27], [52], [81], [1], [82]) using the centered formula Eq. 0.12 for the first derivative and the same type of relation for the second derivative. Another approach to construct fourth-order compact schemes is to set fj = ( i u ) , where L is the differential operator to be discretized ([63], [25], [12]). For

0.3. High-order schemes

11

convection-diffusion equations, the advantage of this method is that only tridiagonal form of difference equations is to be solved, whereas other fourth-order schemes reduce to block-tridiagonal systems with 2 x 2 blocks. A comprehensive review and analysis of centered compact schemes are beyond the scope of this book. Some of them will however be briefly discussed in Chapters 2 and 5. Here we only note that in general centered compact schemes have a tendency to cause strong spurious oscillations in the presence of steep gradients of solutions. To damp them, an artificial dissipation is often needed which may offer some difficulties while numerically simulating high Reynolds number flows. However, centered high-accuracy schemes seem to be very useful for convection-diffusion problems which correspond to low and moderate Reynolds numbers or other parameters which characterize the role of diffusive terms. The reader concerned may be referred to the original papers mentioned above as well as the monograph [75] for more details. Non-centered compact schemes, the main subject of this book, may be thought of as upwind counterpart of schemes with centered compact approximations. Due to upwinding, they proved to be an efficient tool for discretization of convection terms described by non-self-adjoint operators. On the contrary, the way of discretization of self-adjoint operators which present diffusive terms in convection-diffusion problems is regarded here as not a very important part of a difference algorithm. In contrast to centered compact schemes, the target problems while using the present approach seem to be pure convection or convection-dominated convectiondiffusion problems. In particular, high Reynolds number flows can be successfully simulated using compact upwind methods. All types of compact approximations may be recognized as difference forms of Pade (or Pade-like) approximants applied to the Taylor expansion series for some difference operators. It will be shown that in the non-centered case two types of formulas can be obtained depending on the way in which Taylor expansions are used for the first-order upwind operators. Chapter 1 of this book deals with the basic ideas of the third-order compact upwind difference schemes (or, briefly, CUD-3 schemes) applied to scalar conservation law as a representative example. These ideas marked the beginning of the development of the non-centered compact methods. The prime object of this Chapter is to explain in a simple way the main properties of CUD-3 approximations, namely, their stability, conservation property, dis-

12

Introduction

persion and dissipation. Special emphasis is placed here on the combinations of upwind compact differencing as a spatial discretization with a variety of time-stepping devices. In particular, it is shown how to construct two-level compact scheme which is of third-order accuracy in space and time. Chapter 2 contains some extensions of the elementary theory presented in the previous Chapter. In particular, we focus on the generalization of the compact upwind operators. In fact, it will be shown that any first-order differencing operator written in the flux form with positive numerical viscosity may serve as the generator of the third-order generalized compact upwind differencing. Using this concept, it is possible to control some properties of CUD-3 schemes. One of them, the so-called entropy-consistency, is important when computing weak discontinuous solutions of conservation laws. It will be demonstrated in Section 2.2 that the proper forms of CUD-3 schemes can be used not only as high-accuracy schemes but also as high-resolution shockcapturing methods. In this context, flux correction procedure is discussed which may be useful at some instances as supplementary tool for obtaining wiggles-free solutions. Considerable attention in Chapter 2 is paid to the second type of third-order compact differencing (CUD-II-3). The concluding sections of Chapter 2 contain additional information on CUD-3, In particular, the fourth-order compact differencing is regarded here as a result of symmetrization of CUD-3. Finally, the relation between third-order compact differencing and Pade approximants is discussed. Chapter 3 is devoted to the principles of constructing fifth-order compact upwind difference approximations (CUD-5). Two types of these approximations are described which may be regarded as extensions of CUD-3 and CUD-II-3. The latter type is derived using Pade approximants. It is demonstrated that fifth-order CUD are far more accurate than their third-order counterparts not only due to higher order truncation error but also due to small numerical coefficients in these errors. The main properties of CUD-5 and CUD-H-5 are investigated. In particular, it is shown that these approximations can be incorporated in high-resolution schemes for discontinuous solutions. A general approach to the derivation of arbitrary n-th order compact upwind differencing is described before the end of Chapter 3. In Chapter 4, the extension of compact upwind technique to systems of hyperbolic conservation laws is outlined. To construct positive approximations, the

0.3. High-order schemes

13

diagonalization of the Jacobian matrices is used. Considerable attention is paid here to the investigation of conservation properties and stability of resulting schemes. When constructing positive operators which are not based on diagonalization, the flux-splitted forms of compact upwind differencing are suggested. Finally, highresolution properties of non-centered compact schemes are illustrated by numerical data for Riemann problem. They were obtained using the combination of fifth-order compact spatial differencing and Runge-Kutta time-stepping. Chapter 5 deals with applications of compact upwind approximations to convection-diffusion equations and systems. The central idea here is that CUD operators for convective terms can be combined in many ways with various discretizations of diffusive terms. The examples of such combinations are presented in the framework of CUD-3 schemes. Essential part of Chapter 5 is the illustration of the fact that compact upwind schemes can be extremely accurate when the solutions of governing equations are smooth. Numerical results are described for the Burgers equation and several formulations of problems. In particular, the numerical solutions for small viscosity case obtained when using adaptive grids are presented. In Chapter 6, multi-dimensional case is considered with one-dimensional compact upwind operators corresponding to each spatial coordinate. The attention is paid to the approximate factorization schemes with CUD-3 approximations. It is shown that in two-dimensional case this combination is quite suitable for both unsteady and steady-state calculations. In the context of solving steady-state problems, the defect correction approach is discussed. The choice of low-order operators which play the role of preconditioned in iterative processes is investigated using one-dimensional analysis. Chapter 7 contains some illustrations of using compact upwind schemes in the computations of viscous gas flows described by the compressible Navier-Stokes equations. Several types of algorithms are considered. They include line-relaxation method, approximate factorization schemes, defect correction iterative processes. Numerical examples cover the area of hypersonic slightly rarefied gas flows with velocity slip and temperature jump boundary conditions, external separated flows, internal flows in nozzles, channels and cascades, numerical examples Finally, Chapter 8 is concerned with the applications of compact upwind differencing to incompressible flow methods. Theoretical considerations received the

14

Introduction

most attention in this Chapter. In particular, quite general iterative approach to the reconstruction of velocity fields with given vorticity in three-dimensional case is described, the convergence being very fast and independent of mesh sizes. Some simple numerical examples are presented. To concentrate on the main properties of compact non-centered schemes, we avoid in this book any discussion concerning their implementation in the algorithms for different types of parallel architecture. This important topic can be the subject of an investigation.

Chapter 1 Third-order schemes with compact upwind differencing

This chapter describes the basic ideas of the third-order compact differencing which happened to be the starting point for the development of non-centered compact schemes. These ideas are applied to the nonlinear scalar conservation law as the representative equation. Compact approximations are regarded here mainly as a tool for high-accuracy discretization of the spatial derivatives in time-stepping methods, the time discretization being thought of as an additional device which can be chosen in many ways. Of course, third-order compact differencing may be used in iterative algorithms as well.

1,1

Third-order compact differencing and corresponding schemes

1.1.1

Derivation

of compact differencing

formulas

We start with the trivial arguments which in the early seventies triggered the development of the methods described in this book. Suppose we have sufficiently smooth function u{x) and its derivative / = 8u/dx.

16

Third-order schemes with compact upwind differencing

Introducing the uniform mesh LJ/, {XJ = jh, j = 0 , 1 , N } which cover the interval [xo, X.f/\, we integrate the equation Ox over the intervals

[2,-1, xi\: «,--«*_, = £

f(x)dx.

(1.1)

Using now the quadrature formula for the RHS of Eq, 1,1 which contains the values of / in Xi-i, %i and x , we obtain i+l

4

u, where / ; =

= (5/,_, + m ~ /i+0/12 + 0(h ),

(1.2)

n

f(x,).

Similarly, the integration over the interval [xj, Xj+i] gives u ,-u, i+

4

=

/ , + 5/, )/12 + 0(A ).

+ 8

(1.3)

+1

Note that Eq. 1.3 can be considered as the Adams formula for ordinary differential equation dv/dx — f(v,x). Of course, there are many other ways to derive Eqs. 1.2 and 1.3. For example, we may start from the special case of the Hermite formula ft/i-i

+0ifi + ft/i+i = mm^i+mm+mmt'U

(1-4)

Substituting Taylor expansion series for u,_i, u ; i , fi-\, fi+i and requiring that /, = {dujdx) + 0{k }, we obtain the coefficients o,-, 0; [i — 1,2,3) as functions of one parameter. Choosing a as a free parameter, we get +

3

I=Ii

2

1 + ha

2

1 - hcx

1

2

a

ft=

+

;

ha

2

2

1

> = -^k—' *—%r* sx * - f Setting q = 0 in Eq. 1.5, Eq. 1.4 reduces to the well-known equality

ha

2

,, ., (i.5)

2

4

which gives / = ( 3 u / c ) i ) | , + 0(fc ). I=I

However, henceforth we will be mainly interested in non-symmetrical formulas Eq. 1.4 with o / 0. As will be shown, these formulas may offer certain advantages over their centered counterpart given by Eq. 1.6. 2

i.l.

Third-order compact differencing and corresponding schemes

17

Setting c<2 = 1/k and cv — —1/h, we obtain Eq. 1.2 and Eq. 1.3 respectively. It will be evident from Chapter 2 that more general situation may be considered when non-zero values of a is regarded as some upwinding parameter. 2

2

Let us write now Eqs. 1.2, 1.3 in the operator form. A _ j = hA fi, U

A+u,- =

+

hA-fi,

where A and A _ are the operators of forward and backward two-point differences while A- and A+ are the operators of quadrature formulas for the right ( [ i , , and the left 3,]) intervals respectively. Thus we have +

A _ , = u, -

Af

u

A u +

;

= u

i + l

+ ;

- ut,

= ( 5 / ^ + 8/, - /,+,)/12,

A.fi = ( 5 /

1+1

+ 8/; - /.-0/12-

(1.7)

The third-order compact differencing relations can then be presented as /; = A^'A^/h

= (du/dx)

3

I=Xi

+ 0(k ).

(1.8)

In further discussion grid functions indices will possibly be omitted for brevity. It follows from the equalities Eq. 1.8 that to obtain the values /, it is necessary to invert the operators A+ or A- i.e. to solve linear system with tridiagonal matrix in the same way as it is done for numerical differencing on the basis of cubic splines. Some boundary conditions are required for the left and right ends of the finite interval [*o,»w]i JV > 2. However, we will be interested further in difference schemes based on the operators A±, A rather than in using these operators for numerical evaluation of derivatives. In the following, Eqs. 1.4, 1.6, 1.8 will be spoken of as compact differencing formulas thus emphasizing that high-order discretization of derivatives is obtained while using three-point stencil. T

1.1.2

Compact third-order schemes

Let us consider scalar conservation law

Upon introducing the mesh ui, X u>>,, : ( = mr, we define some approximation (dufdt) of the derivative dujdt with the accuracy of 0(r ). m

k

Third-order schemes with compact upwind differencing

18

]

Using formally the operator A 'A- or / L A + for the approximation of the derivative with respect to x, the family of difference schemes can be written in the form lo,, \ *+l *-l *-l +

m

m

where f = f(u ), and ffj(j = 0,1,2, ...,k - 1) are some weight factors. Choosing these factors, the desired order of accuracy with respect to the step T can be achieved. In doing so, the truncation error of Eq. 1.10 with respect to the step h will always be of order 0(h ). To verify this, it is convenient to make use of the following symbolic representation of grid functions A±f and A / obtained as a result of expansion of sufficiently smooth function f[x) into Taylor series at the point x = 3

T

A £AV

±

/ = ( / T ^

= (j

T

r

+ y ^ ) / +0(V),

+ f S'W

3

+ 0{h ).

(1.11)

Here / is the unity operator while and D\ are the operators of first and second x-derivatives at the point x — z,-. Substituting Eq. 1.11 in Eq. 1.10, we obtain

from which the third order of approximation with respect to the step h for the schemes Eq. 1.10 follows. Multiplying Eq. 1.10 from the left side by A±, the equivalent schemes can be written as

Since .4-t are three-point operators, the stencil of these third-order schemes contains three grid points on each time level t = [j = Q,k — 1). We will call them compact non-centered schemes thus emphasizing that they are not symmetrical with respect to mid-point. The choice of the pair of operators A+, A_ or A-, A+ in Eq. 1.10 is unimportant from the point of view of approximation but it is essential for the stability of the method. To explain how to obtain the stable scheme, we consider the pair of twolevel schemes

J.J.

Third-order compact differencing and corresponding schemes

19

Let f(u) - on, a = const. Applying spectral technique, the following expression can be derived for the eigenvalue of transition operator 1 — ar 1 + ara,W (a)

A

T p.-1. k

0<«<2^,

±

1.13 '

Here 5exp(—ia) — exp(ia) + 8 and 12[exp(-ia) - 1] 5exp( — ia) — exp(ia) + 8 are the functions of complex argument satisfying the conditions SRe W = — 3fe W- > 0, 3 m W = 3m W_. Choosing now one of the pairs Eq. 1,7 for which aSte Wi > 0 (/ is plus or minus sign), we get unconditionally stable scheme Eq. 1.10 for c-, = a > 0.5. It is worth reminding that the stability of the scheme is thought of here as its stability for frozen coefficients. WJa)

=

+

+

In nonlinear case, the simple criteria for the choice of the proper pair of operators in the grid point xj = jk is the condition ffafj&cWt

> 0.

(1.14)

We see now that usual upwinding principle should be applied here. Namely, if /'(uj ) > 0 in the grid point x — Xj the backward difference A_ (and operator A ) should be used. Similarly, if / ' ( « " ) < 0 the forward difference A and operator A - are needed. This choice of the orientation of one-side differences A_ and A leads to positive operators approximating the derivative df/dx in Eq. 1.9. Such an approximation will be called compact upwind differencing (CUD). To specify the order of truncation error, it may also be designated by CUD-3. 1

+

+

+

To formalize upwinding principle, it is convenient to present the operators A and A± as the following sums A

= At q

±

0.25A ,

T

A , = 0.5(A ^ A , ) ,

O

0

vhere «V =

+ -) A

A = f Q

i+1

- /(_!,

=

+ \l.r+

g/i+i.

A = fa - 2h + /,-.• 2

t

The operators A and A are the self-adjoint (symmetrical) components of the operators A± and A while the operator A defines their skew-symmetric part. 0

2

T

u

Third-order schemes with compact upwind differencing

20

1

Introducing the parameter s = sgn /'(uj ) and omitting indices of the operators A± and A > the CUD-3 scheme chosen from Eq. 1.12 is given by T

A"

+ ALL—lA

+|i - * ) , - ] ,

>LL = \o- ^' A

a

(i is)

where A = Ao - 0.25sA ,

A = 0.5(A - s A ) ,

o

0

2

m

a = sgn / ' ( u ) ,

a = cr„.

Setting s — 0 in Eq.1.15, we obtain centered approximations used in papers [27], [52], [81], [1], They have the fourth order truncation error with respect to the step k. Since 1

r*

=r+/'(u™)^

1

- u-) + o(\\u^'

j

- «™h ),

the linearized form of Eq. 1.15 can be presented as (A + ^ A / ' ( u " ) ) " "

+ 1 r

""'" + | A r = ^ h

m

+

l

+ (1 - *)
(1-16)

1.2

Compact schemes as approximations of conservation laws

1.2.1

Equations

of balance

Since the operators A define quadrature formula it is possible to derive finitedifference schemes as equations of balance of the fluxes through the boundaries of elementary cells i.e. in the form of difference conservation laws. The scheme Eq. 1.15 can be obtained by so called integro-interpolation method [90] in particular. Suppose, for example, that g — 0 in Eq. 1.1. Writing this equation as div a = 0,

T

a = (u, f(u)) ,

(1.17)

we integrate Eq. 1.17 over the elementary cell tm<«<Wl.

i,-i
(1-18)

Reducing the surface integral to the contour one, we approximate then the equality j> udx - f(u)dt = 0

21

1.2. Compact schemes as approximations of conservation laws

in which the integration is performed over the sides of the rectangle Eq. 1.18. When approximating the integrals over the boundaries x = cons( we use the trapezoidal rule. While doing so along the boundaries t — const we apply one of the quadrature formulas given by the operator A or To be definite, suppose that f'{u ) >0 everywhere and hence the operator A should be used. As a result, we obtain the equality m

+

+

A (u^

-

+

m

+

+ / , - fJlV

- fT.x)r = (1

(1.19)

which is in fact the scheme Eq. 1.15 for o, — o — 0.5 multiplied by kr. Let the computational domain be the rectangle 0 < ( < T = Mr, 0 < x < X = Nh. Then the summation of Eq. 1.19 results in the integral conservation law due to the cancellation of the grid functions uj" and ^>J" corresponding to the common boundaries of cells. It reads X-h

X-K

T

T

J u(T,x)dxj u(0,x)dx + j f(u(0,t))dt - J f(u[X - h,t))dt = 0, 0 0 0 0 where the first two integrals are approximated according to the quadrature formula Eq, 1,2 while the last two integrals are presented by the trapezoidal rule. The scheme Eq. 1.19 can also be cast in the usual numerical flux form for which the boundaries of cells are placed at x = Xj±i/ . We note finally that Eq. 1.19 may be rewritten as 2

where

1.2.2

— A u is regarded as some averaging of u for the cell +

Conservation

< x < X(.

property in the case of sonic points

If the derivative /'(u™) changes its sign thus causing the switching of the scheme from one pair of operators to another, the summation of difference equation of type Eq. 1.19 does not lead to the cancellation of grid functions values in the internal nodes of the mesh. Suppose that the derivative changes its sign between the grid points Xk and Xk+i so that the so called sonic point *, where /'(x«) = 0 satisfies X), < X, < Xfc+j. Then the quadrature formula is not applied to the interval [xt, x*+i] if f'{n-™) > 0 and / ' ( u j ^ ) < 0 and the non-zero term fk — fo-i arises when summing over k. If /'(fJT_,) < 0 and /'(u™) > 0 then the balance equations for the cell with the side \xt,Xk+i) are written twice which results in the non-zero term — f . It t

22

Third-order schemes with compact upwind differencing

can be shown that in both cases the numerical sources of the form u

u

Jk+i ~ Ik , . k ~ k i r ± h T arise inside the computational domain. Their intensity can be seen to have the order 0(h ). If in the vicinity of the sonic point the higher derivatives of the exact solution exist and are not too large, then such local non-conservation is expected not to influence seriously the quality of numerical solution. However, it is not the case when the exact solution is not smooth. The violation of conservation laws may then cause serious errors. Slight modifications of the operators A and A are needed to guarantee the cancellation of the fluxes of the vector a given by Eq. 1.17 through the common boundaries of cells. One may set for example 3

A = yt -0.25A s, o

o

A = 0.5[A - A _ ( T C

1 / 2

s)A ]. +

(1.20)

Here 7i denotes the shift operator along the axis x so that Tifj = f(ij + Ih). Hence, the grid function A_(T,/2s)A / at the point x = z, is given by +

A.t^^lA+Z, = s

1+1/5

(/,-

+1

- U) -

- fi-i).

(1-21)

If Si = const for ( = i then the operators Eq. 1.20 coincide with those given in Eq. 1.15. In all cases the difference scheme Eq. 1.12 with the operators Eq. 1.20 can be presented in the form m

- nf

1 + Qj+1/2 - Qj-ttf + frlU+itt ~

= 0,

where + ( ( <))j]. s

= g(("i)j+i - («t)/) Qi+i/2 = 0.5(/j+i + / j ) - s 1

fi = "Wf )

+ (i - # 0 ,

) + 1 / 2

(/

W.h =

J + 1

u

- /j), 1

"7* -*?.

Here s — 1 corresponds to the pair of the operators A , A _ and s = — 1 corresponds to A-, A + . While summing difference equations over indices j and m the values of grid functions in all grid points except near-boundary nodes are cancelled. +

_1

3

Using Taylor expansions one can easily see that ft~'/4 A/ — df/dx -f 0(h ) for the grid points in the vicinity of which the function s does not change its sign. One may also observe a decrease in accuracy near sonic points where the truncation error is of the first order.

1.3. Dispersion and dissipation properties of CUD-3 operators

23

More appropriate form of the operator A in Eq. 1.20 is Afj = Aofi - Q.2o(s (I

+ T,) - $t_ {l

i+1/2

r/z

+ lU.jJ/j

which provides second-order accuracy near sonic points and third order elsewhere. The reduction in the approximation order near sonic points does not significantly influence the accuracy in the £ -norm if the number of such points is not large in comparison with the total amount of nodes. 2

1.3

Dispersion

and

of

operators

CUD-3

dissipation

properties

To estimate the dispersion and dissipation properties of CUD-3, it is worth comparing the exact solutions of convection equation and its semi-discretized form. Let us consider the simplest convection equation (hi du — + c — = 0, at ax

c = const.

(1.22)

Its semi-discretization via A and A operators reads l

du„ldt + cA~ AuJh

= 0,

(1.23)

where A and A are the operators from Eq. 1.15 with s = sgnc. Looking for the solution of Eq. 1.23 in the form (/(()«'*'"'. obtain the ordinary differential equation ^ + dt

^W {kk) In 0

k = const, we

+ i^-W^kfi) (7 = 0, h

(1.24)

where 1

W (o) = 96s*(a)sin' a/2, 0

W,(a) = 24*(a) sin a(7 - cos a), ?

3

*(a) = [(8 + 4cosc.) + 35sin o]"' Comparing Eq. 1.24 with the equation U' + ickU — 0 obtained from Eq. 1.22 after substitution of U(t)e' , the following expression may be derived for the numerical phase velocity c" c = cW,(a)/a, a = kh, kl

w

Third-order schemes with compact upwind differencing

24

where c = c<ms( is recognized as the exact phase velocity of the harmonics with the wavenumber k. The term c\Vn(kk)/k in Eq. 1.24 which is nonnegative for all values of a characterizes the dissipation and hence the damping of harmonics. For not-too-short waves (kh

1) the following estimates hold

c" = c[l + 0 ( q ) ] ,

= ch-'d=0{a ),

4

l

ch- W (a) 0

3

Wt^

= tt'oM =6-

These expressions mean that for long and medium waves in comparison with the step h when the product kh is not large, phase error which is caused by discretization .A~'A//i is insignificant and has the fourth order with respect to kh. The function c"(a)/c where a = kh as well as the dissipation d are shown by solid lines in Fig. 1.1, The phase velocity curve c"(o)/c for second-order centraldifference approximation of the derivative du/dx is also presented by dashed line. As can be seen from Fig. 1.1, phase errors for .4~'A operator are small in much wider range of kh than that for the central-difference operator. The behavior of the dissipation d(kh) is the important property of the operator .4~'A. Referring to Fig. 1.1, the dissipation and phase error are negligible in the same range of kh. Hence, we may expect quite accurate representation of the harmonics except those corresponding to the shortest wave permitted by the grid. In the high-frequency region (JT/2 < kh < x), where the phase velocity strongly deviates from the exact value, the dissipation rapidly increases with kh thus providing the efficient damping of spurious oscillations. Therefore, the compact differencing ,4"'A contains a built-in filter which does not perturb the physically relevant components of solutions and extinguish those components with the mesh size h. Though CUD-3 schemes are not monotone in the strict sense (due to Godunov's theorem), no additional devices such as artificial viscosity, limiters etc. are needed in many practical applications. It will be shown in Chapter 2 that when calculating unsteady discontinuous solutions of inviscid problems, wiggles inevitably appear. However, they are well localized near the fronts and often does not prevent the interpretation of numerical data. Of course, standard techniques such as flux limiters can be used to remove them completely. For comparison, the phase velocity curve e"(o) for centered fourth-order discretization 0.5.4.0'Ao (s = 0) is also depicted in Fig. 1.1 by dotted line. According to Fig. 1.1, phase errors for fourth-order schemes are comparable with those for

o

TI/2

n

Fig. 1.1: Dispersion and dissipation of A 'A (solid lines), 2nd order central differencing (dashed line) and fourth-order compact differencing (dotted line)

CUD-3 but such schemes do not contain the dissipative mechanism which could damp spurious short-wave components permitted by the mesh. It is the main disadvantage of the approximations A$ A that often results in the impossibility of their application without any artificial dissipative mechanism. The form and parameters of such mechanism are not always obvious and sometimes can be the topic of special investigation. 1

0

1.4

Difference equations

1.4-1

Analysis

of

conditioning

Supposing that computational domain is the rectangle 0 < T < MT, 0 < x < Nh, we consider now typical nonlinear difference equations arising when applying CUD-3 in the form of the two-level scheme with weights Eq. 1.15.

26

Third-order schemes with compact upwind differencing Using index notation, we rewrite Eq. 1.15 for /'(u™) > 0 as

*'•=«*.

3= l,$,...,N-

1,

where the grid function dj is supposed to be known. 1

1

1

To solve Eq. 1.25, it is natural to linearize the function /jfuj " " ) in terms of known grid functions. Without loss of generality, suppose that the simplest linearization is performed /r

+ i

a

~ «;(«r' -«?)+ft'

>=fffih

where Oj is a known grid function. Thefinite-differenceequations Eq. 1.25 can be presented then as (5 - 2 / V , K - i + (8 + 2 / £ > - u , = I2d J

j+

h

(1.26)

A

where R = 6£n7'(u™)/ - I f / ' ( u f ) < 0 , Eq. 1.25 reads s

- Wj., + (8 + 2 t f > ; + (5 - 2R )u HI

>7 = ;

i+l

= 12dj,

(1.27)

6<,T\f'(u?)\/h.

Since Eq. 1.26 and Eq. 1.27 have the same structure and can be obtained one from another by reversal numbering of the grid points, we consider only the first of them. For preliminary investigation of the system of equations Eq. 1.26 for j = 1 , 2 , N — 1 and some boundary conditions at j — 0 and j = N, one may use the results described in [43]. A more detailed discussion concerning this subject will be given in Chapter 5, Let us consider first the case when function Oj(uJ") = a, satisfies the discrete analogue of Holder condition a* - a, < D

k-l N

u (1.28)

Such situation arises particularly when the solution of initial problem Eq. 1.9 is continuous. Following [43] we write for Eq, 1.26 the characteristic equation 3

- q + (8 + 2Rj)q + (5 - 2/£>_i) = 0

(1.29)

27

1.4. Difference equations which has the roots 3, = 4 + Rj -

/ v

2 1 + 2 ( / 7 - / i _ ) - 6 f i + fl?, i

i

l

r

J

^ = 4 + ^ 4 / 2 1 + 2 ( f l - / 7 _ , ) + 6/fj + /?J. 1

J

J

If the inequalities \q \ < 1 and |f | > 1 hold with sufficiently large /V, the systems Eqs. 1.26, 1.27 are well-conditioned in the sense of [43J, i.e. the sensitivity of their solution to the perturbations of Rj and dj does not increase with JV. More precisely, suppose that the coefficients on the LHS of Eq. 1.26 or Eq. 1.27 are bounded by some constant M independent of N. Then these systems are thought as well-conditioned if x

2

< Mmax{\if\, |V>|, m?x\dj\}, where ip and ip are the RHS of boundary conditions at j = 0 and j = JV respectively [43]. Setting k = j and I = j — I in Eq. 1.28, one may see that the inequality \q \ > 1 is satisfied due to Rj > 0. To verify the inequality < 1 it is convenient to consider two cases: 3

(i) 5 -2Rj-,

> 0 and g, < 0;

(ii) 5 - 2/?j_! < 0 and q > 0. :

Taking into account Eq. 1.28, it is easy to see that \qj \ < 1 in both cases and thus the system under consideration is well-conditioned. When computing either discontinuous solutions or solutions containing sonic points (/'(ti) =i 0) difference Holder - continuity Eq. 1.28 may be violated in some nodes, it is convenient then to use the condition [43] 8 + 2Rj > | 5 - 2ftj_i| + 1 +

6 > 0,

(1.30)

Inequality Eq. 1.30 holds for 5 - 2/t,_, > 0. If 5 - 2/tj_, < 0 then the following limitation is sufficient to meet this condition: T

max O-j-t - 0.j k<7 i

1

(1.31)

Thus for discontinuous case the system formed by the equations of type Eq, 1.26 or Eq. 1.27 is well-conditioned if not a very restrictive inequality Eq. 1,31 is satisfied.

28

Third-order schemes with compact upwind differencing

It should be noted that if the grid function a, which is close to zero changes its sign due to spurious oscillations, it is advisable to use the following definition of the switching parameters s —0

if

s = sgnnj

|oj| < e, otherwise,

where e is some tolerance. Certain information about the solution of difference equations can be obtained by setting Rj = const = R, |aj| = n and dj = 0. Then the solution is given by c

C

«i = i 9 i + M< (1.32) where C\ and c are some constants. When R < 5/2, that is ar/h = K < 5/\2a, we have 3

4 - V 2 T < 9 i <0.

(1.33)

If Eq. 1.33 holds the solution Eq. 1.32 may contain an oscillating component the amplitude of which rapidly decreases due to \qt\ < 1. Note that the solution is os ci 11 at ion-free if K > 5/12
1.4-2

Solution

procedures

The natural way to solve three-point difference equations is the Gauss elimination technique which is stable for a well-conditioned system. Since \qt\ < 1 and \qj\ > 1, the boundary conditions for each end of the considered interval should be prescribed. These conditions are the same as those in the case of conventional three-point implicit schemes. Generally, they can be written as

(1.34) where ft and 7; (t = 1,2) are some coefficients. The proper forms of these relations may be obtained taking into account the incoming and outgoing characteristics in particular. The choice of boundary conditions Eq. 1.34 can influence the stability of the whole algorithm. The estimation of this stability can be obtained using, for example, Babenko-Gelfand condition [43J. In some cases the iterative technique for solving system Eq. 1.25 may be of use. Supposing for example that y»'(uj*) > 0, (j = 1,...,JV— 1) one may write

J.5. CUD-3 with different lime discretizations

29

where k is the iteration number. If the function is linearized with respect to k— 1 iteration, then system Eq. 1.35 becomes linear, its matrix being two-diagonal. Such system can be easily solved. Supposing constant coefficients, the eigenvalue A of the transition operator is readily found to be i

=

t

w

|

%

g

^

'

* * * * *

«

m

where / i = Gara/h, from Eq. 1,36

a — / ' { u ) = const. The following estimate can be derived 2

|A| < [min{13,v'9 + 24R+ 16/? )]~\ Therefore the iterations Eq. 1.35 converge very rapidly. If it is desirable to solve the system Eq. 1.25 in the non-linearized form then the iterative process Eq. 1.35 with the linearization of functions fj may be more profitable since it combines "nonlinear'' and "linear" iterations, the matrix to be inverted being two-diagonal. lii,k

!

We note that for some functions / (for example, /(ti) = u ) the direct solution of nonlinear equations is also possible.

1.5

C U D - 3 with different time discretizations

1.5.1

Three-level

schemes

So far, the simplest two-level scheme from the family Eq. 1.10 was considered, namely the scheme with weights. Purely implicit (a = I , U\ = 0) and CrankNicholson schemes (o = tr, = 0.5) are special cases of this scheme. Of course, other types of time discretizations are possible. We outline now some of them as examples. 0

a

m+l

Third-order scheme with (dufdt)

= (u

m + I

m

-u )/r

Using the grid tiJ , let us introduce the Adams operator ,4, similar to the operator A r

Atf=

(5/

m + l

+8/

- / - ' ) / 1 2 ^ - J fdt.

m

m

(1.37)

Integrating Eq. 1.9 over the domain t < t < < i , Xj < x < Xj+\ or ( < £ < t < i < Xj for / ' ( « " ) < 0 or /'(uj ) > 0 respectively and replacing the integrals by quadrature formulas Eqs. 1.2 or 1.3, we obtain by virtue of commutativity m

ra+

1

m + !

m

30

Third-order schemes with compact upwind differencing

of the operators A and A, the scheme m+

A^——^-

+ A (A/h)f '

=A Ag.

l

(1.38)

t

Clearly, this scheme is the member of the family given by Eq. 1.10 corresponding to ( T = 5/12, CT| = 8/12 and c = -1/12. 0

3

Using the index notation and supposing f'(u) > 0, we rewrite Eq. 1.38 as 0

"j-i

12 5

/

r

B

"j-i ,

"j 12

T

-_fx*

1

"j _

"y+i " j + i 12 r

r

sir-/r-,

12 A 12 where rfj denotes the RHS of Eq. 1.38.

i / r - / n ' 1 2 / i

ft

=

a

*'

To investigate the stability of the scheme Eq. 1.38, one may use spectral technique and analyze the roots of the equation A

"

1

+

(y|

A

+

il

_

m )

r

W

{

a

)

=

0

T

'

=

a T l h

=

c o n 3 t

'

a

=

r

i

v

)

( L 3 9 )

in which 3te{rW) > 0. The simplest way to derive the stability criteria is to solve Eq. 1.39 numerically. The dependence of the spectral radius on the CFL number |a|r/n which is obtained in such a way shows that the scheme Eq. 1.38 is conditionally stable. More precisely, X < 1 if \a\rjh < 1. Hence, this scheme is appropriate only for unsteady problems when the choice of time step T — n is in agreement with physics of phenomena. Some increase of operation count as compared with those for conventional explicit schemes may be compensated by high accuracy manifested itself as small phase and amplitude errors for physically relevant scales. Note that the scheme Eq. 1.38 uses in fact Adams-type time stepping. Schemes with {du/dt)-"*' = ( 3 »

m + 1

m

1

- 4u + u" -')/2r

Consider the family of three-level schemes A [ ^

+

' - ^

+

"""j

+

| ( ^ « +

a

i

r

+

c r ^ i=4.

(1-40)

where o + f | + f j = 10

=

2

3

lf ero — 'l, — o~2 0 we have the scheme with truncation error 0(T + h ). The eigenvalues A of the transition operator for this scheme can be written as 2±(1 3

7

+

-2R-2SiY' 2fl 25, ' +

W

1.5. CUDS with different lime discretizations

31

where notations R = arfSteW, S — ar^smW are introduced for brevity, r = r/h and W is complex-valued function from Eq, 1.13 satisfying the condition aMe W > 0. Let us show now that |Ai,j| < 1. Denoting the absolute value of the expression {1 — 2R — 2Si) by Y, two possibilities may be considered: Y < 1 and Y > 1. In the first case the absolute value of the numerator of Eq. 1.41 satisfies the inequality ,2

|2 ± (1 - 2R - 2Si)' \

7

< 2 + TV < 3, 2

2

while the absolute value of the denominator [(3 + 2R) + 4£ ] is not less than 3 because R > 0. Thus the absolute value of the RHS of Eq. 1.41 does not exceed 1. Now let Y> 1. Writing Eq. 1.41 in the form

A

2 ± y F ( c o s p / 2 + isinp/2) —2+2R+2Si—•

=

r , *-***»"$-m/y]* 2

one may deduce after simple manipulations that the inequality \X\ < 1 holds if 2

/!

*,, (y-, R) = Y - V=f4[(l -2R + Y)/2\' 2

+ 4 + 16R > 0.

This condition is met for <& because Y > I. Consider now $ j . Taking into account that R > 0, Y > 1, we deduce that ^(V^O) > 0, since 3

1 /

v -) w = 2 V - l, - [ (f,, l +, K / 2o]l"- ^ > 0

dy

and $,(1,0) = 0 . The positivity of $i(F,0) certainly results in positivity of *i(K, R) by virtue of - 16 + 2[(1 - 2R + Y)/2]

OR

> 0.

Thus three-level scheme Eq. 1.40 is unconditionally stable. Using the weight coefficients cr,- ( i = 0,1,2) in Eq. 1.40, the scheme with truncation error 0(r + h ) can be constructed. The weight factors c , C\ and cr which cancel 0 ( r ) and 0 ( T ) terms in the Taylor expansion series for Eq, 1.40 can be easily found to be 3

3

0

2

s

2

2

1

It can be shown that the scheme Eq. 1.40 with such coefficients is conditionally stable.

32

Third-order schemes with compact upwind differencing

1.5.2

Explicit forms of third-order compact schemes

The unconditionally stable schemes defined by Eq. 1.10 can he used to obtain both steady and unsteady solutions. In the former, the first-order two-level scheme with a — 1 can be quite efficient. If unconditional stability is not of prime importance when solving unsteady problems then explicit schemes based on CUD-3 approximations may be of interest.

It should be noted that explicitness means here only that the discretization of space derivatives is performed at m-th rather than at (m + l)-th time level. Several considerations should be emphasized in context of CUD-3 explicit schemes.

(i) Formally, all known explicit time stepping may be combined with CUD-3 approximation. Due to the positivity of this approximation, one may count on the conditional stability of such schemes if the same time stepping provides conditional stability for usual space discretizations. (ii) The inversion of three-point operators is still needed for obtaining the solution at m, 4- 1-th time level. Just for the sake of illustration, consider the simplest explicit scheme for Eq. 1.9 m+l

m

2 + -A-'Afiu ") 1

=g J

m

(1.42) m

To implement Eq, 1,42, the grid function z = A~ Af(u ) should be evaluated as a result of solving tridiagonal system Az = A / . Though the inversion of A may be preferable as compared with that of implicit operator A + ( r / / i ) A / ( u ) , the operation count in both cases is approximately the same. m

(iii) The dispersion and dissipation properties of CUD-3 may be somewhat masked by first- and second-order time stepping if unsteady solutions are of interest. Consider for example, two-level scheme Eq. 1.12 with er = a, Oj = 1 — a and f(u) — cu, c = const, g — 0. Substituting the function = X exp(ion), a = kh,, we obtain 0

m

(1.43)

1.6. Two-level 0(r 3 where

l,xl

+ h3 )

scheme

33

and () are given by

l,xl

= [1 - 2(1 - O')KWo - (1 - 0')2K2(Wc?

(1 1

() = -

o

+ O'KWo)2 + 0'2K2Wl

+ Wn]1/2

KWI [1 - (1 - 20')KWo - 0'(1 - 0')K2(Wd

'

arctan -;:---;-;----::-~=;:_-~'--__;_=7:':':;;_____:c=

+ Wnl '

In these expressions the functions Wo(o) and WI(o) are defined by Eq. 1.24 while K = cr / h is the CFL number. Comparing Eq. 1.43 with relative evolution of the exact solution Eq. 1.22 which IS

u(t + r)/u(t) = e- ickT = e- iKo , the relative phase speed c' / c = () / K may be estimated for small

0

as (1.44)

For these values of

0

the estimate ( 1.45)

also holds. Upon inspecting Eq. 1.44 and Eq. 1.45 one may see that the phase and amplitude errors are greater than those for semi-discretized Eq. 1.24. However, they are minimal for Crank-Nicolson scheme (0' = 0.5) for which

The estimates Eqs. 1.44 and 1.45 are obtained setting Wo = 0(0 4 ) and WI = 0+0(05 ). To preserve the fourth-order phase and amplitude errors of CUD-3 approximations, one should use at least third-order time stepping. One of the possible options is the combination of CUD-3 and Runge-Kutta method.

1.6

Two-level 0(7 3 + h3 ) scheme

1.6.1

Derivation of two-level scheme

The above considered three-level schemes with truncation error O( r3 + h3 ) are only conditionally stable. However an unconditionally stable scheme which is based on

Third-order schemes with compact upwind differencing

34

high-order space and time discretizations may be of interest. Such scheme can be particularly useful when constructing marching algorithm for steady problems in which one of the space coordinates plays the role of time. To derive such a scheme [116], we begin with approximating Eq. 1.9 by J

m

m

+ A (A- Af) /h

=A ,

t

(1.46)

ig

where A is the operator defined on the mesh ui similar to the operator A on the mesh wj,: A,g = ( 5 — + 8g )/12. t

T

1

+

m

defined

m + 1

9

f l

In fact the operator A is adjoint to the operator A, from three-level scheme Eq. 1.38. Obviously, the scheme Eq, 1,46 has truncation error 0 ( r + ft ). But at the same time it contains undesirable term A~' Af * j\2k which makes its implementation difficult. To overcome this obstacle, we need only to use any second-order scheme to exclude this term without any loss of accuracy. For this purpose, let us choose the Crank-Nicolson scheme written in the form t

3

m

m

1

.,m+i _ , , - l

] 1

1+ where F

m

m

= Tlf A,g , 2

m

(/

1

(1.47)

7 $ being the shift operator defined by T§[g(l) = j ( i + r/2).

= A~ Af(u'")/h

and combining Eq. 1.46 with Eq. 1.47 we

1

+

1

A / " " + A-' A / - + ) = F »

,

Denoting Lu obtain

3

x

?rl)u--(/--|rL)u- -^[(/

l

+

£)- (/-rL) "

T

U

+

2r(/ + r L ) - V "

m

= TA g

(1.48)

t

m

m + 1

To evaluate u one should first find preliminary value of u from Eq. 1.47. Substituting it into Eq. 1.46, it is sufficient then to invert the operator [ / + 2/3TZ-] . More precisely, the algorithm is as follows (i) Solve m 1

(l + TL)V ={l - rL)u -

+ 2rF".

(ii) Solve (/ + ITL)U-

=

m

(!-^rL)u ~'

+ ~Lv + TA g™. s

3

3

1.6. Two-level 0{T + k ) scheme

35

While performing the steps (i) and (ii), some linearization is needed for nonlinear operators L . Supposing that it has been accomplished, the inversion of the operators I+TL and / + (2/3)rL reduces to solving tridiagonal systems. Hence, two tridiagonal inversions are needed here instead of one inversion in the case of three-level schemes. To verify third-order accuracy of the scheme Eq. 1.48, we note that it differs from the £>(r + h } scheme Eq. 1.46 by the term 3

3

0=^(l«-Lti" 12

+ 1

),

where v is the solution resulting from step (i). This expression may be cast in the form

W

12

(1.49)

12 1

Multiplying Eq, 1,49 from the left side first by L

,/2 = / + 2 / J « + 0 ( T ) ,

and then by (/ + T L ) , we get

A, = I - - D

2

t

+

2

0(T ),

where D is the operator of the first derivative with respect to t, it is easy to verify that Tp and hence Eq, 1.49 have the order of 0(T ). This means that the scheme Eq. 1.48 is really 0 ( r + k ) scheme. t

3

3

1.6.2

Estimates

3

of stability, dissipation and dispersion

Supposing f{u) = Q U , a = const, one may use the spectral method to prove the unconditional stability of the scheme Eq, 1,48, Performing tedious algebraic manipulations, the following expression for eigenvalues of transition operator can be obtained

_ 1 1

1 + 1**+

£(I V -

- iK v(v + 3

2

l + f A V + f (13*» + 37*») + * A ' M 5 ^ + V )

where K =at(k,v

+ !(,* + + J(»'

a

<

+0)fK '

and u being the functions W (o) and Wi(a) respectively, a = kh. 0

We observe that for arbitrary values of CFL number K the RHS of Eq. 1.50 does not exceed unity and hence the scheme Eq. 1.48 is unconditionally stable.

Third-order schemes with compact upwind differencing

36

Closer examination of Eq. 1.50 shows that as might be expected the dissipation of the scheme is 0 ( a ) when a < 1. For this limit, the amplitude errors may be estimated as P| « , _ v - -K*p.< 4

K

Setting Q = kh = JT, we obtain m 1 1

j _

2

3

1 + ( 4 / 3 ) g * , - (5/9)JY < - (2/3)jif ^ + ( 1 / 4 ) * X 1 + (10/3)K*„ + ( 3 7 / 9 ) * ^ -I- {20/9)K*»l + (AffyK*^'

where v - W (t) = 6. This gives |A| as 3/4 for large CFL numbers thus indicating that high-frequency harmonics are noticeably damped. m

0

5

Keeping in mind that e = 0(a") and u = » + 0 ( a ) we may derive the following expression for numerical phase velocity c* c A>(18 + 42AV + 2 3 / f V ) c = —— arctan if a 18 + 42/fe + l7K p? - S/CV ' 2

4

1

where only Ofa ) terms are retained. The Taylor series expansion for this expression gives

The estimates for |A[ and c'/c differ from those for semi-discretized Eq. 1.23 in that the coefficients in the fourth-order terms are now dependent on CFL numbers. Closer inspection of the estimate for |A| shows that 1 — Kv may be recognized as the attenuation coefficient for low-frequency harmonics corresponding to semidiscretized equation, while — ( 7 / 7 2 ) j i is an additional term arising due to time discretization. Similarly, pjct is exactly the numerical phase velocity extracted from Eq. 1.24, the term (289/324)/t"V being additional phase error. We observe that dependence on CFL number is very strong in both expressions. 4

Before we end this section, we summarize the main properties of the scheme Eq. 1.48 in the form of the following keywords (i) Third order of approximation with respect to r and h. (ii) Three-point stencil in i-direction and two-level two-step time marching procedure. (iii) Unconditional stability. (iv) Small amplitude and phase errors for waves which are not too short.

i.e.

Two-level 0{r

3

3

+ ft ) scheme

37

(v) Damping of high-frequency oscillations. We remark that other types of high-order space discretizations may be used instead of CUD-3 when constructing the operator L . Of course, the present technique may also be thought of as a third-order method for ordinary differential equations.

Chapter 2 Some extensions of basic ideas

The aim of this Chapter is to outline the ideas which can be considered as additional to those described in the previous sections. Some of them deal with conservative forms of CUD-3 operators which are appropriate for obtaining weak discontinuous solutions of conservation laws. Others illustrate the variety of CUD-3 discretizations and additional mechanisms which may be useful in some situations. At the end, another type of the third-order compact differencing is presented. It has certain advantages while obtaining steady-state solutions.

2.1

Generalizations of the CUD-3 operators

As follows from the theory of Chapter I , the first-order upwind operator A(s) can be considered as a generator of CUD-3. In fact, two forms of this operator have been exploited, namely, (2.1) and (2-2) where Sj = ±1 is a switching factor. Naturally, the question arises if it is possible to use for conservation law

38

2.1. Generalizations of the CUD-3 operators

yj c

an arbitrary first-order upwind approximation A as a generator of another threepoint operator A in such a way that G

i(^)-V/( = ^ u)

+

o,y).

The answer to this question is positive. Consider the general form of a three-point conservative approximation of hdf/dx

with a positive grid function Qj+j/2, j = 0, +T, +~2,.... In fact, it is well known that many first-order difference operators fit in this form. Suppose also that <2f+i/a is the restriction of sufficiently smooth positive function Q to the mesh Xt+i/i = 0 + ' /4}A and / ' ( « ) 5^ 0. Then the Taylor expansion series which is similar to Eq. 1.11 has the form

G

Uf h where Q =

1

=(j\

^D Q + ^Dl] DJ + 2 " 6 r

3

0(k ),

Qlf'{u). G

Now we try to find the operator A A° = (l-

which has the expansion

± .Q + ?-Dl\+0(h>). D

(2.5)

Eq. 2.5 is similar to Eq. 1.11 of Chapter 1, where the switching function s was used instead of Q. Obviously, the appropriate form of A° is A°gj = A gj - j<3j+i/s(Sj+i - ffj) + j ^ - i / S t f * - fl-i) 0

which reduces to Ag, with A defined at the end of Subsection 1.2.2 if we set Qj+j/2 = Sj+i/a-

The family of CUD-3 schemes then becomes the third-order compact approximations generated by Eq. 2.4 with

40

Some extensions of basic ideas

Of course, a more general choice of Sj can be considered. For instance, one may put = fcsgn f'j /2} where k > 0 is either constant or the restriction of a smooth function k(u,x). As a further example, Lax-Fridriechs operator can be mentioned. In this case +1

Qj+1/1 = Qf+1/2 = ^

Qf+1/2 - 2 f

2

~ >

r

where r = rjh. As it can be seen from Eq. 2.7, some difficulty arises if / j / <S 0 in the vicinity of sonic point. Clearly, it violates our assumption about finite values of Q/f'(u). The possible way to circumvent this limitation is to set Q = Q(f'(v)}, Q(y)/y being finite when y —• 0. Obviously, if Q(y)/y is sufficiently smooth as y —* 0, then third-order accuracy is preserved everywhere. + 1

2

G

The simple example of continuous Q(y) with finite ratio of Q(y)/y is Q {y) = \y\. For this choice of Q(y) the first-order flux q becomes G

u

q%Mi = o &c/i+. + h) -

- >)

2

8

(-)

and A

G

=

G 5

+ 1 / 2

-<,f_

CIJ

3

Comparing the numerical viscosity Q °by Eq. 2.8, one can see that G

CUD

Q - Q -

3

1 / 2

.

(2.9)

(Eq. 2.6) and the viscosity defined

2

= 0{h )

if u(x) is sufficiently smooth. Hence, the Taylor expansions for conservative A(s), Eq. 2.2 and A are the same up to 0(h ) provided that s is the restriction of a smooth function. It means that (A°, A ) with A = A ~ is a third-order differencing pair. We have in this case G

3

G

G

Gj+l/a ~~ l/j+i/sl/ZJ+l/i — j - H / 2 — Q s

CUD

7

f ' )

3

CUD-3

As will be explained in the next section, this pair is one of the candidates for entropy-consistent version of CUD-3 schemes provided that fj+ij^ is defined t>y some interpolation procedure in the expansion region.

2.2. Applications to discontinuous solutions

Fig. 2.1: Results fop Burgers equation and initial distribution Eq. scheme CUD-33-NC.

2.1Uwhile

using n on-conservative

2.2

Applications to discontinuous solutions

2.2.1

Comparison

of non-conservative

and conservative

forms

All forms of compact operators described in Chapter 1 and foregoing sections can be used when computing smooth solutions. We can count here on convergence Theorem which guarantees that 0(k } truncation error and the stability of a scheme result in convergence of the order 0{k } both for linear and nonlinear problems. However, it may not be the case if weak discontinuous solutions are looked for. 3

3

For further discussion, let us consider again the scalar conservation law Eq. 2,3 with a convex flux function }(u). We are interested in unsteady solution of Eq. 2.3 with initial data u(z, 0) = ua{x). If f'(u.) ^ 0 everywhere in the computational region then all forms of CUD-3 schemes are conservative and can be used to compute weak discontinuous solutions of Eq. 2.3. However, as it will be further discussed, the presence of sonic points where f'(u) = 0 violates the conservation properties of the operator pair generated by Eq. 2.1. The numerical experiments with the different forms of CUD-3 are described in [120]. Due to unsteady nature of the solutions to be obtained, 0(T + At ) scheme 3

3

Some extensions of basic ideas

42

Eq. 1.38 was used. It was labelled in (120) as CUD-33. Firstly let us consider the non-conservative option of CUD-33 corresponding to Eq. 2.2. It will be designated by CUD-33-NC. In the numerical experiments the switching near sonic points was regulated by the tolerance parameter e by setting sgn/'fu) 0

if \ffy\
One of the typical examples is shown in Fig. 2.1 for the inviscid Burgers equation. In this case f(u) — u /2 and 2

1 - cos 2™ if \x\ < 1 0 otherwise.

^

1 Q

j

As can be seen from Fig, 2.1, the solution moves forward due to non-negativity of initial data. For some t• < % it consists of two shocks, one of which is weaker and travels behind the other stronger shock. For t greater than the critical time T. the two shocks merge to form a single shock that moves forward. These results were obtained for h = 1/40, CFL = 0.5 and t = 0.2, 0.01. For comparison, the solution computed using Godunov scheme [40] with the fine mesh (h = 1 /200) and CFL = ] is also presented in Fig. 2.1 as an "'exact" solution. It can be seen that the position of the leading shock is not correct, the error being smaller for e = 0.2 than that For t = 0.01. On the contrary, the trailing shock moves with proper speed. The explanation of this phenomenon is quite obvious. The trailing shock is captured by conservative scheme (/'(u) > 0) whereas the stronger one encounters at transitive conditions due to spurious oscillations near the sonic point (/'(u) = u = 0). This certainly leads to the loss of conservation property of the scheme. Another observation is that some wiggles may exist ahead of the shocks. It is also understandable since phase velocity of the short wave harmonics may exceed the exact one if CUD-3 operators are used. It may be noted that the wiggles generated by the scheme are well localized and do not prevent the interpretation of the results.

The example presented in Fig. 2.1 as well as other computations of [120] shows that one should avoid using CUD-33 -NC for computing discontinuous solutions. The good choice then is the conservative option of CUD-33 based on Afs) defined by Eq. 2.2. It is designated in [120] by CUD-33-C. Fig. 2.2 presents the results for the same initial data Eq. 2.10 obtained while using CUD-33-C. The positions of the leading and trailing shocks are seen to be absolutely correct. The numerical experiments

2.2. Applications to discontinuous solutions

43

T3 In ilia I dala Solution with 6=0.2 Solulion with 6=0.01

Fig. 1.1: Results for Burgers equation and initial distribution Eq. 2.10 while using conservative scheme CUD-33-C. show that they are independent of CFL and t. However these parameters may influence the amplitudes of spurious oscillations.

2.2.2

Entropy-consistent

forms

It is well known that weak solutions of nonlinear problem Eq. 2.3 are not unique when sonic points exist in rarefaction regions. However, uniqueness is guaranteed if these solutions satisfy the entropy inequality. They are often referred to as physically relevant solutions. The simplest example of the solution which is not physically relevant is rarefaction shock defined by

with constant U. While using CUD-33-C in the double-shock case, there were no problems with the existence of spurious rarefaction shocks. It is clear however that the generating operators Eq. 2.1, 2.2 and therefore the corresponding CUD-3 schemes admit the exact solution of the type Eq. 2.11. Hence, one can expect that prohibited solutions may be obtained with these schemes for some special initial data.

Some extensions of basic ideas

44

Initial dais Solution with CFL=I. Solution with CFL=0.S

F i g . 2.3: Results for Burgers equation and initial distribution E q . 3.12 while using conservative scheme C U D - 3 3 - C .

The example of computations extracted from [120] is presented in Fig. 2.3 for the following initial data if |jf| < 1/2 otherwise.

(2.12)

The results were obtained for diiferent values of CFL and e, the shock positions being insensitive to these values. The computed solution as expected shows a steady-state rarefaction shock at i = —1/2. It generates in turn spurious compression shock moving leftward. Note that wiggles are practically absent for this particular choice of t. The entropy consistency of the first-order schemes is well investigated. For example, Godunov's scheme is entropy-consistent. Osher [73] introduced a family of so-called E-schemes which were known to converge to the physically relevant solutions. Tadmor [101] showed that any scheme with numerical viscosity which is not less than that of entropy-consistent scheme is also entropy-con si stent. In all these and other results rather strong numerical viscosity non-vanishing in the sonic points plays the role of the sufficient condition for entropy inequality to be satisfied. However, the presence of the first-order viscosity terms is far from being a necessary condition for existing physically relevant solution in the case of sonic rarefaction. The examples are CUD-3-type schemes with some entropy fixers. To construct entropy-consistent versions of CUD-3, one can adapt the following

2.2.

Applications to discontinuous solutions

45

strategy: e

c

(i) Entropy-consistent first-order operator A should be chosen instead of Eq. 2.1 or 2.2. The action of this operator on the grid function of the type Eq. 2.12 should not vanish which implies its non-zero numerical viscosity at a sonic transition. EC

EC

EC

4

(ii) Three-point operator A which gives [A )-'A = kD + 0[k ) is needed. It can be readily obtained as a special case of the generalized operator A described in Section 2.1. z

G

Following this strategy we chose the first-order operator as the operator defined by Eq. 2.7. To prevent vanishing numerical viscosity at the sonic point, we set if Uj < u j+i otherwise.

(2.13)

Clearly, f,+i

if

Uk

=

— U,

k <

j

and

~ fj-i

~ |/j+i/ |(wj+i

+ j/J_i/ |("j -

2

"j-l)

2

^0

= U, k > J. EC

CUD

3

CUD

3

As was explained in Section 2.1, A = A ~, where A ~ is the operator from CUD-3 conservative pair generated by the operator Eq. 2.2. Of course, (A )~* A f is the third-order differencing of f(u) if /'(u) does not change its sign while its local truncation error is of the second-order in the vicinity of sonic points. EC

BC

Fig. 2.4 shows the results of computations for a rarefaction shock as initial data. It can be seen that rarefaction fan, the physically relevant solution, is obtained instead of the spurious stationary rarefaction shock. Another entropy fixer and numerical examples are presented in [120].

2.2.3

CUD-3

schemes with flux correction

In the numerical examples presented above, wiggles are seen near the compression shocks, their amplitude being dependent on the type of CUD-3 scheme and initial data. This fact is quite understandable because CUD-3 is high-order differencing

Some extensions of basic ideas

46 j3

Initial dflla Solution with C F L - ] . Solution with C F L - 0 . 5

Fig.

2.4:

Results for Burgers equation and initial distribution E q . 2.12 while using entropy-

consistent scheme C U D - 3 3 - E C .

without any artificial devices like additional numerical viscosity, limiters etc. Due to well-known Godunov's theorem, it cannot be absolutely monotone. The presence of wiggles is not embarrassing in many practical cases. They are well localized near discontinuities and do not usually spoil numerical solutions. Moreover, they are practically absent while numerically simulating viscous flows provided that Mach numbers are not too high. Neglecting artificial filtering of wiggles seems to be quite reasonable if one wants to have "pure" solutions without any cosmetics. No questions arise in this case whether some details of the phenomena under investigation are masked and if so, to what extent they are masked. However, there are situations when CUD-3 schemes fail to operate while dealing with nonlinear problems. Such situation may be encountered if high Mach number problem is to be solved. The negative values of density and/or internal energy in one or two grid points may be rather embarrassing in this case. The idea of flux correction introduced by Boris and Book [14] and developed further by Zalesak [133) seems to be quite suitable if wiggles removing is badly needed. Using CUD-3 schemes in the flux form, it is possible to compute high-order

2.2. Applications to discontinuous solutions

47

T3

Fig.

2.5: Results for Burgers equation and initial distribution E q . 2.10 while using entropy-

consistent scheme C U D - 3 3 - E C with flux correction.

fluxes 9ji_i/2 defined by u™

+1

1

—u 1

-

m

+ my*

-

=%

i?-i/2

i < j < N

- 1 ,

where

After introducing low-order explicit flux 1^ ^ from a monotone scheme +l

—

"

-

+ vj+l/2

the antidiffusive flux ff /2

2

L

9j -1/2 = ° .

c a n

u

9j+I/2 = 9 j ' - H / 2 ( j -

1 1

j+l)'

°e defined by

+L

9J+1/2 — 9 > 1 / Z ;

9j+]/2'

Using Zalesak's interpretation of the original idea of Boris and Book, this flux is then limited by the following formula jf+i/a

=

3

i+i/2

m a

0

4 '

m i n

1

2

s

1

J

i

a

f e

a

[l^ / l' -'+ / ^ '+5 " /+i) . j-i/s(«i

(2.14) where s

j + 1 / 2

= sgn^

+ ] / 2

and uj" =

fif*'.

Some extensions of basic ideas

is

The flux-corrected CUD-3 solution is now given by +

u™ > =u?-T( f - f_ ). a +l/2 q

(2.15)

l/2

Fig. 2.5 shows the results of flux correction Eqs. 2.14, 2.15 for the double-shock case Eq. 2,10. The scheme used is the conservative form of CUD-3, namely CUD-33EC. We see that pre- and post-shock oscillations have been completely annihilated without smearing the shock. We also note that the action of the flux correction is limited to the several grid points only where wiggles are noticeable.

£.2.4

Steady-state computations: comparisons with first- and secondorder schemes

Some information about the accuracy of CUD-3 scheme as compared with several well-known methods was obtained in [77], In this work, the computations were carried out for the following problem

t

M u(0,i)=A

-*•>+•£•

«(;i,i) = B,

M

u(x,o) = (z). Wo

where A , B and 0 are some constants. The exact steady-state solutions of Eq. 2.16 were obtained in [77] for the cases " = 0, 0 ^ 0 and f j t Q, 0 = 0 with some constraints imposed on input data A , B , 0. Using the exact solution as reference one, the accuracy of numerical methods was investigated by evaluating L error defined by 2

1/2

E = 2

Lj=l and the maximum norm E

m

= max|u(ij) — t>j|.

Here N is the total number of grid points, u(xj] and Vj are the exact solution at the grid point Xj and computed solution respectively. Since only steady-stale solutions were of interest, CUD-3 approximation of nonlinear terms was used with first-order time-stepping, i.e. the scheme Eq. 1.16 with tr = i . The other methods were

49

2.2. Applications to discontinuous solutions

10"

0 01

0.O2

0.04 F i g . 2.6: L j - error,

E,.

(i) Godunov, first-order scheme (ii) Jameson, second-order scheme [55] (iii) MacCormack, second-order scheme [69] (iv) Harten, TVD2, second-order scheme [50] If u = 0 then as it was shown in [77], the exact solution exists if „

2

2

1 B - A 2G(1)-G(0)'

G(1)-G(0)/0,

where G(x) — / g(x)dx. For the proper choice of A and B, it has the form of the jump from one branch of smooth function to another, the branches being defined by l/a « ± 0 ) = ± 20(C{x) - G(0)) + A'Depending on the initial data, a discontinuity occurs at x £ [0, 1] as a jump between the positive to negative branches of the exact solutions.

50

Some extensions of basic ideas

The computations reported in [77] were carried out for k = 1/25, 1/50, 1/100 and g(x) — sin JTI, X It was noted that oscillation completely vanish when steady state was obtained using CUD-3. The reason is quite obvious: the grid function u in this case satisfies 3

where A(s) is two-point backward and forward difference for positive and negative branches respectively. Numerical data extracted from [77] are presented in Fig. 2.6 and Fig. 2.7 respectively in the form of convergence curves Ei(h) and E (k). One can observe the striking difference in the accuracy achieved for a given mesh between CUD-3 and other schemes. As it was stated in [77], with 25 grid points four digit decimal accuracy is realized from CUD-3 calculations. m

2.3

Discretization of equations in non-diver gent forms

The use of the conservative form of initial equations is natural and convenient when constructing difference schemes on the basis of the above considered operators. Such

2.3. Discretization of equations in non-divergent forms

51

schemes turn out to be three-point which facilitates solving difference equations. For equations presented in non-divergent form, the straightforward application of the operators A± and A does not result in the system with three-diagonal matrix. Consider the equation T

du %

d (2.17)

Using CUD-3 operators A±, A with the simplest discretization of the derivative du/dt, one can write the following scheme for Eq. 2.17 with truncation error 0(k } with respect to the step h T

3

"

m +

' "

U m

r

+

,

m+,

+aZ 'Ai A u /h

= /„

T

m + !

(2.18)

When a = const, this scheme coincides with the above considered CUD-3 scheme. Hence the choice of the pair of the operators A and A is regulated by the parameter s = sgn a. When the coefficient a(x,t) in Eq. 2.17 is variable, simple multiplying both sides of Eq. 2.18 by the operator A does not allow to obtain three-point equations. In this case, we introduce a new grid function q„ = A~ Au™ */h satisfying the equation i

4(S»)ft, = A ( s > :

+ 1

+

A

(2-19)

in which s = sgno„. The scheme Eq. 2.18 becomes now n

u

m

+ 1

+

r Q

-+lg

n

=

r

( u

+

£-+1)

(2.20)

+1

Substituting the function u™ expressed from Eq. 2.20 into the R.HS of Eq. 2.19, the difference system with respect to q reads n

Aq + (r/h)A(a- ) n

= ( r / f t ) A « + /"+').

Qn

(2.21)

Inverting the tridiagonal operator on the LHS of Eq. 2.21, one can obtain the grid function q . The solution function u™ ' can then be extracted from Eq. 2.20. +

n

52

2.4

Some extensions of basic ideas

Another form of third-order compact upwind differencing (CUD-II-3) Difference

operators

So far we have been considering compact differencing of the form 1

kD ^

A- (s)A(s)

x

4

+ Olh ).

Another possible type is k D ^ ^

4

+ R^Q. + Olh ),

(2.22)

where ifj, and QK are three-point operators. To obtain these operators, we start with the Taylor expansion series for A(s) which can be cast in the form D = h-A(s) x

+ ^D 1

2 T

- § 2 £ + 0(h% b

(2.23)

where s is supposed to be a constant. Instead of straightforward discretization of D\ and D^, we make use of the chain of equalities

2

All we need now is to discretize D and D at least with the first and second order respectively. Of course, some arbitrariness exists while choosing these discretizations. T

Approximating D by A(s )//i with the other constant parameter Si, we represent Eq. 2.23 as x

:

where £3

=

A(s) + ^ ( / + ^ A { s , ) ) " ' A jh 2^ 3s ;

(2.25)

and TE stands for truncation error. Hence, R

H

= / + f A(s,),

QH= -&2. S

For any grid function u the action of the operator L can be computed in the following way: 3

2.4. Another form of third-order compact upwind differencing (CUD-ll-3)

53

(i) Solve the tridiagonal system (/+^A(3i))w = A > ;

(2.26)

(ii) Compute L u - { A ( S ) U + 0.5su)//i. 3

3

Using Taylor expansions, it is easy to verify that TE = 0(/i ). Moreover, the following estimate holds for a sufficiently smooth tlfxj du £swj = 3-

,1s, 1 \, d*u T2~7 ll^Td* 3

+

I=I.

a

I

< I.

+

(2.27)

Upon inspecting Eq. 2,27, we observe that the truncation error for L is greater than that for CUD-3 if \s\ = 1. However, it can be decreased by increasing |s| and decreasing S,. 3

The next important point is the ability of L to act as a positive operator in difference schemes. To estimate 4 ° ' , the self-adjoint part of L , we calculate the inner product (L u, u). In doing so, let us consider again Eq, 2,26 setting R\ — I + ( l / 3 i ) A ( s i ) . We use then the chain of equalities 3

3

3

h(L u, 3

«)

= (R~ ' [fl A(s) + i & j l u , «) = ([jfeAjft + f A a k R,v) h

A

(2.28) =

([RlR A(s)+iR^}v, ), h

V

where v = R^u and therefore u = R v. h

To verify the positivity of L , one should calculate the self-adjoint part of R' RhA(s) + (s/2)i?jA;. It can be readily accomplished to give 3

h

(2.29)

where it = S i / s . We observe now that if k > 0, then the first term in the brackets on the RHS of Eq. 2.29 is a positive operator since ( — A ) > 0, The second one is also a positive operator due to ( — A ) < 4. Hence, the inner product in Eq. 2,29 is positive and sgn(L u, uj =sgns. 2

2

3

54

Some extensions of basic ideas The above results can now be summarized in the form of

Theorem 2.1. Let L be defined by Eq. 2.26. Then it is a third-order differencing operator with truncation error given by Eq. 2.27. Let further that sgnS] = sgns. Then i is positive (negative) if s > 0 (s < 0). 3

3

The differencing formula Eq. 2.25 with the proper choice of sgns when applied to the convection-type equations will be referred to as the third-order compact upwind differencing of the second type (CUD-II-3).

t4.2

Some properties

of

CUD-II-3

Difference equations. The linear system with three-diagonal matrices is to be solved for computing the action of L on the known grid function (Eq. 2.26). It can be written in index notation as 3

^{l-*l)%+t-r- (l +

3 7 ) % - = A f l &

'j = U-,N-\.

(2.30)

Boundary conditions are needed to solve Eq, 2.30. To provide them, it should be noted that vj — h D + 0(h ). It means that some approximations of D at the end points x = x and x = IJV are admissible. 2

2

3

2

a

It can be deduced from Eq. 2.30 that this system is well conditioned since |l +

^j>

+

moreover, its matrix is M-matrix if Si > 1. Thus we see that solving Eq. 2.30 creates no problems, the quality of the solutions being expected quite satisfactory. Dispersion and dissipation. The phase errors for semi-discretized convection equation — + a L u = 0, a = const at namely the ratio of the scheme phase velocity to the exact one, can be easily calculated as 3m W(a)fa, where H7(a) is defined by 3

kL exp(ian) 3

= W(a)exp(ian),

n = 0 q=l, q = 2 , 0 < a < 2JT,

a = kh.

2.4. Another form of third-order compact upwind differencing (CUD-II-3)

55

The result is SinW(a)

sine*/

jjj

),

21

1

\

, , a 1 . 1 = sin' - .

J

Using Taylor expansions for sine* and sin (o/2), we observe that

As in the case of CUD-3, the phase errors are seen to be quite small for small dimensionless wave numbers a = kh. The attenuation of harmonics due to the action of £ per time hja may be characterized by exp(—d), where 3

2a/ d= dleW(a) =

2

where / = sin (a/2). The use of Taylor expansion yields d=

4

12

18|a

a

+ 0(Q ), 6

thus indicating negligible amplitude errors for small a. The graphs of 3 m W(a)/a and 3ie W(a) are shown in Fig. 2.8. Comparing this figure with Fig. 1.1, we see that CUD-3 and CUD-II-3 curves give approximately the same range of a for which difference solutions are very close to the exact one (0 < a < T/4). Note that the shortest wave dissipation d(ir) is equal to 4sk/(3 + 2k) as compared with 2s for CUD-3.

2.4-3

Conservative

forms

The action of the operator L at any internal grid point x = Xj can be presented for 3

S\ — s — const

as

where _ u hq /2j+l

i+1

+ Uj ^

s, 2^

„-l\Uj+l ~ ttj ' *~~2

_ '

1 3~7 *

Some extensions of basic ideas

56

2 0

T c'/c, d

1.5

F i g . 2.8: Dispersion and dissipation of C U D - U - 3 for Gi = s = 1 (solid line), s j = 0.5, # = 2 (dashed line) and s i = 0 . 2 5 , s = 4 (dotted line).

Thus we can see that b is suitable for the approximation of conservation laws provided that the slope of characteristics does not change its sign. In a general case some modifications are needed to provide the conservation property of CUD-II-3. It can be accomplished in the following way. First we replace the operator sAj by A ( 7 _ i / s ) A _ . We remind that the latter looks in the index form as 3

,

+

2

A (r_ +

1 / 2

s)A_/, = s

J + 1 / 2

( / j i - / , ) - s,_, (/ - / , _ , ) . +

/2

;

As a consequence, Eq. 2.2 should be used instead of Eq. 2.1 when defining A(s) in Eq. 2.22. The next step is to modify the inverse operator in Eq. 2.22. Following the strategy of Section 2.2 we write down the Taylor expansion for the conservative A(s), namely A ( s ) , to obtain c

Di =

+ \DZSD:

- jDl

3

+ 0(k }.

(2.31)

Eq. 2.31 may then be cast in the form D - j-A (s) = ^ D ( / + y^Y'D; l

x

c

l S

3

+ Oik ).

(2.32)

Our aim now is to construct second-order conservative discretization of the operator on the RHS of Eq. 2.32, thus preserving 0(h ) truncation error for D approximai

x

2.4. Another form of third-order compact upwind differencing (CUD-II-3)

57

tion. It can be done in several ways. For example, we may write 0

0

~ ST *)"' *

=

C

*-*&+{T_ s)(l

+ ^A (S,,S))~'A_ +

m

0(k*)

and c

hLi = A ( s ) + i A ( J - _ +

c

1/jS

)(7 + iA (s,,s))"'A.,

(2.33}

where A {&%,a) is given in the index form by (Li \s s)v ,/2 u

—

1+

c

c

We note that A ( s i , s ) in contrast to A ( s ) acts on the grid functions which are defined in the mid-points of our mesh wj,. We designate the set of these mid-points by Let Uh and (A be sets of grid functions defined on ui^ and i i respectively. Then A_ and A in Eq. 2.33 may be considered as the operators which perform the mappings £//, —1 t/j, and Uh —• Uh through the relations i*j_i/2 = A _ U j = tij — u,., and wj = A U j _ = v - i > j _ respectively. n

+

+

1 / 2

1 / 2

j+l/2

Now we have c

l

(7'_ s)[/ + A / 3 ] " A - :


1/3

A (T_ +

C

1 / 2

_ 1

s)[/ + A / 3 ] A _ :

U ^U h

h

and

We also observe that if s = sgn f'(u) then s, and S j in Eq, 2.33 are practically always non-zero which excludes singularities in A ( s , s i ) . However, care should be taken if s = /csgn /'(u), where k is some function which gives a smooth transition of s in the vicinity of sonic points. Finally It should be noted that the third order of L as in the case of CUD-3 is preserved everywhere if s is the restriction of some smooth function. + )

c

c

Considering again Eq, 2.33, the following sequence can be adopted for computing the action of L%: (i) Solve c

(7+ A (a,,a)/3)« -. J

1/2

= u -«,-.„ i

3= 1

N;

(2.35)

.58

Some extensions of basic ideas

(ii) Compute for each grid point x — Xj, j — 1 , N — 1 c

L uj

c

- (A (s)uj + 0 . 5 s

i+1/2

u

- 0.5s _ Uj_i )//!. J

i+1/2

1/2

/2

Of course, some boundary conditions are needed to perform step (i). It may be seen that if s = const then L = L and Eq. 2.35 leads to the same result as that obtained via the algorithm Eq. 2.26. 3

2.4-4

Some comments on CUD-II-3

3

schemes

If L (or L%) is considered as an upwind space discretization operator then rich variety of schemes may be constructed for unsteady equations using different types of time stepping. From implementation point of view, there is no difference between CUD-3 and CUD-II-3 as a part of explicit schemes. The inversion of tridiagonal systems is needed in both cases for computing the actions of spatial operators, the operation count being proportional to the number of grid points. Due to the positivity of these operators, any explicit scheme with reasonable time stepping is conditionally stable in L norm for initial value problems. 3

2

The difference between CUD-3 and CUD-II-3 becomes apparent when implicit schemes are considered. For instance, suppose that the simplest two-level implicit linearized scheme is used for scalar conservation law Eq. 2.3. It can be written in CUD-II-3 case as 1

/ + T-Lj/'tV ))

m

+ L f(u )

= 0,

3

(2.36)

m

where the parameter s is defined by sgns = sgn/'(u ). Scheme Eq. 2.36 is unconditionally stable if frozen coefficients are assumed. Unfortunately, the inversion of the implicit operator in Eq. 2,36 is more time-consuming as compared with this inversion in CUD-3 case. The reason is that the difference system obtained via multiplying by (/ + A(si,s)/3) in Eq. 2.36 is a pentadiagonal system rather than a tridiagonal one. More reasonable application of CUD-II-3 seems to be that in the case when the main concern is the steady-state solution of Eq. 2.3. Instead of Eq. 2.36 we may then write m

a+,

U

/ + TL f'(u y ~ '" i

i

m

+ L / ( u ) = 0, 3

(2.37)

2.4. Another form of third-order compact upwind differencing (CUD-II-3)

59

where Ll is the easy-invertible first-order upwind operator. Clearly, b.(s) is the proper candidate for L 1 . Using Fourier analysis, it can be shown that scheme Eq. 2.37 is unconditionally stable if Ll = b.(s) . If T ---+ 00 then Eq. 2.37 becomes

which may be recognized as a defect-correction type of a scheme. As this takes place, 1 - Ll1 L 3 , the transition operator of iterations, is a contraction operator which provides the convergence of Eq. 2.37, the rate of convergence being independent of mesh size.

2·4·5

Flux-splitted forms of third-order CUD

In the previous chapter and sections, the general idea of changing upwind direction was used to preserve the positivity of compact approximations while crossing sonic points. Another well-known and widely exploited idea is the flux splitting (see, for example [99], [67]). When applied to the scalar conservation law Eq. 2.3, it can be presented as the decomposition

+ r(u)

j(u) = r(u)

(2 .38)

with oj+ lou> 0 and oj- lou < 0 everywhere in the computational region. Considering L(s) as one of the third-order CUD operators, we discretize oj(u)lox as

Lj(u) = L(s)r

+ L(-s)r,

(2.39)

where s is supposed to be positive. If j+ = au, a = canst> 0 and j- = bu, b =

canst < 0 then Lj(u) = [aL(s) where aL( s)

+ bL( -s)

+ bL(-s)]u,

is a positive operator.

As a matter of fact, the positivity of this operator is not necessary associated with the positivity of or lou and the negativity of oj- lou. All we need here is the inequality

or ou

ojou

-- - -- > 0

or

a - b > O.

To prove this, we present L as L(O)

> 0,

6U

Some extensions of basic ideas 1

0

where i * ' and i ' ' are skew-symmetric and self-adjoint parts of L respectively. Then we can write i/(u) = L

( l !

+

{/ d-r) + ^

(

0

)

+

(/ -r)-

For the frozen coefficients case the operator (a 4- fiJZi"' is skew-symmetric while the operator s(a — b)L^ is self-adjoint and positive if 0 — 6 > 0. Of course, this reasoning is quite general and L may be regarded as any upwind operator. There are many ways of presenting /(u) in the form Eq. 2.38. Quite natural one is to put

+

/

= \{fW)

+ Cu),

l

/ " = -(f(u) -Cu),

C = const > 0.

(2.40)

Using Eq. 2.40, we obtain (

£ / ( « ) - £ " ' / + |(7L %.

(2.41)

Upon an inspection of Eq. 2.41 one may see that the action of £, the fluxsplitted form of third-order CUD, consists of two parts. The first part is the fourthorder skew-symmetric approximation of Of/di while the second one is the thirdorder dissipation. Note that the latter term looks as sL'°'/(u) if no splitting is used. To provide positivity of L , C should be any positive constant. Therefore df /du is not necessary positive. However, C may influence the diagonal dominance of the matrix to be inverted when computing the action of the operator L . +

The advantage of flux-splitted third-order CUD is that no switching is needed while crossing sonic points. Hence, conservation property is preserved without any loss of the approximation order. Moreover, it follows from Eq. 2.41 that standing rarefaction shock is not the exact solution of difference equations and L is therefore entropy-consistent. +

In closing this section, we remark that sufficient smoothness of f (u) and f~(u) is needed to preserve third-order accuracy if another choice of splitting is used instead of Eq. 2.40.

61

2.5. Symmetrization of the CUD-3 operators

2.5

Symmetrization of the CUD-3 operators

2.5.1

Fourth-order

compact approximations

As was emphasized earlier, CUD-3 schemes as a family of upwind schemes are appropriate for convection-dominated problems, the representative examples being either discontinuous inviscid or, preferably, high Reynolds number flows. However, if exact solutions do not exhibit rapid variations (the example is low Reynolds number flows) then centered schemes seems to be advantageous. The advantage of centered compact schemes as compared with CUD-3 schemes is due to the fact that high accuracy of these schemes is accompanied by the reduction of operation count. From the algorithmic viewpoint, the simplest form of fourth-order compact differencing may be obtained by simply setting s = 0 in all CUD-3 formulas. Then the operators A and A reduce to A and A / 2 which are exactly well-known operators for convective terms in fourth-order schemes ([27], [81], [52], [1]), 0

0

For zero diffusive terms, one of the versions of these schemes may be written as the scheme Eq. 1.15 with s = 0. 1

m

A + 0.5—Ao^'lw" ))

+ Q.5-~v{v ) = Mf-

0

(2.42)

Obviously, the implicit scheme Eq. 2.42 is unconditionally stable for a > 0.5. It is also clear that using the pair (/In, A ) instead of (A, A ) decreases the amount of arithmetic operations while implementing the scheme Eq. 2.42. 0

The fourth-order analogue of 0(r ,,771+1 _

A

0

m-l

3

3

+ k ) scheme Eq. 1.38 reads j

+ T^Ac^iT) = ^ ^ h 0

T

m

D I

/ ,

(2.43)

where -4 / = g /

m + ,

0[

m

1

+ - r + -/ - .

A

The truncation error for Eq. 2.43 is 0(T + h*) while its stability is only conditional. Such kind of time stepping was mentioned, for example, in [64]. The numerical experiments [72] with the scheme Eq, 2,43 showed that as it might be expected, strong spurious oscillations arise when computing discontinuous solutions of inviscid Burgers equations. However, striking accuracy of fourth-order compact schemes applied to incompressible Navier-Stokes equations was reported [45]. In fact, majority of fourth-order compact schemes which have been mentioned

Some extensions of basic ideas

62

in the introduction were designed for solving convection-diffusion equations. One may expect that the scope of their CFD applications is limited by relatively small Reynolds numbers.

2.5.2

CUD-3

with MacCormack-type

time stepping

Another way to avoid switching of one-sided pair of the operators is to use them alternatively for successive computational cycles. Such a procedure may also be considered as symmetrization of a sort. In this Subsection, we will combine the idea of MacCormack time stepping [69] with <4~'A approximation. In doing so, the aim in particular is to illustrate the compatibility of non-centered compact differencing with any reasonable time discretization. Consider again the scalar conservation law Eq, 2.3, To advance the numerical solution from t = ( to ( = t + i , we denote by (.4, A ) an arbitrary pair of operators (<4+, A _ ) or (A_, A ). Let the other pair be (A, A ) . The first step now can be written as Au + r&f(u) = Au ", r = r/h, (2.44) m

m

+

1

while the second step is m+

• 4 u ' = 0.5 [Af

m

+ Au- r A / ( i ) ] .

(2.45)

To investigate the approximation order of the scheme Eqs. 2.44, 2.45 it is convenient to multiply Eq. 2.44 by A" and Eq. 2.45 by A~ Setting N = A~ A/h and N = A~'A/h, the equality Eq. 2.45, in view of Eq, 2,44, can be written as l

u

m

m+i = m

m u

l

- ^ ( ; V + #)/(£). 2

m

Since f{u) ~ f(u ) + Oo(u - u ) + 0(r ) where a = f'(u ), the difference according to Eq. 2.44 is equal to - T I V / ( U ) . Therefore we can write 0

+ ^ ± * i / ( « " ) - § ! > + XfaNW)

= 0.

u-u

m

(2.46)

Clearly the operator (N + JV)/2 is the fourth-order accurate differencing operator. One can also see using Taylor expansions that 0{T) terms cancel when Eq. 2.46 is compared with the exact Eq. 2.3. Hence the truncation error for Eo,- 2.46 is 0 ( r + A ). s

4

2.5. Symmetrization of the CUDS operators

63

Analyzing the stability of Eq. 2.46, we set / ( t i ) — a t i , where it is a constant. Excluding the intermediate function u, we obtain m

1

u" *' = 0 . 5 ( / + « ) w

n

1

m

m

=

Ru ,

where Rt = A-^A-arAj^A

+ ar&y'A,

r = r/h.

As all operators in fl, commute, the following inequality holds < | | ^ - M | | | | ( A - arA)(A

+ WrA)T%

(2.47)

-1

In this inequality ||/1 «4|| = 1 since (A'-'AU, IIA-'AH = s u p i

A-'AU) j

—

(A'-% ^

=

(«, u l

s

u

p

A-^V)

\

"

v

[A-'v,

=

A

u

A-'V)

by virtue of the fact that the operator A is normal (A'A — AA~). Taking into account now that the second norm on the RHS of Eq. 2.47 is equal to unity as a consequence of Kellog's lemma [71], we get the estimate 11*11 < 0.5(14 ||flij|) = l which is evidence for the unconditional stability of scheme Eq. 2.46. Comparing the scheme Eqs. 2.44, 2.45 with symmetrized scheme Eq. 2.42, one may see that the former requires one extra inversion of three-point operator. However, it possesses some dissipation mechanism to suppress high-frequency noise which is absent in scheme Eq. 2.42. To show this, we consider the third term in Eq. 2.46. If / ( u ) = au, then i i = (/ + TNa)' ^ so that 1

+ N)aNf(u)

= -^(N

1

+ N)N(l

s

m

m

+ T-A'Q)~ o u - G u .

If a — const, the following relation holds for inner product (Gu, iij (Gu,

U) = - ^ ( # , ^ 0

,

2

+ A ,)a u, (/ + rJVo)jj),

where v — ( I + riVa)~'u and A'o, JVj are the self-adjoint and skew-symmetric parts of the operator JV respectively. The RHS of this equality can be presented as 2

2

3

- ^ ( ( M f ^ v , v) + r(N, [ T V + rf/V - N>) ]v, a

v).

64

Some extensions of basic ideas

Clearly, the first term of this expression is positive by virtue of Nf = (iVf)' < 0. The second term is equal to zero since the operator JV) is skew-symmetric and the operator in brackets is self-adjoint. Thus the operator G is positive and hence it is responsible for scheme dissipation. If unconditional stability is not needed, then one can use alternating operators A and A for constructing explicit scheme which is similar to MacCormack scheme, it may be presented in the following manner. As the first step, u is calculated from the equation m

u = u™- rNf(u ), m+l

and then finally u

N = A;'AJk

(2.48)

is computed as m

= 0.5(5 + u - T W / ( U ) ) ,

A> =

A-'Ajk.

Excluding u with the help of Eq. 2.48, we may write this scheme as w

+

w

" ' - " + I(JV 4- % f > ~ ) -

T

-N Nf{vT) a

= 0.

(2.49)

2

One can easy verify that its truncation error is equal to 0 ( T 4- h*). The second order with respect to T arises because the last term in Eq. 2.49 approximates (r/2) ( c ' u / 3 i ] _ in Taylor expansion of the solution u(x,t). 2

2

i

(

Since N = —JV", the inequality (-NN) > 0 holds. The scheme Eq. 2.49 is explicit scheme with a positive operator, hence it is conditionally stable. To implement this scheme one should invert the operators A and A with constant coefficients.

2.6

Third-order compact differencing cretized Pade approximants

as dis-

At this point, it is worth emphasizing that close relations exist between compact differencing and Pade approximants. CO We remind that Pade approximant of the Y, C** is given by

*=°

1 + E ****

2.6. Third-order compact differencing as discretized Pade approximants

65

the truncation error being The coefficients a* and dj, can be readily obtained from the relation ( l + E hz") £ « « * = £ li=l

Jr=0

+ 0(*" " ) +

+ l

*=0

which gives the linear system with (m + n + 1) X (m + n + 1) Toeplitz matrix. To derive third-order compact differencing, we start from the expansion for the operator A. Writing it in the form D, = A-'A(s) + ^Dl

- |iC5 +....

D, = ±

t

(2.50)

we formally set Co — n A , ci — [sj2)D and c = —(1/6)Z? . Suppose now that one wants to approximate Eq. 2.50 to the third order by _ 1

3

x

2

2

(l + b,h + b h y'a , 2

(2.51)

0

where 6j and b are some differential operators. Using standard procedure to evaluate a , 6| and b as a coefficients in Pade approximants, we obtain 2

a

Since c =

2

2

A(s) = D - (sh)/Z Dl + 0(h ), one can easily verify that x

f +U + M

1

= 1- y f i - + f ^

+

3

0(h ).

2

Approximating now D and Dl by 0.5A /'» and A j / n respectively, we finally get x

£„=[/-

o

3

A + ^ A , ) " ' A(s)/A + 0(fc ). 0

The operator in the parentheses is readily recognized as CUD-3 operator A(s). To derive CUD-II-3 formula, we use again the expansion Eq. 2.50. This time, however, we want to apply Pade approximants to

&«-{«+. to obtain Eq. 2.25.

Chapter 3 Fifth-order non-centered compact schemes

This Chapter deals with the fifth-order compact approximations generated by upwind differences. Their main properties are very similar to that of CUD-3 described in the previous chapters. However, they have an advantage of possessing higher order truncation errors with small numerical coefficients before them. This makes them significantly more accurate than any conventional six-point approximation. The fifth-order compact upwind differencing (CUD-5) was proposed by the author in 1985 [115]. The first CFD application of CUD-5 was described in [37]. Other forms of CUD-5 derived via Pade approximations were presented in 1991 [119]. Their testing was carried out for several model problems. The aim of this Chapter is to give the reader the initial information concerning this type of little-known discretization. The author hopes that the ideas presented here may be useful for constructing high-order sophisticated algorithms.

3.1

Fifth-order compact upwind differencing

Let u(x) be a sufficiently smooth function restricted to the mesh ^ : {xj = jh], h — const. As a direct extension of the CUD-3 relation A{ )m S

}

= A(s)u /h, 3

f%i =

+ 0{h%

C7

3.1. Fifth-order compact upwind differencing

where {A(s), A(s)} is the CUD-3 pair defined by Eq. 1.15 in Chapter 1, we will find a triad of three-point operators -4i(s), <4 (s) and A(s) in the relation s

4i(a]ft»j + hAi(s)Mj = A(s)uj/ft.

(3.1)

Here *=i

k=i 2

and m-j, Mj are the approximations of (dujdx)j and

2

(d u/dx )j.

Following the idea of the undetermined coefficient method [48] we want to obtain a , i j t ( i = 1,2,3) for which the approximate relation Eq. 3.1 between the first and the second derivatives of u becomes identically accurate if u — x, x , i . Substituting these functions into Eq. 3.1, we derive the following linear system / fa, \ 1 1 1 0 <j 0 U ) -2A 0 2ft 2 2 2 d 3A* 0 oh -6ft 0 6fc d (3.2) -4h 12ft 0 4ft 0 12ft* i, d* b d 5ft* 0 oh* -20ft 0 20k -6A 0 6ft Wh 0 30ft* yds) t

2

5

2

2

3

3

3

1

3

3

2

S

5

5

4

where d* = A(s)i*/ft. Determining the coefficients at and A* (fc = 1,2,3) from Eq. 3.2, we then rewrite Eq. 3.1 in terms of the operators A and A as 0

5

M = Ao(«)u/A.

(3.3)

S

If we now want to obtain from Eq. 3.3 as (du/dx)j + 0(A ), it is sufficient to use any representation of the form Mj = (d u/dx )j + 0(ft ). The suitable representation is 2

Mj= U-r^i)

2

Aiu/h.

4

(3.4)

Let u be a known grid function. To determine m from Eq. 3.3, one should obtain Mj by first inverting (7 4- A /J2) and then inverting three-point operator I 4- 7Aj/30 — 3sA /16 with the known grid function 2

0

A (s)u/k 0

+h

Fifth-order non-centered compact schemes

6K on the right-hand side.

Substituting Eq. 3.4 into Eq. 3.3 and multiplying the result by (7 the following relation can be obtained

Aj/12),

(3.5) 16 I ^2 40 The approximation properties of m defined by Eq. 3.5 can now be easily established using Taylor expansion series in the vicinity of the grid point x = xj. They can be summarized in the form of

Theorem 3.1. Let u € satisfying Eq. 3.3, 3.4 or Eq. 3.5:

A^.+SJ' Then the following estimate holds for

A look at Eq. 3.6 shows that mj is a very accurate fifth-order representation of due to the factor 1/1920 which is several order of magnitude smaller than that of the usual six-point formulas for du/dx. (dufdx)j

The coefficients ot and 6* in Eq. 3.1 can also be obtained via technique which is similar to deriving Pade approximants of power series. It can be described as follows: Setting z = kD

r!

we write the relation

z = (/ + 0, A + b A y' 0

where

2

2

2

[A(s) + z (a I 0

+A ai

0

+ a A )\ 2

2

5

+ 0(z ),

(3.7)

3.1. Fifth-order compact upwind differencing A(s] = 0.5(A -sAa) 8

and an, at, 6* (i; = 1,2) are unknown coefficients. To determine a* and b^, we consider the following representation for the variable z 1

2

aoz + 2(0.5 + a,z ) E £^r, z= f

- 0.5s)(z + 2 E 7 ^ 7 ) ^— ^ - ^ + 0 ( z ) . (3.8) 2

+

1

5

™

The left-hand side of Eq, 3,8 may be regarded as the series E c*^ with cj, = 1 k=0

for k = 1 and = 0 for any other k, the right-hand side of Eq. 3.8 being some approximation of this series. In contrast to the conventional Pade approximants, the unknown values an, ai, a , i j , 62 are the coefficients of some linear combination of z* rather than the coefficients of z themselves. Nevertheless, the technique of determining these coefficients is the same: one should multiply Eq. 3.8 by the denominator of its RHS and equate the terms with z*, k — 1,2,3,4. Once the values of at and bi, are known, all we need is to replace z in Eq. 3.7 by its proper discretization. The suitable one is (/ + A / 1 2 ) ~ ' A which is h Dl to the sixth order with respect to h. Note that the representation of Ao and A in Eq. 3.7 is more accurate than this discretization of Dl- Hence, 0(h ) formula for mj can be derived using a more accurate formula for D . 2

k

2

2

2

2

2

e

2

3.1.1

Properties

of CUDS

operators

Rewriting Eq. 3.3 as

where £ 5 = B~'C and

hC = s(-A )Q/ + ^ A j + A ( | j + ^ A j , 2

a

0

2

(3.9)

it can be shown that £ is a positive or negative operator depending on the sign of a

s. Suppose s > 0 (the case s < 0 can be treated in the same manner). Then the following inequalities hold: C>0, (3.10)

Fifth-order non-centered compact schemes

70

B'C > 0,

(3-11)

B~'C>0.

(3.12)

The first of them is obvious because s ( - A )((3/8)7 + (1/16)A }, the self-adjoint part of C, is positive as the product of two positive self-adjoint commutative operators s(—A ) and ((3/8)/+(1/16) A ) . In order to establish Eq. 3.11 it is sufficient to calculate the product B'C and consider the self-adjoint part of it, (B'C?)' '. In doing so, one should use the properties of A and A and the equality A , = 4 A + A in particular. The final result is 2

2

2

2

0

2

0

2

2

2

2

By virtue of ( - A ) > 0 and A , < 4 ( - A ) we have 2

2

3

(-A )>0,

& <4(-Al) 2

0

and therefore the positivity of (r?'C)' ' is established. The inequality Eq. 3.12 immediately follows from Eq. 3.11 due to the commutativity of B and C: l

(B~ Cu,

u) = (Cv, Bv) = (B'Cv, v) > 0,

l

where v = B~ u. We summarize now the obtained results in the form of

X

Theorem 3.2. The operator i = B~ C, where B and C are denned by Eq. 3.9 is positive or negative for s > 0 and s < 0 respectively. 6

*

It is clear now that stable schemes can be constructed using the operator L with the proper choice sgns. Let us consider, for example, semi-discretization of the equation du du a = const. (3.14) dt + dx = 0, It has the form du — + ai u = 0 (3.15) s

b

3.1. Fifth-order compact upwind differencing

71

with sgn s = sgn a. The stability of the scheme Eq. 3.15 in the L norm is evident. It can be established for the maximum norm as well. 2

Henceforth the use of the operator i defined by Eq. 3.9 will be called the fifth-order compact upwind differencing (CUD-5). 6

The dispersion and dissipation properties of the CUD-5 can be easily seen from the exact solution of Eq. 3.15 with initial data u(0,Zj) = exp(ifcfej); ,ai

Uj = u(t)e ,

a = kh, j = 0,^1,^2,...,

o < a < 2JT.

(3.16)

Substituting Eq. 3.16 into Eq. 3.15, we obtain Uj{t) = cexp(—Z()exp(iaj), where Z(o) are eigenvalues of L% corresponding to its eigenfunctions exp( — ioj): hL exp(—iaj)= b

Z(a) exp(—iaj).

Then the phase errors for Eq. 3.15 can be estimated as c" Sim Z(ct) — = —, C

,, a = kh,

O

,„ , (3.17

where c" is the ''scheme" phase velocity and 9m(Z) is given by ^ a )

= s i n

Q

( l - ^

+

^ - ^ p ) / 2

f

l

(3.18,

with

After using some algebra, one may deduce from Eq. 3.17, 3.18 that 6

^ = 1 + 0(a ). It means that phase errors are negligible for sufficiently long waves as compared to the mesh step h. The calculation of the measure of the dissipation d = |s|Sfe Z(a) which characterizes the attenuation of the harmonics per time hj\a\ gives <<=j^(l-
72

Fifth-order non-centered compact schemes C*/C

1.0

kh nil

Fig,

3,1:

Dispersion

(solid

line) and dissipation (dashed line) of L$.

where / and g are defined in E q . 3.18. One can see that d = O(a'). Hence, the dissipation is also negligible if waves are not too short. Figure 3.1 shows the graphs of c'/c and d as functions of a = kh. for |s| = I , It is evident from Fig. 3.1 that c'/c s: 1 and d = 0 for a wide band of wave lengths except those which are comparable with h (TT/2 < kh < jr). This domain is larger than that for CUD-3 schemes.

3.2

Other forms of compact upwind differencings (CUD-II-5)

3.2.1

Using Pade. approximants

The direct extension of the technique which was used for constructing CUD-II-3 (see Section 2.4) is the including of additional terms in Taylor series expansion. It can

3,2. Other forms of compact upwind differencings (CUD-ll-5)

73

be written now as 3

OB = A-'A(s) + f / -

s

+ £ j W J * - g ^ * D j j |W3* + 0 ( n ) ,

(3.19)

where non-zero parameter s is supposed to be constant. The idea is to replace the expression in parentheses by Pade approximants which preserve the fourth-order truncation error. Two Pade approximants will be of interest to us. They are

and

Using standard technique, we multiply equations 3.20 and 3.21 by 1 + b\y and l+ii!/+&2>/ respectively to obtain the linear systems for unknown a; and 6; (i = 1,2). The solutions of these systems are given by 2

2A

O, = a

2 1

= -2/15*,

2

2

1

a = a ' = (5s - 4)/60s ,

6, = b]' = l/5s

2

(3.22)

and 2

2

2

6, = | ' = (21s - 20)/15s , Q

b\ = b\ = 2s/5 ,

S

1

5

2

2

b = b / = (5s - 4)/20o,

g = 4 - 3s .

2

(3.23)

2

Here we assume that 4 — 3s ^ 0 for Eq. 3.23. Now the expansion Eq. 3.19 may be replaced by 0 , = h-'A(s) + (s/2)/ (AO )AD + 0(h% J

2

I

where l

P = R- Q,

fi-J

+

fcC'^W*. jt=]

k

Q = I + Za^(kD ) I

(3.24)

k=l

and m, n. are 2, 1 or 1, 2. k

The next step is to discretize (hD ) in Eq. 3.24. We observe that it is sufficient to approximate hD and h?D\ with the third and second order respectively to preserve 0(h ) truncation error for P However, there is some arbitrariness in the choice of difference operators Ri, and Q^, which are the approximations of R and Q. Using this arbitrariness, we restrict ourselves with those discretizations which require three-point inversions for computing the action of the operator R^Qht

T

4

Fifth-order non-centered compact schemes

74

To derive the final forms of Q and Rh, suppose now that Q^K the fourth-order approximation of Q, is already obtained. Then we have to find the fourth-order approximation of the equality h

Rf = Qu,

(3.25)

J

where / = Pu = R~ Qu and u is a sufficiently smooth function. In doing so, we want to obtain a three-point operator fly,. This leads us to the equality (/ + 0.5Ar' A + C " A ) / N

O

2

N

=

flfW

(3-26)

Here h denotes the restrictions of the functions u and / . Unfortunately, Eq. 3.26 is only third-order approximation of Eq. 3.25 due to A / = hD f + 0(h ). Hence, an additional term is needed on the RHS of Eq. 3.26 to compensate the third-order term on the left-hand side. This term can be obtained from the estimate 3

0

x

4

hDJ^^h-^Dlh+Oih ), 2

where / = / > « = ( / - (k/Zs)D^u + 0(h ). Now the modified RHS of Eq. 3.26 can be written as (Q{^ + ^ A ^ + OO ), (3.27) 1

where /\ u>,fk

3

3

is any approximation of D u.

3

Equation 3.26 is not a unique discretization of Eq. 3.25 with three-point operator Ru, It follows from the fact that one can add the terms 71 A / ; , and 7 J A J / / , to the LHS of Eq. 3.25 and compensate them by adding some terms to the RHS, preserving 0(k ) truncation error. The compensation can be carried out using the following relations which are valid for / = Pu A

4

A „ / = ^kDt-Z-h'Dl

4

+ ^Dlju

F T

2

l

A . / * = {h Dl-

3

3

h Dlj

+ Oik ),

4

U

+ 0(h ).

(3.28)

(3.29)

Taking into account Eqs. 3.27, 3.28 and 3.29, Eq. 3.26 can be rewritten in the form 7 + (0.5&7'" + 7 . ) A + ( A r + 7 , A 0

2

7i

72

2

,1

7+( r'"+270^,+K'"+7 -^7 )" ^+(y-g+^r'")A D. a

2

1

2

3

uh, (3.30) h

3.2. Other forms of compact upwind differencings (CVD-II-5)

75

where [•](, is the discretization of [-] and 71, 72 are arbitrary constants. Now using Taylor expansions, it can be shown that Eq. 3.30 is the fourth-order approximation of Eq. 3.25. To derive the fourth-order accurate difference form of the RHS of Eq. 3.30, one should use the third-, the second- and the first-order formulas for hD , h D\ and h?D\ respectively. The following expressions for the operators R and Q can be written as a final result: 2

x

h

Kk = I+ (0.567'" +

7 l

) A + (!?'" + -ft) A , 0

2

QK = I + ( o T + 2 , U - i > i ) A ( s ) + L £ - ~ n

h

,

7

1

+

1

2

- £

7 l

) A

2

(3.31) 6

+(^-i7 +l r)A A{s ). 2

3

Here the third-order compact differencing is used for hD with the parameter S\ while h?D is approximated by A A ( s ) with the parameter s . r

3

3

2

2

The constants 71 and 72 in Eq. 3.31 have the following remarkable property: the coefficients oj* , 6;' in these formulas can be replaced by a-' , b{ with special choice of *yj and 73. For example, let 71 and 72 be determined from the equations 2

l

+ 0.56|' = 0.5if ,

7 l

% +

which is equivalent to using if" and with 71 = 72 = 0 in the operator R&. Then simple calculations give the following result: 1

La , „

oj' +

27,

2,1

= a, ,

a

1,2 2

,

+

2,1

2

72

-

r-7i

= a

2

>

Ti

"2"

72

~ 37

,

6 '

=

6

1,2.1 1

This means that 0 ' , b ' can be used with 71 = 72 = 0 instead of a]' , b}' with j , and 72 defined above. s

of

1

2 1

2

2

Keeping in mind this property, we will henceforth assume that the constants and S f a r e defined by Eq. 3.22 with the superscript of af \ of' being omitted. 1

Upon inspecting Eq. 3.31, one can see that /?/, and Qt, depend on the five parameters s, si, S j , 71, 7 . Of course, the number of free parameters can be increased while using other ways of discretization of (hD ) , k = 1,2,3 in the operator Q. Supposing that ftfc and Qh are defined by Eq. 3.31, it is possible to use the existing degrees of freedom to impose some additional requirements to the pair R/,, Qh- But before doing so, let us consider another approach to discretizing the operators R and Q given by Eq. 3.24. 2

k

I

76

Fifth-order non-centered compact schemes

Up to now, we have used the third-order compact differencing to approximate hD„ only in the operator Q, Another possibility consists of using it in the operator R as well. In this case [R-'Q]^

l

Inserting I — A~'(s)A(s) J

1

= (l + b,A-'{s )A(s )y\l+

i]

[R~ Q}[

a A- (s,)A(s,)

1

+

l

a A ). 2

2

between the operators R^' and Qh, we obtain

= (A( )

+ hAiti))'

Bl

1

(A{si) + a,A(s )

+

s

a A(s,)A ). 2

2

Now the pair of operators Rh, Qh can be written as

Q h

=

1 +

( t - ?) ° { y a

+

+

-

* )

A

i

-

s

j

A

?

j

A

<3 32)

°-

-

It is easy to see that Eq. 3.32 is not the particular case of Eq. 3.30 with some special choice of f, and 72. Indeed, Eq. 3.32 is another family of [R~'Q][ \ the fourth-order discretization of P' ' ' which depends on two parameters. 4

2 1

The final step in constructing the fifth-order compact differencing operator is the fourth-order discretization of Dl in Eq. 3.19. It is readily carried out using the operator (7 + A j / 1 2 ) A^/k Finally, our operator becomes -1

L

h

2

= \A(s) + [3/2)R?Q {I

+ Ai/ny'Azj/h.

h

(3.33)

The differencing defined by Eq. 3.33 will be called fifth-order compact upwind differencing of the second type, or CUD-II-5. The approximation property of L can be summarized in h

2

Theorem 3.3. Let s(4 - 3s ) jf 0 and u(x) € C,^_^ the estimate L u\ h

= D u\ \z=z, z

-CffDivl

, x

Then L u satisfies h

1/1 < 1,

(3.34)

li=i , J+

where C = s

1 . fa f 288 24

13 v SiQi a 15s> 48 24

2

a

71 / 4 2 M5s

s, 12

s\ b 4•< 12 2

2

72 / l _ S g \ 12>2 5 , '

(3.35)

3.2. Other forms of compact upwind differencings (CUD-H-5)

77

for fi , Qh defined by Eq. 3.31 and fc

C =a

I ii ^ 144 s 900s

2

(3.36)

for fh, Qh defined by Eq. 3.32.

The proof follows from the results of substitutions of Taylor series expansions on the left-hand side of Eq. 3.34.

The comparison of the truncation errors coefficient of CUD-5 (Eq. 3,6) and that defined by Eq. 3.36 shows that the latter is significantly greater than the former if S\ = 1 and s a; 1, However, one can observe that C in Eq. 3.36 can be decreased by increasing s and decreasing s . In any case, the truncation error with C defined by Eq. 3.36 is much smaller than that of the usual six-point fifth-order formula. The same is true if C is defined by Eq. 3.35 provided that the proper choice of ft and 72 is used. 3

t

3.2.2

Families of compact differencing

operators

One possible strategy while choosing free parameters is the reduction of operation count. In general, the following steps are needed to compute L^u with a known grid function ti and R , Qh defined by Eq. 3.31. h

(i) Solve the tridiagonal system

(ii) Solve the tridiagonal system ,4(s,)u (iii) Solve the tridiagonal system

t2)

= A( )u<'>. Sl

78

Fifth-order non-centered compact schemes

(3.37) The final result is (3|

L u=

[A(s)u + su ]/2/x.

h

Now it is clear that step (ii) can be avoided if 01 + 27! = 0, that (8 i f % = 1/15:8. Another observation is that the terms with five-point operator A A ( s ) on the righthand side of Eq. 3.37 are canceled if 71/2 — 7 /3s + b,/6 = 0. This gives us the fixed value of 72 = s(3 i + &i)/2 and eliminates the free parameter s . As a result we have the following forms of three-point operators Rh and Qh which depend only on s: 2

2

2

7

&

2

+

(3 38)

Only two three-point inversions are needed if C in Eq. 3.35 can be written then as

and Q

h

i]

are used. The coefficient

2

£7<" = ( l / 1 2 0 - l / 1 3 5 s ) . S

Considering the case when Rh and Qt, are defined by Eq. 3.32, we wish to avoid using the five-point operator A A ( s ) . This results in a = 0 and therefore s = 4/5. Again, only two three-point inversions are needed, the difference in step (iii) being due to the different coefficients on the LHS and the RHS of Eq. 3.7 respectively. Three-point operators Rh and Qh are given now by 2

2

2

(3.39) while the coefficient C in Eq, 3.36 becomes 1

gl ) = s(s,/144s + 1/720),

L[

2>

s = 2/Vo~.

To distinguish the two families, the operator Lh will be denoted by £ f and lor Rh and Qh defined by Eqs. 3.38 and 3.39 respectively.

It is obvious that the present strategy of using free parameters is not unique. Of course, some other possibilities to obtain desirable properties of the operator Eq. 3.33 exist. For example, if five-point operator is admissible on the RHS of Eq. 3.37 then the choice of additional parameter can be used for decreasing the truncation

79

3.2. Other forms of compact upwind differencings (CUD-I1-5)

error coefficient Eq, 3,35, This and other ways to control the behavior of schemes by free parameters will not be considered in this boolt since too many variations exist in the framework of the basic idea, some of them being problem-dependent. Considering again the algorithm (i) - (hi), it can be seen that boundary conditions are needed for each step of the algorithm. They are readily available provided that the exact values of the end-point second derivative of u as well as their fourthorder approximations are known. It is supposed that they can be either extracted from the particular problem formulation or computed using a proper high-order formula. Compact high-order differencing at the end points is described, for example, in [81].

3.2.3

Positivity,

dispersion and dissipation

As in case of CUD-5, the important property of the compact differencing operator Eq. 3.33 is its ability to be positive or negative depending on the sign of the parameter s. However, in contrast to CUD-5, this property is not unconditional and imposes certain limitations while choosing s, S\ etc. In general, it is easy to obtain the necessary conditions of positivity or negativity of LK for s > 0 or s < 0 respectively. Let us denote the self-adjoint part of AA as i } , ' Calculating Fourier transform of Lj,, the eigenvalues X(L^), can be estimated as the real parts of the eigenvalues A(i/,). Considering these real parts for long-wave limit, one can deduce that they are positive (negative) if the coefficient C in Eq. 3.35 is positive (negative). Similarly, the short-wave limit yields 0

4 8 / 8 \ /4 8*j\ 1 -s;6; - - D ; - 6 a i s , + 4a + ,1 (4s - — - 12s,J + 7 2 ^ - - - — J > j j 2

2

as a necessary condition of i j , > 0. The derivation of the sufficient conditions for Lh > 0 can be carried out by further analysis of the real parts of the L eigenvalues. The same results can be obtained by calculating the scalar product (L^u, u), where u E L . Rewriting L as K

2

hU

- \&o + | { i - ^ ' Q A ) ( - A ) , 2

R = (l + A / 1 2 ) % , h

2

ft

(3.40)

it can be seen that by virtue of ( — A ) > 0 and the commutativity of all operators 2

involved, 7./, > 0 for s > 0 if and only if / — Ffj^Qh > 0, The latter inequality means that

Fifth-order non-centered compact schemes

80

((ft, - Q )v, km) =

(3.41)

- RlQ>,)v, v) > 0,

k

where v — R^'v. Here again we have used the commutativity of the operators in Eq. 3.40. It should be noted now that and R are polynomials with respect to A and A and that A^ = —4A + A | . Taking into account these facts and neglecting the skew-symmetric part of RlRh — R'^Qh, it may be shown that Eq. 3.41 reduces to the operator inequality h

3

0

2

3

(\C(-A )

k

+

2

£c (-A ) } ,v)>0, k

2

(3.42)

V

where C is defined by equation 3.35 or 3.36 and c = c*(s, Si, s , fx, 72), k = 4,5,6. k

a

Introducing the eigenvalues of ( —A ), it can be easily seen that Eq. 3.42 will be satisfied if C and are such constants that 2

u , ' > o ,

» e [0,4].

3

(3.43)

2

Returning now to the operators t$* and L ' defined by equations 3.38 and 3.39 respectively, the inequality Eq. 3.42 becomes ft

(3-44)

> 0

(iiT-i)c-^ i^-57-y) 3+

for

&nd L

k

A

> 0

(3.45)

respectively, \s\ being equal to 2/t/E in the latter case.

Considering y as the eigenvalue of ( —Aj), the operator inequalities Eq. 3.44 or 3.45 can be reduced to Eq. 3.43 with c = cs = 0. 5

Upon inspecting Eq. 3.44, we may conclude that the inequality is satisfied for s > 0 by virtue of A < 4 ( - A ) if s > 5/3. Analyzing the inequality Eq. 3.45 one can also see that it is satisfied if s,/s > 5/24, where s = 4/5. 3

2

3

2

2

We are now in a position to formulate

Theorem 3.4. Let s > 0 (s < 0). Then L$ > 0 [L$ < 0), t = 1,2 if 2

s >\

for L<»

3.2. Other forms of compact upwind differendngs (CUD-II-5) and

7 s

l

.

r( )

c

5

f

24

81

2

°

r

£

*'

2

where s = 4/5 in the latter.

2

Now it is clear that and Ljj ' can be used as upwind operators in a difference schemes for advection equations. Henceforth they will be referred to as the fifth-order compact upwind differencing of the second type (CUD-II-5). Since the approximating properties of L$ and i j ; ' are already established in the foregoing sections, we will omit indices h as an indication of their difference nature. Of course, many other options are possible while using various sets of free parameters in Eq. 3.30. The choice of the operator t ' ' and i ' ' does not necessary mean that they are the best. 2

1

2

1

Considering again the operators If ' and let us discuss their dissipation and dispersion properties. One may expect that these properties do not differ from those of CUD-5 operators too significantly, the phase and amplitude errors being of the orders 0(k } and 0(h ) respectively. The quantitative estimates can be obtained while considering semi-discretized problem 6

s

u(0, nh) = exp(ifc/i7i),

n = 0, +1, =p2,

where a = const, sgn s = sgn a. The relative phase velocity c.jc and the dissipation d can be easily derived as

[l|

where Z ( a ) is given by hlS^e™ = Z « ( a ) e

i t m

.

5

Denoting / — sin (a/2), we obtain Jfe Z<'' = 2s; f 15 1

32 135s '' 2

3^9s

2

25'

/ft

Filth-order non-centered compact schemes

82

3 Z " > = sinojl +2([l 4

-

m

j|j*]fl*J.

where

Similarly, »e 2

( 2 )

= 2s/

3

9'

where / ,

22-,,

,28

4 ,

17

- 11

4 ,

4 -v,,

2

and A is the ratio 5[/s, s — 4/5. Using the above relations, one can easily obtain the same estimates as in the case of CUD-5, namely, 6

= 1 + 0{o ) and 6

= 0(a ).

3.2.4

Conservation

laws

treatment

Now let us consider the application of CUD-5 and CUD-II-5 to the nonlinear conservation law a a* Suppose that there are no sonic points in the computational domain. Then dj(u)jdx can be discretized by L$ or L ' \ i — 1,2 so that sgns = sgn f'(u). In this case the actions of the above operators can be cast in the familiar numerical flux form. Considering for example £j , 1 = 1,2, we observe that 5

Other forms of compact upwind differencings (CUD-II-5)

F i g - 3.2: Dispersion and dissipation of t.\

For s = 1.3 (solid line) and s = 1.4 (dashed line)

above; dispersion and dissipation of t ^ ' for s i / s — 0.75 (solid line), S\/s 2

s i / s = 2. ( s m a l l dashed line) - below.

83

= 0.4 (dashed line) and

Fifth-order non-centered compact schemes

84 where

The following algorithm can be used for the calculation of ? j j (j — 0, ...,JV) is known.

+ 1

/2

if the grid function

u

(i) Solve

)

(/ + A i / j 2 ) « f = / i - / , J f

»j* = Qkuj .

R vf

(iv) Compute

j = l , . . . , i V - 1.

11

(ii) Compute (iii) Solve

i

= vf>.

h

q

H

m

= 0.5(/

i+]

+ fj) - DJfe(&H - / / -

vf). 1

It is understood that some boundary conditions should be imposed on u' ' and u' '. We see now that if there are no sonic transitions, the operators L and £jp can be used in stable conservative schemes. Here the stability for frozen coefficients assumption follows from the positivity of these operators provided that s — |s|sgn /£, while conservation property is the direct consequence of the flux form Eq. 3.46. 3

s

Unfortunately, the presentation Eq. 3.46 is no longer valid for the grid points where / ' ( i ) = 0. Two options are possible in this case. One of them is simply to ignore the loss of conservation property and "switch" the parameter from s to —s, the stability and the fifth order accuracy being preserved. This procedure is quite justified if smooth solutions are looked for. Here we may count on the Theorem "approximation of order k + stability —j convergence of order k" However, there are situations when correct numerical flux balance is highly desirable. The simple example is the shock transition with sonic points in unsteady inviscid calculations. As was shown in Section 2.2, ignoring the numerical conservation law may cause a spurious shock position. To preserve unconditional conservation property, the modification of the generic operator A(s) was used in CUD-3 schemes. Unfortunately, the direct extension of this principle to the fifth-order compact differencing is not clear due to more complicated high-order terms in the Taylor expansions for the modified A(s). So we come to flux splitting as the second option while considering sonic transition. Its main advantage is the possibility of using constant parameter s for each part of the splitted fluxes.

3.2. Other forms of compact upwind differencings (CUD-II-5)

85

As the simplest version of flux splitting we consider here the following discretization of df(u)/dx which was discussed in Section 2.4.5 9fM dx

•i L , ( / ( i i ) + Cti) + ^ ( / ( u ) - C u ) = Lu, •• 2

(3.48)

where C > 0 and £j, i are the operators L or Z^'\ i = 1,2 with sgns = 1 and sgn s = —1 respectively. r

s

According to Eq. 3.48, left- and right-sided compact differencing is used for f(u) + Cu and / ( u ) — Cu respectively, each of them being conservative by virtue of constant values of the parameter s. Denoting self-adjoint and skew-symmetric operators by indices (0) and (1) respectively, we can write z* = | * | £

(0)

+ £

( , )

.

and therefore Lu = L^f+^CL^u.

(3.49)

Obviously, = Wj&

mf(u)

+ 0(h*),

[

L % = O(h')

in our case. We can also state that L > 0 for linear f(u) and any positive constant C. Thus positivity of (/(u) 4 Cuj and negativity of | / ( n ) - C u J are not necessary to ensure the positivity of L . We summarize the results for the flux splitting in the following form:

Theorem 3.5. Let the operator L be defined by Eq. 3.48 or equivalently by Eq. 3.49, where Lt or L are one of the operators Z. , L['\ i = 1,2 from Eqs. 3.9, 3.38, 3.39 respectively, correspondingly for s > 0 and s < 0. Then the following assertions hold: T

5

(i) i f u e C f ^ - ^ + A j then

where (? = 1/1920

for i

8

,

Fifth-order non-centered compact schemes

SG

2

for

if\

C = 1/5/288+ 1/720

for

L.[ \

C = 1/120 - l / 1 3 5 s

2)

(ii) L > 0 for linear /(u); (iii) Lu,- = c

J + ] / 2

- g _ i , where 3

/3

9 i + 1 / 2

= (oj

+ 1 / I

+ o;

+1/2

) / 2 and ^

+ 1 / 2

, o;

+1/J

are

given by L , ( / ( ) + Cu). = ^ u

z, (/(u) - Cii)^ = r

+ 1 / 2

- q-_ , 1/2

- o;_ ; L/2

(iv) standing rarefaction shock is not the exact solution of steady-state equation, namely /(u) = 0.

Now we can see that using flux splitting preserves approximation, stability and conservation properties of the fifth-order compact operators. Morever, entropyconsistent solutions may be obtained for sonic rarefaction case. As to the role of the constant C in Eq, 3,48 or Eq. 3.49, we note that its positivity is sufficient to meet the requirements of Theorem 3.5. However, the magnitude of C influences the dissipation properties of L. In particular, if C —+ 0 then L tends to a skew-symmetric operator generating spurious oscillations of solutions.

3.3

Fifth-order compact schemes for scalar conservation laws

3.3.1

Explicit schemes.

Numerical example: unsteady

discontinuous

solutions of the Burgers equation Explicit schemes with fifth-order compact upwind approximations do not require special treatment. Indeed, the positivity of the spatial operators guarantees the stability of these schemes provided that CFL condition is met.

3.3. Fifth-order compact schemes for scalar conservation laws

87

1.0

0.5

0.0

0.2

0.8

1.0

-0.5

-1.0

Fig. 3.3: Results for Burgers equation withoutfluxlimiter. The variety of explicit CUD-5 and CUD-II-5 algorithms can be obtained by introducing various time-stepping procedures. The distinctive property of such schemes is that the inversion of tridiagonal operators are needed when calculating the discretized space derivatives at the preceding time levels. The increase of operation count per node as compared to that for conventional explicit schemes can be compensated by possible reduction of the total number of grid points. If unsteady numerical solutions are of interest, then high-order time discretization is essential for preserving small phase and amplitude errors inherent to CUD approximations. The obvious candidate is high-order Runge-Kutta time stepping. Using fourth-order Runge-Kutta procedure, the numerical experiments were carried out with the inviscid Burgers equation. They were similar to those described in Chapter 2, the aim being to investigate the shock-capturing properties of the conservative CUD-5 and CUD-II-5 schemes. Since the numerical solutions obtained by these schemes exhibit similar behavior, we present here only the results for CUD-II-5 Runge-Kutta method with the operators defined by Eq. 3.38, The numerical solutions of Eq. 2.16 with 0 = v = 0 and initial data given in the interval [0, 1] by u{x,0) = 0


if

if

0.6 < x < 1

0.4 < x < 0.6,

To preserve the conservativity everywhere in the computational domain, the flux-splitted form of the operator L (s) defined by Eq. 3.48 was used with s = 1.4 and various values of the constant C. 5

Figure 3.3 compares the CUD-II-5 numerical solutions with the exact one at time 6GV. For these computations, ft = 1/100, r = 0.3ft. Note that not counting the wiggles near the shock, agreement with the exact result is rather good. Another observation is that the rarefaction fan is properly resolved thus indicating entropy consistency of the scheme. Other numerical experiments showed that the values of C influence mainly the number of the nodes where wiggles are noticeable, the dependence of this number on C being pronounced only for small C (C < 1). To remove the wiggles, flux limiters can be readily applied. Fig. 3.4 shows the results computed for the same parameters but with Boris and Book limiter acting only near the shock region. We see that the wiggles are practically annihilated by the use of flux limiters. Hence, high-accuracy fifth-order CUD schemes may also serve as high resolution schemes.

3.3. Fifth-order compact schemes for scalar conservation laws 3.3.2

Implicit

schemes

Schemes w i t h direct inversions. To discuss the implicit schemes with fifth-order compact differencing, let us consider the simplest two-level scheme with CUD-5 or CUD-II-5 operators which we denote here by L: I + BTLflv-))

m

-

+ Lf(u )

= 0,

(3.50)

0 < o~ < 1. This scheme is unconditionally stable if a > 0.5 and approximates scalar conservation law Eq. 2.3 to the order 0(T + (a — 0.5)r + A ). The truncation error for Eq. 3.50 is 0(k ) if the steady-state solution is achieved. 2

5

s

1

In contrast to CUD-3 case, the inversion of 7 + orLf'iu ") cannot be reduced to the solving of tridiagonal systems. However, three-point vector equations with 2 x 2 matrices can be written if the CUD-5 operator i is used. It can be done in the following manner. 5

Setting

where v = u

m+1

m

— w , the scheme Eq, 3.50 can be cast in the form of two equations:

R v + -&(s)(f'(u™)) h

ro

-roQ^M

= -r[A( )-Q (J S

h

l

n

+ A /12)- A ]f(u ), 2

(3.51)

2

( / 4- A /12) At - A ( / ' ( y > ) = 0, 2

(3.52)

2

where r — r/k acid Qh, Rh are given by 3 R =I - - s A h

0

+

7 -A>,

m

The RHS of Eq. 3.51 is the action of L on the known grid function f(u ) multiplied by r. It can be computed according to the algorithm described in Section 3.1. The next step is solving Eqs. 3.51, 3.52 which can be written in the matrix-vector form: h

where A), Bj and Cj are 2 X 2 matrices. The block-tridiagonal system Eq. 3.53 can be solved using standard blocktridiagonal Gaussian elimination provided that appropriate boundary conditions are

90

Fifth-order non-centered compact schemes

available. The boundary conditions for v follow from the problem formulation while that for M can be extracted either from exact equations and boundary conditions or from one-sided high-order compact formula for the second derivative. Another way to solve Eqs . 3.51 and 3.52 is to use internal iterations for each time step. One of such iterative processes looks as follows. Suppose

V(k)

is the known k-th iterate of u m+!

-

um. Then we obtain M from

Eq. 3.52 via three-point scalar inversion and substitute the result into Eq. 3.5l. Using another three-point inversion, we can finally calculate V(k+l). This process can be presented as

v(k+ l ) = ru [Rh

+ ru6(s)f'(u m

)r

Qh (I

+ 62/12 fl6d'(um)v(k) + g,

(3.54)

where g is a known grid function . Supposing 1'( u m) = a = const and setting rl = rua, we rewrite Eq. 3.54 as Vk+l

= G l G2V(k),

where

Gl =

(I + 62/12flQh,

G2 = rl(Rh

+ rl6(s)fl62.

To investigate the convergence of the iterations Eq. 3.54, the estimates for IIGlll and IIG211 are now needed. By definition,

IIGlll2 = sup (Glu, Glu) #0 (u, u) or

2

IIGll1

1 = 64 sup v;o!O

([s2(1 - ~62)2 - Hs6~jv, v) ([ t;.j2) , 1+

l~

v, v

where v = (1 + 62/12t l u. Taking into account that 6~ = 46 2 + 6~ , (-6 2, v, v) < 4( v, v) and 62/12j2v, v) > (4/9)( v, v), we can write

2 IIG l l1 2 < 9s sup [1 . 256 v

+ (75 + 64S225

2

) (-6 2v, v) (v, v)

([1 +

+ (~_~) (6~v , v)]. 36

225s 2

The final estimate for IIGlll can be given now by if if

(v, v)

3.4. Steady-stale algorithms The estimate for

[IGJII*

91

reads

where / f = ^ A H ( / + ^ A

2

)

2 +

r ? | s | ( / l A ) ( - A ) + ( ^ - A ) V A ) . (3.55) +

I

2

s

0

We see that all operators in the denominator of Eq. 3.55 are positive. Hence,

It is clear now that 5 8

. *

||G|||||G || 2

,

r 5

is bounded by

64 -125

,,

3 / 8V

1 +

4096 o625s^

°

t h e r W , S e

-

Hence, the iterative process Eq. 3.54 converges with the rate independent of the mesh size and CFL number provided that s is not too small. Of course, using Eq. 3.54 makes sense only if a relatively small amount of iterations is needed to maintain unconditional stability of the scheme Eq. 3.50.

3.4

Steady-state algorithms

If the time-stepping accuracy is of little interest, then all the three types of fifth-order operators can be incorporated in a certain type of implicit schemes. The main idea here is to replace / + crLf'(u ) on the LHS of Eq. 3.50 by a more simple operator just as it was done in the case of CUD-II-3 scheme (see Section 2.4). Again, the obvious choice of this operator is m

m

/ + ri,/'(u ), where L , — A(«i)/A. In general, the parameter sj is not necessary equal to that of the operator L(s). Now the scheme Eq. 3.50 is replaced by the scheme / + rL /'( 1

m u

) j " *'~

U

+ L / f y ) = 0.

s

(3.56) s

The truncation error of Eq. 3.56 is 0(T + k ) and reduces to 0(h ) for steady-state solution. Setting r = co, we obtain £,(

m + 1 U

-

r a U

, = -L/(y,

(3.57,

Fifth-order non-centered compact schemes

92

which is the iterative scheme of defect-correct ion type. We remind that the operator L in Eq. 3.57 can be thought as one of the operators i , defined in the previous sections. 5

The stability of the scheme Eq. 3.56 can be analyzed using several techniques, for example, spectral or Fourier methods. For this purpose we write the operator inequality L" + TL'LI{S)

> ^L'L

(3.58)

which is sufficient for the stability of Eq. 3.56 in the residual norm || Since L > 0 in our case, the condition Eq. 3.58 is met for arbitrary T if l

L-L,(s) >

-L'L.

(3.59)

Setting L = Li(s) + C7, Eq. 3.59 can be written as LIZ,, > G'G,

(3.60)

where commutativity of L | and G as well as the relation (L\Gv, v) — (G'LjV, v) are taken into account. The analysis of Eq. 3.59 can be performed by means of the same techniques as were used in the previous sections. The first step here is to write Eq. 3.60 in the form

!>.0,

(3,1,

where a are the coefficients which are the functions of s and s,. The next step is to replace Eq, 3.61 by algebraic inequality introducing either eigenvalues of A j or inequalities for the elementary operator A ; or its powers. k

Firstly let us consider CUD-II-5 operator Eq, 3.38. Omitting details of derivation, we reduce Eq. 3.60 for L — £ p to the following algebraic form: S * ' )

1

- + ( l - 4 #

!

+ 7

+ <*))

>

0

'

0<*<1.

where

.

/

1 \i.f

8

244

8 ,

(3-62)

3.4.

Steady-state algorithms

93

1J

2

Since the positivity of i £ impiies that 1 - 5/3s > 0, the inequality Eq. 3.62 holds if the second term of its LHS is positive. Simple analysis shows that it is just the case. Let us consider now another CUD-II-5 operator, namely I$\ Eq. 3.39. We remind that it was generated by the first-order operator A(s) with the fixed parameter ± s , [s| = 2/T/E. While choosing the operator L \ in Eq. 3.56, we put L) = A(s)/ft thus retaining only S\ as a free parameter. Of course, another possibility is L \ — A(s )/ft where s ^ s. 2

2

The inequality Eq. 3.59 for L = L \ after tedious algebraic manipulations can be cast in the form 2

p{x)k + 2q(x)k + w(x) > 0,

0 < x < 1,

(3.63)

where k = Si/s and p(x) =

(l-x)x

4/

13 15

1 \2/l

q{x) = -x -(1-x) B(X) = -x

2

I

\2

1 , ' 3

/

i

I

1\2/1

1 ,\

4 , 25^

1 ,! 3

16

2

1 l1 \2, \ i I

4 \\

1

I16 D

75 1\2

1 \

2

16 225

:

A look at Eq. 3.63 shows that p(l) = 0 and hence the admissible values of k must exceed uj(l)/2p(l) = 5/24. It is just the lower limit of k dictated by the positivity of i ^ ' 2

We omit here the algebraic analysis of Eq. 3.63 which shows that Eq. 3.63 holds if 5/24 < k < fc . mar

The value of k„ can be obtained numerically as a result of solving the system of two equations. Rough estimates gives k as 9. ar

mac

In addition to the stability of the scheme Eq. 3.50, another requirement should be imposed when computing the action of L 5 ' The important point of this procedure is solving the tridiagonal system with the matrix 2

94

Fifth-order non-centered compact schemes

which arises while inverting the operator RK from Eq. 3.39. This system is well-conditioned in the sense of [43] if 2

1, 3 + 5*

1

Si

/ 1

k = k= | £ As can be shown, k < k

+

Si \ .

1

Si

/Si

1 .\

(3.65)

si 1 J ^ | / [ = ± _ ; 1 « .59. 3

and therefore the admissible values of k are

mal

-

< k < k.

We note that in the case of L ' ^ the tridiagonal system for Rj, is well-conditioned for all admissible values of s. s

Now the results of this section can be summarized in the form of

Theorem 3.6. Let the parameters s and S\ be chosen in such a way that the operators L = }'(v. }, i — 1,2 in the scheme Eq. 3.56 are positive for / ' ( u ) = const and sj/s < k. Then the scheme Eq. 3.56 is unconditionally stable for the frozen coefficients approximation. m

m

Comparing three types of fifth-order compact upwind schemes, namely CUD-5 and two options of CUDTT5, we can thus state that: • the numerical constant in the truncation error of CUD-5 operators is less than those in the truncation errors of CUD-U-5 operators; • CUD-5 implicit schemes for scalar equations admit the inversions of blocktridiagonal matrices with 2 x 2 blocks which is not the case if CUD-II-5 operators are used; • the realization of CUD-II-5 schemes for steady-state calculations can be reduced to the inversion of three-point operators.

3.5. Compact upwind differencing of arbitrary n-th order

95

These observations show that the choice of the fifth-order operators depends on the particular problem to be solved. Considering other possibilities of using free parameters in a general form of the operators R^ and Qj, defined by Eqs. 3.31, 3.32 we omit here further investigation, addressing it to the readers who are interested in this problem.

3.5

Compact trary

upwind

n-th

differencing

of

arbi-

order r

3.5.1

Method of attack

So far, third- and fifth-order compact upwind differencing has been derived starting from the generic upwind three-point operator A(s). However, the question naturally arises whether it is possible to construct other forms of CUD-3 arid CUD-5. Another question is how to construct (if possible) CUD-n, that is n-th order compact upwind differencing. To answer these questions, let us first examine the structure of the third- and fifth-order formulas of both considered above types. According to Eq. 3.5, the fifth order differencing CUD-5 can be presented in the form 7 + A f l , ( A ) + A„S,(Aj) z 2

2

= A fi(Aj)-aA @ (A | D

where R\(x), Si(x), rameter and

a

1

1

)

(3.66)

Q\{x) are linear functions of r , s is the switching pa8

To obtain CUD-3 formula from Eq. 3.66, it is sufficient to replace / f i ( i ) , 5 i ( i ) , P (z), (Sit ) by constants, namely 1

:

Fifth-order non-centered compact schemes

9C

In the case of CUD-II-5, it is easy to verify using Eq. 3.33 with Eq. 3.38 or Eq. 3.39 that

=.A ^CAj)-*A (3 (Aa3, f l

a

(3.67)

a

where Sk(x), Rk(x), Pk(x) and Qt(x) {k = 1 or 2) are fc-th order polynomials. Again, third-order compact differencing of the second type can be obtained by low ering the order of polynomials. Comparing Eqs. 3.66, 3.67, we see that both have the same pattern, the difference being only due to different order of the polynomials on the right-hand sides. We further observe that in spite of this difference, both relations are obtained by specifying four parameters. For example, all coefficients in Eq. 3.67 can be readily defined once three-point operators Rfo Qt, given by Eq. 3,38 or 3.39 are known. These operators have the form Q

h

= I + Q, A

0

Rh — 7 + 6, A

+ Q AJ, 2

0

+ 6 A , 2

2

where aj, Oi, i i and 6j are the coefficients which were determined while deriving CUD-II-5 approximations. We emphasize here that the number of parameters obtained in the process of the derivation of compact differencing is exactly the same as the number of the loworder terms in Taylor expansion series which are to be annihilated to give high-order accuracy. We finally note that in our case the operators acting on z in Eqs. 3.66, 3.67 are designed to be the products of three-point operators. The only reason of it is the simplicity of their inversion. Having discussed the structure of the third- and fifth-order formulas, we are in a position to explain the method of attack when designing n-th order compact upwind differencing. We start from (2r + 3)-point non-centered formula for the first derivative hD « A = z

R

1

-P (A )A r

2

0

- sQ (A )A , R

2

3

(3.68)

where P [x) and Q (x) are r-th order polynomials and P (0) = <2r(0) = 1. In fact, the well-known numerical differentiation formulas can be presented in such a form. Clearly, the first and the second terms in Eq. 3.68 are the skew-symmetric and selfadjoint parts of the operator A , ( s ) respectively, s being the switching parameter. r

r

r

3.5. Compact upwind differencing of arbitrary n-th order

97

Let us introduce now (2p + 3)-point averaging operator R, = / + fip(Ai)A

+ 5 (Aj)A ,

a

p

0

where Rp[x) and 5 ( i ) are p-th order polynomials. To obtain n th order approximation to h(d/8x), where n < 2(p + r} + 1, we require that for fixed i p

=

J

,

+ l

fl; A (s)

+ 0( " ).

r

2

Since we are interested in dissipative approximations, n is assumed to be odd. To determine the coefficients of Hp, S , P, and Q , we substitute into the equalp

r

ity • V

s

n

l

& (s) + 0(z + )

(3.69)

r

the expansions Ao = £ c _ 2 1

J l Z

2

'-',

A = f> ,2 ' 2

1=1

2

1=1 n

Equating then the terms with 2, 2*, ...,2 on the RHS and LHS of Eq. 3.69, we obtain the system -4g = f,

(3.70)

where the components tfi, g , o f the vector g are the unknown coefficients of R?, S , P , Q f is the known (2(p + r) + 2)-vector and A is a matrix with n. rows and 2(p + r) + 2 columns. This procedure is quite similar to that of obtaining the linear systems with Toeplitz matrix when deriving Pade approximants. 2

p

r

r:

Supposing first that n — 2(p + r) + 2, it can be shown that n x n matrix A does not degenerate and the vector g can be uniquely defined. It means that only one approximation to h(d/dx) exists. If n < 2(p + r) + 2, then 2(p + r) - n + 2 degrees of freedom can be used in different ways. For instance, one can require that the operator fl^, be presented in factored form, the factors being three-point operators. Another choice is to fix some coefficients or use them as free parameters to control the properties of approximations. In all cases care should be taken to prevent degeneration of the n x n matrix when obtaining n coefficients. Returning to Eqs. 3.66 and 3.67 we see that 6 and 8 coefficients are involved in these formulas. Hence, 2 and 4 degrees of freedom were used in CUD-5 and CUD-II-5 relations respectively. To derive fifth-order differencing without degrees of freedom it is sufficient to consider the operators R„ and A with tridiagonal and pentadiagonal matrices or vice versa r

98 3.5.2

Fifth-order non-centered compact schemes Examples:

5-th and 7-th order compact upwind

differencing

without degrees of freedom Following the above described strategy, we introduce the operators Ri=I

+

+ a A + i ^ A , + tjAj, 2

2

A

A , = | ( / + « i , A ) - sA (/ + 4 A j ) . 3

2

Substituting the expansions Ao _ 2 ~

Z

f! 6

+

_fl £ 120 7! "'

+

+

2 A 2

=

J

+

^

+

+

+

l 2 360 and setting n = 7, we obtain the linear system

-

a, = -s, Oj = T + d i , b

-oi + 6 12

61

'

= - —s - d s, 12 2

1 1 , 120 4

1 1 ,

1 1

1

1, T20 ' + 4 ^ = - 3 6 0 " 6 ' a

s

dlS

The solution of Eqs. 3.71 generates seventh-order compact upwind differencing with pen tad i agonal matrices 8

kD, = 7 V A , ( s ) + 0(/i ), +

1 To'

\ )

+ A

/ 31 22

\\2%

+

29 \ -A . 3880 j 2

A,(s) = A „ ( / + ^ A ) - $6*\1 + ^ A ) . 2

2

(3.72)

We summarize the important properties of approximation Eq. 3.72 in the following

3.5. Compact upwind differencing of arbitrary n-th order Theorem 3.7. Let u e C[* „ )

99

Then the estimate holds

iJii

where fli and Ai(s) are defined by Eq. 3,72, Moreover, the operator Rj~'Ai(.s) is positive (negative) if s > 0 (s < 0).

Setting bi = 6j = 0, we get three-point operator which we denote by Rg. Fifthorder compact upwind approximation can then be obtained considering the first four equations of Eq. 3.71 with 4j = b = 0. The solution of this system leads to the operators 2

(3.73) such that 1

hD = Ri A,(a) + 0(h'). t

The reasoning which is similar to that in the case of Theorem 3,7 leads to Theorem 3.8. Let u t£ C

^ . J ^ + J ^ .

Then the estimate holds

where >?o and A,(s) are defined by Eq. 3.73. Moreover, the operator R^' A ( s ) is positive (negative) if s > 0 (s < 0), ;

When computing the action of the operator R^'A (s), only one tridiagonal inversion is needed. Another advantage of this approximation is that it can be t

100

Fifth-order non-centered compact schemes

easily implemented in implicit schemes just as it was done in the case of CUD-3. However, CUD-5 and CUD-II-5 approximations are more accurate than / i ^ ' A i ( s ) due to the coefficients in their truncation errors. Concluding Chapter 3, we remark that the answers to the questions put at the beginning of this section are quite evident. Of course, one can design different forms of third- and fifth-order (in general, n-th order) compact upwind differencing. So far we operated on the premise that matrices to be inverted should be tridiagonal. However, the operators with pentadiagona! matrices may also be a good basis for constructing Pade-type upwind approximations.

Chapter 4 Hyperbolic systems

In the previous chapters compact upwind differencing was used for constructing positive approximations in the scalar case. To meet the same requirements for hyperbolic systems, standard techniques may be exploited. The first way is to reduce vector equation to the set of the independent scalar equations by means of diagonalization. The second one consists in matrix or flux splitting followed by applying "left" and "right" CUD operators. The application of the former technique which leads to the matrix-difference compact operators is explained in great detail in the first section of this chapter for the case of CUD-3. This case is the prototype for other forms of CUD. They can be readily obtained by simple generalization of the corresponding scalar operators. This procedure is briefly sketched in Section 4.3. Finally, the ideas of flux splitting are presented as a tool which may be used for constructing positive approximations for all forms of compact upwind differencing.

4.1

C U D - 3 schemes for vector conservation laws

4-1-1

Matrix-difference

operators obtained via diagonalization

When considering systems of equations, the question arises which of the pairs of the operators A±, should be used to discretize space derivatives in each equation of system. In general, this question may be answered in the same way as in the case 103

Hyperbolic systems

102

of the usual schemes with one-sided differences, i.e. by the diagonalization of the matrix of the corresponding linearized system. The diagonalization allows to choose such pairs of the CUD operators A± and A for each scalar equation that lead to the unconditionally stable implicit schemes with positive operators. ¥

Consider the system

where u, f, F and g are p-vectors, and functions f, F and g are sufficiently smooth. The corresponding linear system may be written as

where L — df/du and Q — dF/du are Jacobian matrices. We suppose that the _1

scheme Eq. 4.2 is hyperbolic and the matrix L Q = H h&s real eigenvalues A, 1

(i —

P).

In order to derive the structure of compact third-order scheme we first assume that the matrices L and Q are constant. Applying then the transformation u — Sv, where the eigenvalues of the matrix K are the columns of the matrix S, Eq. 4.2 becomes „ •

where A = diag{Xj, A , X ] independent equations 2

e

and K = 5A5~'. Thus we come to the system of

T T ^ T ? - *

>

M

in which and /m are components of the vectors v and S~'gi respectively. The transformation of hyperbolic systems to the form Eq. 4.3 is widely used when constructing schemes with one-sided differences (see, for example [41], [59]). Each of the equations in Eq. 4.4 is a particular case of above considered Eq. 1.9 and can be approximated by CUD-3 operators A and A . As a representative example of time discretization, we consider again two-level scheme with weight factors A - 0.25sgn k A ) ^ 0

0

" ^

(A

+ °

"

= (Ao- 0.25sgn AiAo)^,),

(4.5)

where the bar denotes the weighted values of the functions v and g,, namely, v =

g. = ^ r ' + (i-^)gr+

4.1. CUD-3 schemes for vector conservation Jaws

103

Writing Eq. 4.5 in the matrix form and using the old variable u = Sv we obtain the following scheme after multiplying all terms by S U

- 0.25WA ) ^ - 1 ^

A

+

o

V

° 2k

T

M

A

^ Q

= fa - O.SSAf Aojgj.

(4.6)

In Eq. 4.6 M = SDS~' and D is the diagonal matrix with elements sgnA,. We rewrite now scheme Eq. 4.6 in the following equivalent form f* - O . 2 5 M A V

) L - ' L ^ ^ 1+

0

'

*=™±**qa

Zn

T 1

= [ 4 -fJ.25JlfA,)X" £gi. J

e

After multiplying the above equality by L , scheme Eq. 4.6 reads (A

a

A

- 0.25MA )L"

+ "~ ^

o

A 2

2

g » = [Ao - 0 . 2 5 M A ) g i , 0

(4.7)

-

where M — LML '. The matrices M and M in the schemes Eq. 4.6 and Eq. 4.7 play the same role as the scalar factor sgn ^'(u™) in the scheme Eq. 1.15 for scalar Eq. 1.9. Obviously, for p = 1 these matrices become sgn X%, Slightly modifying the operators in the scheme Eq. 4.7 but preserving its general structure, we write now the discretization of nonlinear equation Eq. 4.1 as B

+ C.F - B

x

l g

,

(4.8)

where f

m

m

= f(u ,z),

F = ffF

m+1

m

1

+ (l - o - ) F ,

m

F" = F ( u , z ) ,

the following options being possible for the operators B and C z

B

X

= A

B -At>k

0

- 0.25A M,

C = ^

M) A+],

(4.9)

M ) ( / + r , ) - ( T _ , M ) ( 7 + ZLij],

(4.10)

0

0.25[(T

1 / 2

E

#

r

0

= 4 -0.25JWA , O

0

1 / 2

/ 2

C =^[A -A_(7-, R

[ A - A _(T

z

O

/ 2

Jtf)A ], +

= — [AO-WAJ].

(4.11)

104

Hyperbolic systems

It Is supposed that mid-point grid functions corresponding to Zj+1/3 are defined in the following way TM !/t

= (M

+ Mi)/2 = ( M ( u , ,

j+1

J +

I j + 1

) + Af(u„s,))/2

or TM

= M,

m

^.i.fS

Analysts

+ l / 2

= M((u

of non-conservative

-l-u,)^,^,/,).

J + 1

and conservative

forms

If the eigenvalues of the matrix K = L Q do not change their sign in the computational domain then the scheme Eq. 4.8 has the third order of approximation with respect to the step h. In fact, supposing sufficient smoothness of the solutions of Eq. 4.1, the following representations for the operators B and C hold x

B,f = [l-^D M

+ jD y

t

C f=

x

[l-^DzM+

x

+

jD^DJ

r

3

0(k ),

3

+ 0(h )

(4.12)

for the operators Eq. 4.9, 4.10 and BJ=

(/—MO. + y/^f + O^),

C f = (j-^MD x

+ j&tyDJ

X

for the operators Eq. 4.11. Here and further

3

+ Oik )

(4.13)

above M is omitted.

Substitution of Eq. 4.12 (or Eq. 4.13) in Eq. 4.8 followed by multiplication by h„

2

.,

h .,\-i

or by

9 leads to the equality (f

B + ]

m

-f )fr

+ Df t

3

= + 0(h ). s

We see now that local truncation error for the scheme Eq. 4.8 is 0((a — 0.5)r a

3

r +ft ).

4.1. CUD-3 schemes for vector con serva I ion Jaws

105

if at least one function A,(a;), (j — l,...,p) changes its sign near x = Xj then the elements of the matrix M are discontinuous. If the grid-point i = Xj is not a sonic point then the expansion Eq. 4.13 with sufficiently small h is valid because it does not contain the derivatives of the matrix M. However, the grid functions B f and C S cannot be presented in the form Eq. 4.12 in the case of the operators Eq. 4.9 or Eq. 4.10 since the elements of the matrix M are not smooth. r

r

To investigate the order of local truncation error for the scheme Eq. 4.8, it is convenient to extract the terms with the matrix M. Their sum in the grid point Xj = j h for the operators Eq. 4.9 may be written as

(4.14) where {df/dt) =

- f™)/r.

Obviously, the matrix M can be presented as M = X;(sgnA )A/ *', (

fc

-1

where — SD^S , matrices having the single nonzero element which is equal to unity at the /c-th row and k-th column. Thus the discontinuity of the elements of matrix M is entirely due to the change of function sgnAj, while the elements of matrix A/'*' (k = 1 , p ) remain smooth. Since the part of the scheme Eq. 4.8 which does not depend on M approximates Eq. 4.1 with the fourth order with respect to h, we only have to estimate the sum Eq. 4.14. fc

If the eigenvalues A do not change sign then the products sgnA M' ' are smooth functions of Using Taylor expansions, the expression Eq. 4.14 can be presented as t

t

4M
Hyperbolic systems

106

Assume now that the function sgnA* changes its sign in the vicinity of the grid point Xj. Let, for example »gnA (s^j) — I ,

sgnX (xj)

t

- -1,

k

sgnA*(:c ) = - 1 ;

(4.15)

3+!

all other situations can be considered in the same way with the same final result. If Eqs, 4.15 hold, the part of the expression Eq. 4,14 depending on A/'*' can be written as

«i8((S>-«) «ia((|>- )J <^,^ w+

t

. (4.16)

+

Using the smoothness of elements of the matrix A/'*', one can verify that the expression Eq. 4.16 has the order O((o — 0.5)T + T + k). 2

The local truncation error for the scheme Eq, 4.4 with the operators Eq. 4.10 can be analyzed in the same way. To do this, we assume again that the function A* changes sign in some interval and Eqs. 4.15 hold. Then repeating our arguments we obtain instead of Eq. 4.16 the expression [xj_I,Xj]


J + 1 +

((S)-)

; + 2

^)

(4.17) 2

2

This expression in contrast to Eq. 4.16 has order 0({a — 0.5)r + r + h. ) for the exact solution of Eq. 4.1. We note that the above estimates characterize in fact the local approximation. If one uses L -norm then the order of approximation in the case when the matrix M have non-smooth entries will obviously increase. 2

We see now that the schemes with the operators Eqs. 4.9, 4.10 fail to be thirdorder accurate in the vicinities of sonic points. However, in contrast to the schemes with the operators Eq. 4.11, they possess important conservation property. Indeed, the actions of the operators Eqs. 4.9 and 4.10 can be cast in the flux form

B fj = -4 fj + q , r

0

+ 1 / 2

-

qj.,/2,

where the grid functions q,+i/ and Q 2

J + ]

C*Fj =

- Q,_i/ , 2

/ may be readily obtained. 2

When summing Eq. 4.8 over all grid points the fluxes through common internal boundaries are canceled.

4.1. CUD-3 schemes for vector conservation laws

107

In the case of conservative schemes Eq. 4.8, it is also worth using the so called "negative" norms [90] IV 2

llfll. = f t E | E f ; |

k

or 11*11*--*

where N is the number of grid points. In these norms, due to the cancellation of fluxes the third order can be preserved even when the matrices M are not smooth. We remark in conclusion that the approximation properties of the operators B and C are established while assuming sufficient smoothness of the solution u of the system Eq. 4.1. In fact the solutions of Eq. 4.1 with some initial and boundary conditions are not necessary smooth allowing the existence of discontinuities. If the smoothness violates in the vicinity of the points, where eigenvalues of the matrix K change sign, then estimates of truncation error in the vicinity of these points can be inapplicable. In these cases one can interpret schemes Eqs. 4.8, 4.9 and Eqs. 4.8, 4.10 as difference conservation iaws which are satisfied for discontinuous solutions as well. R

x

4-1.3

First-order CUDS

upwind schemes

for vector conservation

as the generators

of

generalized

laws

It follows from the analysis of the preceding section that the approximation properties of conservative schemes are adversely affected in the vicinity of sonic points. Of course, the obvious remedy is to replace discontinuous function sgn \t(x) by some smooth function s [x) for which t

sgns^i) = sgnA^z). This possibility was already suggested in Chapter 2 for scalar conservation laws. To exert more general form of control over some properties of CUD-3, one may change C given by Eqs. 4.9, 4.10 by another first-order upwind operator written in the flux form. Following the idea of Section 2.1, we first present it as x

d F

W\ =CfF,- = dx ) . 2k_ 1

1

- F , _ , -PsM/itu*.] - « ^ + . , ( u , - - u _ ) l , W

/ 2

>

1

where fiy±i/2 are some positive p x p matrices with smooth entries. Clearly, wellknown first-order schemes may be used as candidates for C°.

108

Hyperbolic systems Expanding C ° F ( u ) in the case of sufficiently smooth functions, we obtain ;

C?F(u) or, introducing Q —

= ( / + jDl)D F

- ^D pD^

r

+ 0{b?)

T

3F/8u,

CfF(u) = ( / - ^ u Q - ' + jDl)D F

+

x

3

0{k ).

Now, the generalized operator B is wanted the action of which on sufficiently smooth function has the first three terms in its expansion coinciding with those in parentheses of the above equation. This action can be readily written when modifying B Z

X

It is easy to verify that

provided that all involved functions are smooth and the matrix Q does not degenerate. The numerical experiments with different types of C [120] for the case of Riemann problem.

4-1-4

Difference

A

and B

A

are described in

equations 1 +

Due to nonlinearity of Eq. 4.8, one of the natural ways to evaluate u " ' is to use linearized version of this scheme. Writing for example

where 8 = ||Li" " — u | | = O(r), we obtain t

,_l

m

( B L + TOC.Q) R

""'

+

~ "

m

+ C.Fln™) = g,

(4.18)

where Jacobian matrices L = dijdu and Q — dF/du are computed in terms of the grid function u™ Obviously. Eq. 4.18 differs from Eq. 4.8 in 0(r ) terms. 1

2

4.1.

CUD-3 schemes for vector conservation laws

109 m + 1

The difference equations for vector grid function u obtained from Eq. 4.18 can be written in internal grid points of the domain \x , sw] as 0

As*pi

l

+ Bmr»

+ Gitif*

= f

j = i , N -1,

jt

where f, is known and Aj, Bj, Cj are the following matrices Ai = (i/ -

G:

T

tfi^Qj-s,

T

M )Q .

+ -(J -

. = ( 1 / + ~M )L, i+l

+l

- -(l +

j+Ui

j+i

Here / denotes unity matrix and r — r/h. The above system is a block-tridiagonal one with p x p blocks. To solve it one can use the well-known algorithm of block-tridiagonal Gauss elimination (see, for example [91]). The boundary conditions for x = xo and x = xpt are necessary to implement the Gauss elimination. These conditions are problem-dependent so it is not worth discussing them. However, we suppose that they are formulated so that difference systems might be well conditioned that guarantee the stability of the algorithm. The simplest estimates of stability of block-tridiagonal Gauss elimination can be easily obtained in our case if the coefficients of the system are constants. Using the diagonalization of the matrices A = Aj, B = Bj and C = Cj {j = 1 , N — 1), we may write 1

SA'S-'up,

+

+ SB'S-'u? '

1

+ SC'S-'u]*

= RHS,

j = l,...,JV-l,

where A' = diag{- - -sgnA, + jfjSj - |Ai|)},

C = diag{~ + isgnAi - ^(A, + |A,|)},

r = r/h,

Ai (i = 1, — ,p) being the eigenvalues of L~'Q. Now the system of equations with respect to p-vectors split into p independent systems with tridiagonal matrices for scalar grid functions. According to the analysis presented in Chapter 1, these systems are well-conditioned.

110 4-1-5

Hyperbolic systems Non-divergent

systems of equations

Consider now the non-divergent system of hyperbolic type du .du , ^ - + g ( u , z , ( ) ^ = f(*, ), t

where Q(u,x,t) is symmetrical p x p matrix and u, f are p-component vectors. Its CUD-3 discretization may be considered in the same manner as in the case of scalar non-divergent convection equation. To approximate the above equation, we again use the operators B = Ao - 0.25MA , r

C = (0.5/ft)(A - M A j ) ,

o

where M = SDS~',

z

0

S is orthogonal matrix the columns of which are eigenvectors

of Q, D = diag{sgn\i}

and A; (i = 1, ...,p) are eigenvalues of Q. Let q„ be vector-

valued grid function satisfying the equation

Now the simplest discretization of the above non-divergent system may be written as m Q" - m t? \ Q = Q{u ,X,l ). +

u

+ 1

+ T

q n

u

m

m

+ T

+

+

m

3

Since q„ = B^C u^ ' = [du/dx)™ ' + 0{h ), the truncation error for this scheme is of order 0{r + h }. Using the definition of q„, one can eliminate u to obtain the following block-tridiagonal system with p X p blocks I

1

m + l

B q + T(7 (
r

M

m + 1

).

Solving this system with respect to q one may then evaluate u

4.2

Stability analysis

4-3.1

Matrix-difference

m + 1

operators

Assuming as usual the frozen coefficients, we suppose that f (u, x) — Lu, F(u, x) = Q/u, g = 0 and Q, L are constant matrices. Besides, in order to exclude the impact of boundary conditions on the stability we assume that for Eq. 4.1 the Cauchy problem is formulated in the unbounded domain t > 0, —oo < x < oo with initial data at ( = 0 which are equal to zero outside some intervals of x.

111

4.2. Stability analysis

the Eq. the the

The unconditional stability of the scheme Eq. 4.8 in the case of a > 0.5 and above assumptions follow from that of the schemes for each equation of system 4.4. Nevertheless, it is worth deriving some operator inequalities characterizing stability of Eq, 4.8 in energetic norms. Investigation can then be carried out in spirit of [89]. Thus we consider the scheme with constant coefficients u

m+i _ m u

B

- + C,ifO = 0,

x

B = An- 0.25A/A , s

C„ = 0,5(A -

0

o

(4.19)

MA )/k. 2

-1

Without loss of generality we suppose that the matrix K = L Q is symmetrical, otherwise it can be symmetrized by some transformations. T

1

Since the matrix K is symmetrical, the matrix S is orthogonal (S = S~ }. It means that matrices M and K can be written as M = SDS , K = SKS We see now that the product MK is self-adjoint and non-negative operator: T

MK T

(MK) T

Really, MK = SDAS ,

KM = SADS ,

f sgnA, DA=AD

= KM,

,T

= MK > 0.

(4.20)

where 'X,

...

T

...

= sgnA J

0

B

(1

0

* I

T

\\

T

so ( M / f z , z) = (SDhS z, z) - (DAv, v ) > 0 where V = 5 z . This means nonnegativity of the symmetrical matrix MK. To simplify the subsequent analysis, we will assume that A< ^ 0 (t = I , ...,p) and MK > 0. Let us establish now some inequalities relevant to the difference-matrix operators of the above schemes. First we define inner product (u, v) of two vector-valued grid functions u and v as the sum (4.21) The Gilbert space of grid functions which vanishes rapidly at infinity with inner product Eq. 4,21 will be denoted by H. First of all, consider the difference operators A , A j which are building blocks of the operators B and C . Their properties which are important for the next discussion can be summarized as follows 0

x

T

Hyperbolic systems

112

(i) The operators Ao and A are skew-symmetric and self-adjoint operators respectively. This means that for any u, v £ H we have 2

( A u , v) = - ( « , A v ) , ( A u , v) = (u, A v ) . 0

0

2

(4.22)

2

(ii) The following inequalities hold for v 6 H, v ^ 0 a

0 < - ( A v , v) < 4(v, v) = 4||v|| ,

(4.23)

2

2

- 4 ( A v , v ) > - ( A v , v).

(4.24)

2

They can be written as operator inequalities -A <47,

-4A >-A=,

2

(4.25)

2

where / is unity operator. Eqs. 4.22 - 4.25 can be obtained using simple algebraic manipulations. 2

More informative form of the relation between the operators A and A can be obtained starting from the obvious equalities 2

A = A - A_, 2

+

A = A+ + A _ . 0

Squaring both sides and then subtracting one from the other, we get 2

A , = 4 ( A A _ ) + A = 4A + A . +

2

2

2

(4.26)

In particular, the second inequality Eq. 4.25 follows from this fact. Note that the operator A can be expressed as 0

Ao = I + IA,. o

(4.27)

Thus it is self-adjoint and due to the first inequality of Eq. 4.25 is positive-definite. Moreover, 4 o > * - | 4 / = i/.

(4.28)

Consider now difference-matrix operators. Obviously, for any self-adjoint matrix operator N = N and difference operator 8 the product N6 is self-adjoint or skew-symmetric in the sense of the inner product Eq. 4.21 if the operator 6 is self-adjoint or skew-symmetric respectively. m

4.2. Stability analysis

113

Indeed, let S be skew-symmetric. Then using commulativity of constant matrix operators with difference ones we can write the chain of equalities (N6u,v)=

£

N6u =

SmNvi = -

iVi

\= —CO

Y,

i = -oo

u.«V6vi = - ( u ,

NSv).

i=-co

Denoting self-adjoint and skew-symmetric components of the operators by indices (0) and (1) respectively, we may write Bjp = Ao, (0}

(C K)

B '> = -0.25A/A , z

o

M

== -MKA /2h,

X

(C K)

3

X

= -KA /2k.

(4.29)

0

The final result of this section may be presented in the form of

Theorem 4.1. The operators B and C from Eq. 4.19 satisfy the following x

z

inequalities B > 1/3,

that is

z

C K > 0, Z

BCK Z

> 0,

Z

[B v,

v) > (v, v)/3,

z

that is

[C K\,

that is

{B- C Kv,

X

z

x

(4.30)

v) > 0,

(4.31)

v) > 0,

(4.32)

where v e H is an arbitrary nonzero grid function

The first inequality follows directly from Eq. 4.28. To prove the second one we note that according to Eq. 4.20 MK — (M K) and the operator - A is self-adjoint and positive. The third inequality follows from the expression for the self-adjoint component of the operator B'C K, namely from T

2

I

{B;C,K)M

= MK [ - 2 A + 0.5( Ao)' - A?/3] /Ah a

2

in which - 2 A j = - 0 . 5 A + 0.5A according to Eq. 4.29. Thus we have o

(B- C K) z

z

i0)

= MK&1/24H > 0

Hyperbolic systems

114

4.2.2

Stability in energetic norms

When investigating the stability of the scheme Eq. 4.19 it is convenient to use energetic norms, i.e. norms based on some self-adjoint positive operators D [89]:

IIvllb =

(Dv,

v).

General theory of stability of difference schemes in energetic norms is given in detail in [89] . Henceforth, the stability of the scheme ( 4.33) in energetic norm

II . IIv

will be thought of as the fulfillment of the inequality

which means that "energy" of the solution of Eq . 4.36 does not increase. We repeat the following two results from [89] (see Appendix B).

(i) If the operators C and A are independent of m and the operator A is positive and self-adjoint , then the following condition is necessary and sufficient for the stability of the scheme Eq . 4.33 in the norm II . IIA

C 2 0.5TA. (ii) If the scheme Eq . 4.33 can be presented as ( 4.34) where the operator A is non-negative, then necessary and sufficient condition for its stability in the norm II . Ilv is (j 2 0.5 , where D is any self-adjoint positive operator which commutes with A . The norm II . IIA may be useful when estimating the stability for steady-state problems solved by stationing technique. In this case, small values of lIu m IIA = V(Au, Au) mean that the L 2 -norm of the residual for the approximate solution of Au = 0 is small. If the operators C and A in Eq . 4.33 commute, then the stability of this scheme with the above assumptions for C 2 0.5T A also holds in the norm

4.2. Stability analysis

115

|| - |jo, where D is any positive self-adjoint operator commuting with A [89]. We may set D = / so that || - | | = || - | | . D

H

Let us estimate now the stability of the scheme Eq. 4.19. One can use any of the formulated criteria for this purpose. Really, this scheme has the form [B + or A) -

^——— + Au

m

= 0,

where B = £? , A — C K. After multiplying by A' it reduces to the scheme r

X

(A'B

+ O-TA'A)

U

m

&

1

~

m

+ A'Au

= 0.

(4.35)

In Eq. 4.35 the operator A'A is self-adjoint and positive while the operator A'B is positive, since ((C K)~Bu, u) = (u, B'C K\i) > 0 as a consequence of the inequality Eq. 4.32. Thus for stability of the scheme Eq. 4.35 in the norm || • \[A'A it is necessary and sufficient to meet the condition X

I

A'B

+ trr A* A > 0.5rA-A.

When 0.5 this inequality is valid for any value of T and the scheme Eq. 4.35 is unconditionally stable in the norm || • \\A-A- Moreover, the operators A'B and A'A commute due to commutativity of constant matrices M and K included in these operators. Hence, the unconditional stability of Eq. 4.35 holds in the norm It * \\H as well. Further, while estimating stability, it is convenient to cast the scheme Eq. 4.19 in the form of schemes with weights of type Eq. 4.34 using the operator N = B~ C . The scheme Eq. 4.19 after multiplying by B~' may be written as l

x

ll (l + OTNK)

m +

m

' — u

+ NKv™ = 0.

(4.36)

The operator NK is positive since the following inequality holds as the consequence of commutativity of the matrices M and K (B-'C.Ku, _1

where v = (r5') u. inequality Eq. 4.31.

u) - (BICrKv,

v),

Its RHS is positive far v f 0 as the consequence of the

The unconditional stability of the scheme Eq. 4.36 follows from the fact that the operator NK is positive for a > 0.5.

116

4.3

Hyperbolic systems

Extensions to other forms of C U D

We want now to demonstrate that the diagonalization of the matrix A for constructing positive compact approximations of A(0/dx) may be readily applied to other operators described in foregoing chapters. Before doing so, we note that so far the i-th independent equation of system Eq. 4.4 was discretized using the switching parameter 5< — s g n A;. Returning to the old variable u, the switching matrix M = SDS~ was obtained in which D = diag{si). l

It is possible, however, to use another switching parameter J, such that s g n s\ = s g n A;. In this case, the modified matrix M becomes M = SDS~\ where D — diag{si). Clearly, the approximation and stability analysis outlined in this chapter also holds for the operators B and C based on M. T

x

Suppose now that the i-th equation of system Eq. 4.4 is semi-discretized by any member of CUD family with switching parameter Si satisfying s g n J, = s g n A^. Denoting this differencing operator by fij(5j), we write *Tg& +

= RHS,,

i = l,,.„p.

Returning to the vector u, these equations can be cast in the form h-jg + S\S-'(Sdiag{S (s )}S-')u z

where u = RHS= S(RHSi,

= RHS,

i

t> , . . . . u y ) ' , A =s diag{^i}, RHS , ...,RHS,) . (3)

T

2

Now let us examine the structure of S diog(St[SA\S~ in the case of fifth order schemes. We remind that 5r(s,) may be presented for each equation of system Eq. 4.4 as ,

M ^ . ) = ^ ( A ( s , ) + Q ,(s ))

(4.37)

= A(5 ) + | « r e { i )

(4.38)

J

i

in the case of CUD-5 or ,

i

in the case of CUD-II-5.

k

i

117

4.3. Extensions to other forms of CUD

Here ft and Qh are thought of as some linear operators which are different for different types of compact operators. Considering first CUD-5 approximations, we note that Rh, Qh as well as A(s,) can be written in the scalar case as h

( i

Rh(s.) = p , > + S i w p ,

Q (s.) = p h

{ 1 1

+wt\

3

m )

=#

+

where the operators P^\ Wjp' {it = 1, 2,3) are independent of S{. For vector equation 4.2 which degenerates into the set of scalar equations Eq. 4.4 these operators may be cast in the matrix form Rh(- ) = p, > + mfK

<ws) = p i " + i w i " ,

(i

S

A(s) = P j " + DWi'\ where D = diagfa],

T

s = (si, —

,s ) . p

Keeping in mind these observations, we rewrite S6 (s)S~ with 5 (s;) given by Eq. 4.37 as

J

T

=S

diag{S (sj)}5~' x

E

5 ^ ( 8 ) 5 - ' = (SRhWS-'y'^A&S-*

1

+

SQ„(s)S- )

to obtain 1

5MSJS- = R (M)-'

(A(M) +

h

-

where M = SDS '

Q (M}), h

is the modified matrix M.

Let us consider now compact upwind differencing of the second type. Expressions for Rj, and Qi, for i-th equation are given this time by )

L

]

Rh(s,) = P? + W?\ s; 2

where Pj?\ W* '

Qh = P? +

s,

^

= 1,2) are independent of s,-. Using vector form, we may write

SRh{s)s-'

= p < -T-SD-'S-'W™

= P,

SQh(s)S-< = pW +

M- W .

! 1

+ A r ' v v f

,2|

i

1

,

m

2

Of course, the existence of M~' is supposed here. We see now that Six{s)S~', where b" {s~i) is defined by Eq. 4.38, can be written in the form 5 ^ ( s ) 5 - ' = A(M) + -MRh(M)-'Qh{M). r

Hence the vector counterparts of all scalar compact upwind differencing operators can be easily obtained when replacing scalar s by the matrix M.

Hyperbolic systems

118

We note that the same results may be obtained using Taylor expansion series of the type Eq. 4.12. This fact was already demonstrated in Section 4.1.3 when considering generalized CUD-3 for vector conservation laws. The positivity of the matrix-difference operators obtained as a generalization of scalar ones may be readily proved. We only have to modify the definition of the inner product and to take into account the commutativity of matrix and difference operators. Then the same reasoning may be applied as in the scalar case.

4.4

Flux-splitted forms of C U D

Though the extension of the flux splitting principle discussed in Sections 2.5 and 3.3 to the case of vector conservation laws is quite straightforward, we repeat for completeness the slightly modified arguments. Before doing so, we remind of the fruitful idea proposed in [99] for the Euler equations written in the vector form du

W(u) +

f

t

# i r Making use of the fact that F(u) = Qu, where Q — 8F/du, decompose F as

( 4

3 9

' > it is possible to

+

F = F + F~, F ±

+

= Q+u,

1

where Q = S A * ^ ' and A = diag{\j] positive and negative respectively.

F" = Q-u, +

+

= A 4- A~, the matrices A and A~ being

Let L (s) be any m-th order CUD operator depending on the scalar switching parameter s. We may write m

= Ms)F

+

+ L (-s)Fm

Clearly, it is the positive approximation to dF(u)/dx is assumed.

+ 0(k").

(4.40)

when constant coefficient case

However, to use scalar right and left differencing operators there is no need to split F(u) into the parts for which Jacobian matrices have positive and negative eigenvalues.

4.4. Flux-splitted forms of CUD

119

In fact, we may write instead of Eq. 4.39

I+\U ^ F

+Gw

) \U +

- )

r{a)

-

G(u) =o>

(4 4,)

where G(u) is some vector-valued function with the Jacobian matrix having positive eigenvalues. However, ^ ( F ( U ) + G(U))

and

A ^

F

F

U

_

)

G

(

U

)

do not necessary have positive and negative eigenvalues. Applying now L (s), s > 0, to F(u) + G(u) and L (-s) to F(u) — G(u), we obtain semi-discretized Eq. 4.39 which can be cast in the form m

m

^

+^F(u)

01

+

^L G(u)=0.

(4.42)

Here we have made use of the fact that any member of CUD family can be presented as

and sLffi being the skew-symmetric and self-adjoint parts of L (s) respectively. m

The linear operators extracted from Eq. 4.41 when linearizing and freezing the coefficients is certainly positive. We also have

The simplest way to choose G(u) is to set G(u) = Cu, where C is the positive constant which controls the high-order dissipation. Another choice is G(u) = diag{c\}u with positive constants c,. The advantages of flux splitting were already discussed in Sections 2.5, 3.3. Its role may significantly increase when applying to vector conservation laws. Clearly, using scalar forms of L(s) instead of matrix forms L(M) may save considerably memory and time to compute the action of the operators approximating dF(u)/dT.

Hyperbolic systems

120

0 0.0 i

02 1

0,4 —

1

0.6 '

1

•

0.8 1

•

1.0 I

-

F i g . 4.1: Density distribution Tor Riemann problem, C U D - 3 , C = 1.5.

4.5

Application to Riemann problem

The numerical experiments described in Chapters 2 and 3 illustrate the properties of CUD schemes when computing discontinuous solutions of inviscid Burgers equation. We recall the main observations as follows: (i) Conservative and entropy-consistent forms of CUD can be used as the part of high resolution schemes. (ii) Existing wiggles are well localized near discontinuity and hopefully do not prevent the interpretation of numerical methods. (iii) Wiggles can be completely removed when applying the simplest forms of flux limiters. Due to the marginal nature of oscillations, flux limiters are not supposed to influence the major parts of numerical solutions Similar conclusion can be drawn from the numerical experiments with discontinuous solutions of the Euler equations. The earliest results of applying the CUD-3

4.5. Application to Riemann problem

121

F i g . 4.2: Density distribution for Riemann problem, C U D - 3 , C — 4.

schemes to Riemann problem were reported in [72], The calculations based on generalized forms of the CUD-3 operators B and defined in Section 4,1.3 are described in [78]. In both cases implicit schemes with non-splitted fluxes were used. x

We present here some results of numerical experiments in which flux-splitted forms of CUD-3 and CUD-II-5 were combined with explicit fourth-order RungeKutta time-stepping. In Figs. 4.1 - 4.4 we show numerical data obtained when solving standard Riemann problem [98] with the initial data given by p, = 1,

u

, = 0,

p, = 1 if

-0.5
and p, = 0.125,

u, = 0,

p = 0.1 r

if 0.5 < * < 1.

Here p, u and p are the density, the velocity and the pressure respectively while the subscripts / and T denote the "left" and "right" states of polytropic gas with specific heat 7 - 1 . 4 .

Hyperbolic systems

122

0.0 J

.

0.2 1

p

04 1

0.6 I

.

-

o.s 1

.

1.0 r-

Fig. 4.3: Density distribution for Riemann problem, C U D - I I - 5 , C ~ 1.5.

I n t h e s e c a l c u l a t i o n s , s e v e r a l v a l u e s of t h e c o n s t a n t C i n t h e f l u x s p l i t t i n g d e f i n e d b y E q . 4.41 w e r e u s e d . T h e i m p a c t of t h i s c o n s t a n t u p o n s o l u t i o n s i s i l l u s t r a t e d b y Figs.

4.1

a n d 4.2.

In these

figures,

t h e d e n s i t y d i s t r i b u t i o n is s h o w n for C

=

1.5

a n d C — 4 , t h e t h i r d o r d e r d i f f e r e n c i n g of o p p o s i t e o r i e n t a t i o n b e i n g u s e d for e a c h flux. As rapidly

might

be e x p e c t e d ,

the

amplitudes

of s p u r i o u s o s c i l l a t i o n s d e c a y

w i t h i n c r e a s i n g v a l u e of C d u e to t h e g r o w i n g of d i s s i p a t i o n .

more

It c a n

also

b e n o t e d t h a t i n t h e c a s e of C — 1.5, t h e l e f t a n d r i g h t d i f f e r e n c i n g s d o n o t r e s u l t i n p o s i t i v e o p e r a t o r s for t h i s e x a m p l e . N e v e r t h e l e s s , the c o m p u t a t i o n s were

quite

stable. W e r e m a r k t h a t t h e s i m p l e s t f o r m of f l u x s p l i t t i n g w a s c h o s e n h e r e o n l y for

the

p u r p o s e s of i l l u s t r a t i o n . F i g u r e s 4.3 a n d 4.4 s h o w s i m i l a r n u m e r i c a l d a t a o b t a i n e d u s i n g C U D - I I - 5 e r a t o r s d e f i n e d by E q s . 3 . 3 3 , 3 . 3 8 . C o m p a r i n g F i g s . 4 . 1 , 4 . 2 w i t h F i g s . 4 . 3 , 4 . 4 ,

opwe

see t h a t t h e g e n e r a l p a t t e r n is the s a m e . H o w e v e r , one m a y see a s l i g h t i m p r o v e m e n t i n t h e p r o f i l e of t h e r a r e f a c t i o n fan a n d t h e s m e a r i n g of t h e c o n t a c t d i s c o n t i n u i t y .

4.5. Application to Riemann problem

123

Fig. 4.4: Density distribution for Riemann problem, CUD-H-5, C = 4. The absence of pronounced difference between the CUD-3 and CUD-5 data is due to the fact that the Riemann problem is the test to estimate resolution properties of shock-capturing schemes rather than to demonstrate their accuracy when computing smooth solutions. Finally, we repeat the calculations presented in Figs. 4,1 - 4.4 but with flux limiters. We note that numerical noise is either completely eliminated or can be considered as acceptable for practical applications. Hence, the CUD approximations designed for viscous flow equations, can be thought of as a tool for shock-capturing computations.

Chapter 5 Compact upwind schemes for convection-diffusion equations Using one of the compact approximations of the previous sections for convective terms, one has many possibilities of choosing difference analogues of the diffusive terms. Instead of considering numerous "inviscid-viscous" combinations, we will focus on the case when CUD-3 is used for the first derivatives. The main attention will be paid to two-level implicit schemes in which implicit and explicit parts contain the same operators. In these schemes the choice of the discrete forms of diffusive terms may significantly influence the complexity of solving difference equations. On the contrary, the schemes for which only the action of convection-diffusion difference operators is needed, this choice is not so important. The latter scheme will be considered in Section 5.

5.1

Discretization of diffusive terms

5.1.1

General

considerations

In the case of convection-diffusion equation, the discretization of convective terms can be of major importance. In fact, it is this discretization that defines the architecture of a method. Really, diffusive terms are in most cases negligibly small in all computational region except subdomains with small characteristic sizes. Thus the structure of solution in significant extent is defined by convection and hence its discretization. Moreover, diffusive terms contain self-adjoint negative operators. 124

5.1.

125

Discretization of diffusive terms

Hence, their proper finite-difference analogues do not impair the stability of the algorithm and often improve difference solutions. The most simple way to approximate convection-diffusion equation using CUD is to discretize the diffusive terms explicitly. Then the scheme will look like convection CUD scheme with the RHS containing some difference analogue of diffusion. Similarly, one may use explicit and implicit discretizations of convective and diffusive terms respectively. Of course, great variety of schemes which are based on fully explicit time stepping can be constructed. All these schemes are at least conditionally stable. We will not consider these simple options. Instead we will focus on unconditionally stable implicit schemes suitable for high Reynolds number computations. In these schemes the presence of implicitly treated diffusive terms can make the realization of algorithms somewhat more costly. To illustrate this, let us consider the equation du

dipfu, x)

d

.

, du

„

..

Following the idea of compact schemes for convection equation, we write its simplest difference analogue

m

where A and A is upwind pair and 8iu ^

is some approximation of the term

a . .du —E(U,Z — Ox OX at i — ( + i - Multiplying Eq. 5.2 by A, we see that the difference equations will obviously be three-point only if the operator A&2 is three-point. But tridiagonal structure of the operator A8 and at the same time high-order truncation error for the operator 5 are incompatible due to small number of free coefficients which can annihilate low-order terms in Taylor expansions. The impossibility of reducing difference equations to tridiagonal form may increase operation count when computing m

2

2

Considering the scheme Eq. 5.2 from the viewpoint of stability, suppose that the operator 6 is self-adjoint and negative. Then the scheme Eq. 5.2 is unconditionally stable for constant coefficients if its special case c = 0 possesses this property. In 3

Compact upwind schemes for convection-diffusion equations

126

fact, eigenvalues A of the transition operator for Eq. 5.2 when <{>{») = au, a — const satisfy the equation K(A) + X[a(r/h)W(a)

+ c(r/h')ip{a)]

= 0,

(5.3)

where W(o) and ^(a) (0 < a < 2TT) are eigenvalues of the operators /1"'A (see Eq. 1.13) and S satisfying conditions SeW > 0, Kei/> > 0, 9m V = 0, and R(X) is some function which depends on the discretization of {du/8t) * • Eq. 5.3 can be rewritten as 2

m

x

R(X) + A(T-//i)H>(a) = 0, where W{a) = W{a) + (e/ah)ip(a). Since fte W > 0 if 3te IV > 0, the stability of Eq. 5.3 for ( > 0 follows from Its stability for ( = 0. Thus one can construct many versions of unconditionally stable schemes for Eq. 5.1 in which the derivatives d