Numerical Analysis of Ordinary Differential Equations and Its Applications

+ 3) never be satisfied for even s-P 1

A physicists group ( A B E et al. ) presented a discrete variable method for ODEs which can be found to be equivalent to the Newton-Cotes type collocation method. They called it HIDM, which sounds slightly unsuitable. (See Appendix.) 2.3. A Series of Methods of the Same Stage Number To increase the order of consistency for the s-stage collocation method, we will make a gradual change of the distribution of the interpolating points £ o , . . . , £ which causes the change of the distribution of the collocation points Q , . . . , c . Let ipk(t) be the polynomial of degree (s — k) given by a

s

1

k = s — 1.

for

Moreover, define the polynomial <£>,,*(£) of degree s + 1 by

%M

n

= J{? (t-i)

t +

V*(o}

(* = o , i , . . . , 5 - i ) .

(i6)

Note that the identity
+1

,;+

k

k

0

t

<*(i) = 0 is consistent of order s + k + 1 ifs-k

is odd, and of order s + k + 2 ifs-k

(17) is even.

21

Table 1: The order of consistency for (s, k). s\k 0 1 2 3 4 5 ••• 2 4 4 3 4 6 6 4 6 6 8 8 5 6 8 8 10 10 6 8 8 10 10 12 12

Proof is similar to the one for the previous Theorem. The only difference is that repeated application of integration by parts enables us to attain the exactness of the quadrature for polynomials of required degree. Thus we omit it. Theorems 4 and 5 bring a table showing the increase of the order of consistency in this series. Refer to Table 1. Since the stretching of variable by r = 2t - 1 implies the equivalence of i ^ s _ i ( r ) and the Legendre polynomial of degree s through Rodrigues' formula, we can readily show the following. Corollary 1 The s-stage collocation method determined by the equation <.-i(t) = 0 is nothing but the s-stage Gaufi-Legendre (Butcher-Kuntzmann) method. On closing the present section, we have established away to generate a series of s-stage collocation method starting from the Newton-Cotes type ((s+l)-st or(s+2)-nd order) up to the Gaufi-Legendre formula (2s-th order), increasing the order of consistency two by two. 3. ,4-Stability of the Collocation R K M e t h o d We employ the Butcher array for the formula parameters of RK. Let A, b and e be the matrix and the vectors given by A — (a,j) (1 < i, j < s), b = .. ,b } and = ( 1 , 1 , . . . , 1) , respectively. The stability factor R(z) of the RK method by Eq. (2) is known to be given by T

3

T

det (l-zA R

^

=

\et(I-zA)

T

+ zeb ) -

18

< >

3.1. Derivation of the Stability Factor In the s-stage collocation method, A and b are uniquely determined by the set of the abscissae . .,c provided that they are distinct (Th.2). Hence we can conjecture s

22

that the stability factor Eq. (18) should be expressed only with the abscissae. Define the polynomial 3=1

;=o

which is the derivative of w,(t) in Th. 3 divided by its leading coefficient to make it monic. The coefficients p ( j = 0 , . . . , s — 1) can be represented by the fundamental symmetric functions of c i , . . .,c,. We introduce the Vandermonde matrix V and the diagonal matrices C, S as follows: ;

1 c, ••• c p 1 cs ••• eT

1

1

V =

1 ft

eT

,

C = diag[ci,cj,---,c.]

and

S = diag

4'

r 1

1

(20) Then, the simplifying conditions B(s) and C(s) are equivalent to the matrix identities 4

given by the following ( D E K K E R - V E R W E R ) :

B{s) :

T

T

b V = e S\

and

C(s) :

AV = CVS.

(21)

Let W be the diagonal matrix given by lV = d i a g [ l , l , 2 ! 3 ! - . - , ( s - l ) ! ] . 1

(22)

1

A direct calculation leads to the identity 0 1 0 l

l

1

wv~ cvsw~

= WV-'AVW-

-Po/s\

1

=

0 1

-(s-2)!p _ / ! -(«-l)!j>,_i/»! a

2

S

The right-hand side matrix is the companion matrix for the polynomial

«_3 ft, +

+

... i u i l ! f e ^ . +

+

,_i£

J

h

l

SI 3=0 :

that is, the identity d(z) = det [zl -

l

l

WV- AVW~ )

holds. Taking the identity 3

A(z) = det(Z- zA) = z det(-I into account, we arrive at the following Lemma.

1

-WV^AVW- )

,

i

(23)

23 Lemma 2 The denominator polynomial A(z) of the stability factor R(z) is given by AO) =

z'd{\).

Next, we will consider the numerator of R(z). Eq. (21) yields T

A-eb

T

=

1

{CV-ee )SV- .

(24)

The (j,j')-th component of the matrix CV — ee is given by cj - 1. Then, putting 7; = C; — 1 (i = l , . . . , s ) , we introduce another Vandermonde matrix U and the diagonal matrix T by T

u =

' 1 7i ' 1 72 • • 75"

1

and

r = diag[7 ,7 ,--,7,j. l

2

(25)

. 1 7. " • i f * Then we can obtain T

(A - eb )U

T

1

= (CV - e e ) 5 V - [ / = TUS.

(26)

The shift of the variable t in 7r,(t) by one derives the polynomial p,(t) by

p (t) = ir,(t + l) = f + 1

T

(27)

fi

'£ > <

3=0

whose coefficients introduce another polynomial

S<

8!

(28)

S\

Similar to the previous Lemma, we have the following. Lemma 3 The numerator polynomial N(z) of the stability factor R(z) is given by N(z)=z'e(h

z Due to the above Lemmas, R(z) is represented by R(z) = N(z)/A(z). This expression for R(z) was already obtained by N O S E T T - W A N N E R (their Theorem 4). However, our proof is a self-contained algebraic one applying matrix notations, while theirs is based on the real analysis. The maximum magnitude principle implies the necessary and sufficient conditions for the ^-stability of the collocation RK method as follows: 5

(51) All the roots of A(2) belong to the right half-plane C

+

of the complex z.

(52) \N(iy)\ < |A(%)| for every real y. (The imaginary unit is written by i.)

24 3.2. A-Stability of Symmetric Methods Practically occurring formulae have a special feature for the abscissae. Definition 3 If the location of the abscissae c i . c j , . . . , c, is symmetric with respect to 1/2, the collocation RK method of these abscissae is called as symmetric. Theorem 6 If the collocation method is symmetric, it is A-stable iff all the zeros of A(z) are in C +

Sketch of Proof. The symmetricity means that if c is a root of JT,(I), then 1 — c is a root of 7r,(t). Thus we have (—l) jr (l — f) = ?r,(t), which impbes p,(t) = 7r ((+ 1) = ( — l J ' T T ^ - t ) . This identity yields a relationship between r^ and pj, which implies the equation N(z) = A(—z). Therefore the condition (S2) follows immediately. • J

s

a

By the Theorem it suffices to investigate the location of roots of only A(z) whether the symmetric collocation method, which includes the class given in the previous Section, is ^-stable. For instance, a sufficient condition can be given by the following proposition. Theorem 7 If all the roots of the truncated Taylor series expansion of exp(z) of order m

+

are in C , then any symmetric collocation method of m-stage is A-stable. Proof. We note that &(z), which can be written as

through Eq. (23), is composed by the following two polynomials

in the sense

6

Grace's theorem (e.g. P O L Y A - S Z E G O p60) states that every zero z of A(a} has the form z = -6^ where 9 is a certain root of g(z) and £ is a suitably chosen pointa in a circular domain including all the zeros of f{z). We readily see k

|

g(z) = J[(l - c ) = (1 - d z ) ( l - c z) • • • (1 - c,z), jZ

3=1

2

25 Table 2: ^-stability of the aeries of collocation RK u U 50 Q ]

6!

o 0

0

7

oo

0 0 0

1

oo

Q Q 0 o

s

oo

0 0 0 0 0

4

OO

9 o 10 M 11 XM 0 12 X o 13 X 14 X 15 X X 16 X X 17 XX X IS X 19 X X 20 X X 21 X X

a

OO

X

o 0 0 0

2

OO

a

0 0 0 Q 0

OO

2 3 4 S

o 0 0 0 0 o 0

Q 0 0 o o 0 o o 0 0 0 •

0 o o o o 0 o o o o 0 o

o o 0 o o 0 o o o o 0 o

0 o o o o 0 0 o o o o o

0 o o o o 0 0 0 0 0 o o

X •

X X •

8 ——

o o Q o o 5 0 Q o o Q o o

o o 0 o 0 o 0 0 Q 5 0 Q

o 0 o o 0 o o Q o o o

11 12 u

14 15 16

o o o o o o o o 0 0

0 o 0 0 0 0 o

0 0 0 0 0 0 0 o o

o 0 o o o 0 o o

0 0 0 o 0 o

o 0 0 0 o

-rr

0 5 o o

0 0 o

19 20

0 0

0

+

which implies 0, = 1/c, {j = Hence every zero of A{z) lies in C under our assumptions. • The investigation of ^-stability should be, however, carried out for each cases. For the Newton-Cotes type methods, the collocation points are given by Eq. (14). The polynomial ir,{t) is then given by 7f,(t) =

r V , ( 0 where

4. Acknowledgement The authors are indebted to W. HUNDSDORFER for his suggestion of the equivalence of HIDM to implicit RK formula. They are also grateful to T . K O T O , C H .

26 SUZUKI and T . WATANABE for their stimulating discussions. 5. References 1. K. Abe, A. Ishida, T. Watanabe, Y. Kanada and K. Nishikawa, HIDM — New numerical method for differential equations, Kakuyugo-kenkyu 57(1987) 85-95 (in Japanese). 2. K . Burrage, High order algebraically stable Runge-Kutta methods, BIT 18(1978) 373-383. 3. J.C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18(1964) 50-64. 4. K. Dekker and J.G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations (North-Holland, Amsterdam, 1984). 5. S.P. Nersett and G. Wanner, The real-pole sandwich for rational approximations and oscillation equations, BIT 19(1979) 79-94. 6. G. Polyaand G. Szego, Problems and Theorems in Analysisll (Springer-V., Berlin, 1976). 7. J. Schneid, Stability properties of collocation methods, BIT 28(1988) 184-187. 8. H.A. Watts and L.F. Shampine, A-stable block implicit one-step methods, BIT 12(1972) 252-266. 9. K. Wright, Some relationships between implicit Runge-Kutta, collocation and Lanczos r methods, and their stability properties, BIT 10(1970) 217-227. Appendix. Equivalence of H I D M to the collocation R K Originally HIDM is a discrete variable method for Eq. (1) stated as follows: Let Xj be the step-points with the step-size ft Xj

= a + jh 0 = 0,1

m)

and j/j be the approximate solution at x . Let us denote the fractional points %, TJJ, • - • i.Tm {r)j = a + Sjh), and the approximate solution and derivative at ij,- for y(x) by Yj, respectively. Assume that Yj and Y, are given by 3

i

Yi=£c v>, fk

Here the coefficients {Cjt} and {D } jk

tin) - £c*vM

= o(n

m + l

),

;

(

A

.

i

are chosen so that the relations ^(m)

- i f ; D (x ) jkV

= o(h

k

m + 1

)

hold for any sufficiently smooth function y(x). The series {y } is determined by k

Yi = f(m,Yi)

)

(J-1.2

m).

(Ai)

27

Once the coefficients {Cff,}, {Djk} and the points {m} are specified, then the process can be expressed as follows. Let y, Y, Y be the m-dimensional vectors given by T

y = 0/.,---,Z/ ) , m

T

Y = {Y ,...,Y ) , l

Y =

m

(Y ...,Y f. u

m

From the second identity in Eq. (A.l) we have

+ Vy

hY = dy

0

where the column vector d and the m-square matrix X> are given by - D

' 2>io "

• •

n

D

lm

d =

Thus, provided that V is non-singular, we have 1

l

y = -D- dy

+ hD- Y

0

(AA)

Simiiarly the identity Y = cy + Cy

(A.5)

0

holds, where ' C c =

,

• •

n

C\

m

c = Ci

•

m

Substitution of Eq. (A.4) into Eq. (A.5) yields l

l

Y = (c - CD- d)y

+ kCV~ Y.

0

(A.6)

Let the symbol denote the j'-th component of m-dimensional column vector. Then from Eq. (A.4) y is given by m

l

ft. = [-T>- d] y m

l

+ h\V- Y\ .

0

(A.7)

m

-1

l

If we can establish the identities [ - £ > d ] = 1 and c - CV~ d and (A.7) are written as Vm = ya + h[V- Y] m

= e, Eqs. (A.3)

l

(A.8n)

mt

Y = f [lt,m

l

+ hCV Y)

,

(A.86)

where / is the m-dimensional column vector whose j - t h component is equal to / [vi>Vo + y " j , ) . Thus, substituting h = H/m and denoting x = x , x = Xn+uVo = Vn&m = IM-ii we obtain _1

0

n

m

28 Y, = y + HY,*i fU« k=i n

+

k

K

m

S

-±H Y ), ' l

k

1

where a,> is the (j, k)-th component of CD' /m and bj is the j - t h component of the last row vector of P / m . This is nothing but an impficit RK formula. (2) of m-stage with step-size H{= mh). Next, considering the special case y(x) = 1 for the constraints (A.2), we have the identities m in £ c , . = l and £ f l = 0 (j = l,...,«*}. - 1

( i

These imply c + Ce = e and d + Ve = o. Hence we arrive at the identities 1

e = -V~ d

1

and c-CD~ d=

e,

which are expected in the above. Moreover, the constraints (A.2) and (A.7) mean that if the solution y(x) is a polynomial of degree at most m, the approximations y and Y (j = 1 , . . . ,m) are m

3

exact for y(x ) and — (%), respectively, at the next step-point and collocation points dx of the step-size H = mh. Hence we have a collocation RK formula. From the derivation of HIDM, it is obvious that the equidistant points = a + (jjm)H stand for the interpolating points with the step-size H. This implies that the formula is the Newton- Cotes type mentioned in Section 2, and that the formula parameter is uniquely determined. It also assures the nonsingularity of the matrix V. m

29

F O U R T H ORDER P-STABLE BLOCK M E T H O D FOR SOLVING T H E D I F F E R E N T I A L E Q U A T I O N y" =

f(x,y)

Kazufumi OZAWA Education Center for Information Processing, Tohoku University Aaha-Kv, Sendai, Japan 980-77, Japan E-mail: [email protected]

ABSTRACT A certain type of P-stabJe block method is derived for solving second order initial value problem y" — f(x,y). The block method considered here computes the numerical solutions simultaneously at the two points of x, and is easily paralletisable. Some technique to reduce the local truncation error of the method is also developed. 1. Introduction The second order initial value problems (IVPs) of the form y" = f{*,y),

!/(zo). y'M

given

(1)

are of frequent occurrence in practice, so it is particularly important to develop efficient methods for these problems. To solve the problems one uses the family of linear multistep(LM) methods of the form E%JWi=ft £:fe.

( )

a

Xi = x„ + ih, /, =

2

f(xi,yi).

where h is the step-size and y< is an approximation to the solution y(xi). Hereafter we assume that the function f{x,y) satisfies the Lipschitz condition with respect to y. The LM method of the type (2) is said to be consistent if

where p(() and a(Q are the polynomials denned by

= AC*+&-iC*~'+ •*•+&• The consistent LM method (2) is said to be of order p if the power series expansion of the operator L[y(x); h] := £ a (x jV

+ jh) -

g

ft/0

+ jh)

(3)

30 p+2

satisfies L[y(x); h] = 0(h, ), for all sufficiently differentiable function y(x). In the family (2) the Stormer-Cowell methods(see e.g. Hairer and Henrki") are the most commonly used ones. It is, however, well known that the methods with step number greater than 2 exhibit an orbital instability for the test equation 9

y" = - « V

{«)

where u is real; the numerical solutions generated by such methods do not stay on the circular orbit but spiral inwards. On the other hand, it is also known that the Numerov method, 2-step Stormer-Cowell type method, is unstable for alarge step-size h. 2

!

The interval of H = (w/i) is called the interval of periodicity, if the method (2) with any H within this interval gives a periodical solution. The method having interval of periodicity (0, oo), which is expected to be stable for any step-size h > 0, is said to be P-stable . For the P-stabie method Dahlquist and Lambert and Watson independently established that the attainable order of the method is 2. However, Cash and Chawla showed that higher order can be attained if one or more off-step values are used. The multistep methods which use off-step values are often called hybrid methods. For the certain type of the hybrid P-stable methods QI and Mitsui gave the attainable order. Many high order hybrid P-stable formulae have been derived (see e.g. Cash , Chawla , Hairer , Khiya , Simos and Thomas ). Among these methods Haiter's one is the most simple one, and is given by 2

1J

2

7

13

1

1 5

2

5

10

Vm+l

12

-

17

18

10

Sfc + l h - l = ^ { / V + i + (10 - T)/» +

h - 1

+ 7/-}.

(5)

U = /v*i>*)i ' = " , « ± 1, /. = /(*..*). where the off-step value y is given by a

V* = » . - J^-A'C/m+l " 2 / + / n - l ) . n

(6)

The method(5) is shown to be P-stable provided that 7 ^ 0 , wk je s/n. and is of order 4. In this article we shall develop a P-stable method which computes y „ i using the information available at T„ and can be executed simultaneously with Hairer's method (5) on parallel computers. Perhaps the most commonly used parallel algorithms for the numerical IVPs are the block methods, which consist of two or more LM methods and each of these t

31 methods computes the numerical solution simultaneously at the different point of x. The block methods are, in general, suitable for coarse-grain parallel computations. Although a number of block methods have been derived (see Burrage , Chartier , Chu , L u , Shampine , W a t t and Zhou ), all the methods derived so far are not for the second order equation such as (1) but for the first order equation j/ = f(x, y). The aim of this paper is to develop the P-stable block method for the second order IVPs (1). 1

6

14

16

19

3

20

2. Block method The block method to be considered here consists of two LM methods, one of which is Hairer's(5) one, and the other one is = / i { M / 2 + U - 2 ) + bi(U+i + U-i) + 2&o/ }, 2

J/-+2 - 2y + y n

n

2

n +

fi = f{x„yi), /

i =

n +

where the off-step value y„

(7)

n

t = n , n - l,n±2, f(x„ ,y ), +1

n+1

is an approximation to y(x + k) and is given by

+i

n

2

j/.+i - a(y*+2 + y*-2) + 2by - I / * - , + h { c ( / n

n + 2

+ /„_ ) + 2df }. 2

(8)

n

In our block method a pair of methods (5) and (7) compute y and y simultaneously using the previously computed step values y (i = n — 2, n — l , n ) . Note that the second method (7) dose not use the values y i, / i , which are the values to be computed by the first one(5), since the use of these values in the second one makes it difficult to execute the two methods (5) and (7) simultaneously on parallel computers. First of all, we must determine the coefficients b , bj and b in the second method (7) so that the method is being of order 4, in accordance with that of the first method (5). To do this, we associate with the method (7) the following difference operator n+1

n + 2

i:

n+

n +

0

2

L[y(x)M: L[y{x);h]

:= y(x + 2h) - 2y[x) + y(x - 2k) h {b (y"(x + 2k) + y"(x-2h)) 2

(9)

2

+

b,(y"(x + h) + y"(x - h)) 4- 2b y"{x)}. 0

Assuming that y(x) is sufficiently often differentiable, we expand the operator L about x as the Taylor series L[y(x)-M

= 2(2 - b - fc, -

fe)

0

+

J l 5 6

'"

2 1

4 8

0 f e 2 0

+

3 2

^

+ " (

3 i l

~l a

^ + 0(ft ).

2 b 2

+

4

y"'(^

4

(10)

32 2

The values b and b, for which the terms 0(h } and 0(h*) vanish are given by 0

h-M

jfcft±4

t l | j

(it)

Substituting these into (10) we have

Therefore, under the assumptions that no previous errors have been made(localizing assumption), and that / „ approximates to y"(x i) with the error of order at least ft , we have for the local truncation error of the method(7) + ]

n+

6

T := y(x ) 2

- y

n+2

n + 2

= "

1

5

^

+

1

^

+ 0(k*).

(12)

In order that f is being such an approximation it is necessary that £ „ approximates the solution y(x ) with an error of order at least ft , since f(x, y] has already been assumed to satisfy the Lipschitz condition. To get 5„ i satisfying the condition stated above we must consider the following difference operator L: n+1

+1

6

n+1

+

L[y(x);h}

:= y(x + ft) - a{y(x + 2ft) + y{x - 2ft)) -

&m+m ft2

- * ) +

{ c ( / ( > + 2ft) + y'\x

o

R

y

+

1 3

V ' t ^

( )

- 2ft)) + 2 d / ( T ) } .

Note that the definition of the operator is based on the practical consideration that ;/„ > always subject to the local truncation error when computing i / by (8), even if the localizing assumption has been made. The Taylor series expansion of (13) is given by s

+2

n + 1

L[y(xy,h\

2

= 2(-a-b + \)y(x) + - 1 6 a - 4 8 c + l ,,,, . . + " 2 y i ^

(-4a-2c-2d+l)yW{x)h

W

(14)

1

-360ab - 40a -480c + 1 yW(x)h 360 2

( 6 )

6

+ 0(h*).

The coefficients for which the terms up to order 4 in (14) vanish are given by a

=

-48(i-l- 23 — 8 0 — '

, 4

=

48d + 57 - ^ 0 — '

C

=

8d - 3 — -

<

1 5 )

Substituting the coefficients into (14) we have f

2

:

=

^

+

1

)-

= 3 4 8 d - 2 3 ) - 48. M

+

17

+

^

^

33

Thus, we have a 4th order LM method which is suitable for the second method in our block method. In the next section we will investigate the conditions under which the method (7) is P-stable . 3. P-stable analysis If we solve the test equation(4) by the method (7) then the numerical solution t/i satisfies the recurrence relation 2

2

R (H )y„ 0

3

- 2R (H )y

+2

1

+ MH )*^

7l

2

2

where H = uh, and the coefficients Ro(H ) and given by

and Ri{H )

3(3b - l)(16d - 1) +20 H 60 2, , , 3(36,-l)(16
R (H )

s

2

(17)

are the polynomials in H (3fca-l)(8d-3) H\ 30 4(36 - l)d

2

= 1+

0

= 0,

2

(18) (19)

.

H

2

Necessary and sufficient condition that the y„ defined by the relation (17) has a solution of the form y = e*' [8 = real) for any H > 0, that is, the condition for the method (7) being P-stable , is 9

2

n

2

V H > 0,

2

Ro(H )

(20)

<1,

or equivalent!y 2

2

V H > 0,

{R (H ) a

2

+ R,(H ))

2

(Ro(H )

- Ri(H*))

> 0.

(21)

It can be easily seen that the necessary condition for (21) is d e (-3/32,1/16), since 2

Ro(H ) 2

MH )

2

2

- R^H )

= 2H +

2

+ R H)

=2

l{

+

10

3 ( ^ - l ) (30 y-D-40 , t f

_ (3

& 2

-l)(32 30

t f +

3)

t f 4

2

Moreover, since R + Ri ~ 2 for small H , we are allowed to consider only the case that RQ + RX > 0 and R -Ri > 0. When d e (-3/32,1/16), necessary and sufficient conditions for both of Ro + Ri and Ro - Ri to be positive are 0

0

3*2

- 1 < 0,

(22)

34 and m

;

=

|3(36 -I)(16d-1)-4Q| 2

2

4(36 -l)(32d

+

2

+

3)

<

Q

The discriminant D(d) has the distinct real zeros d, and d (di < d ) 962 - 43 + 20^/-3(36 - 1) 2

2

2

48{36s - 1)

^

'

(24)

9 6 , - 43 - 2 0 ^ - 3 ( 3 ^ - 1 )

* =

« f ^ T )

Since f ( d ) is negative on (di,d ),

25

"

< )

the method (7) is P-stable provided that

2

Here we must investigate the condition for the first interval defined above is being nonempty. We have for any b 2

d i +

i=96(1^) {

1 9

- ^ - ey=i(>r^iy} > o,

which is given by the identity 2

2

2

(19 - 96 ) - ( 8 ^ - 3 ( 3 6 2 - 1 ) ) = (96 + 13) . 2

2

Therefore, if d] < ^ then the interval is shown to be nonempty. A simple computation shows that this is the case when 6 < —4, Moreover, we have for d 2

l i -

d

2

^ 1 2 ( l ^ ( -

2

2

- ^ ^ )

<

0

-

Thus, we have proved that the second method (7) is P-stable, for the parameters 6 and d in the intervals b><--, 1 d, ( 2 7 )

Example 1. Here we solve the equation defined by 2

y" = -<" y, y(o) = i , y'(o) = o, y{x) = casux,

(28)

w = 10

to compare the errors of our P-stable block method with those of Hairer's P-stable method. The errors at x = 5rr, IOJT, 15TT, and 20?r are shown in Table 1 and 2. In this experiment we set bj = -0.112 and d = 0.062.

2

35 Table 1. Errors of Hairer's P-stable method. x = 5JT X = 105T x = 15TT I = 20TT

h = TT/40

h = ir/SO

ft = TT/160

h = jr/320

-3.16E-03 -1.27E-02 -2.85E-02 -5.06E-02

-1.31E-05 -5.27E-05 -1.19E-04 -2.11E-04

-5.22E-08 -2.09E-07 -1.71E-07 -8.37E-07

-2.05E-10 -8.20E-10 -1.85E-09 -3.28E-09

Table 2. Errors of P-stable block method. k = TT/40 X =

5x

x = IOTT

x = 15TT x = 20TT

-5 -1 -1 -1

h = TT/80

75E-01 65E+00 96E+00 15E+00

-3 -1 -2 -5

20E-03 28E-02 88E-02 11E-02

h --= TT/160

h : = TT/320

-1 -5 -1 -2

-5 -2 -4 -8

33E-05 32E-05 20E-04 13E-04

27E-08 11E-07 75E-07 45E-07

Example 2. Next we consider the 2-body problem: Vt = - | , yi(Q) = 1, Jri(O) = 0,

(29)

yi(x)

= cosx,

y2(x) = s i n i .

ID this example, we integrate the equation in the interval x f [0, IOOOTT], and in order to compare the accuracies of the methods, compute the maximum orbital errors (30)

max D
where jfe,* and y , are the numerical solutions corresponding to the exact solutions yi{x-n) and i/2(x„), respectively. The results are shown in Table 3.1 and 3.2. 2 n

Table 3.1. Maximum orbital errors (30) of Hairer's P-stable methods. h = TT/10

h = TT/20

h = TT/40

h = TT/80

h = TT/160

3.407E-04

2.181E-05

1.371E-06

8.584E-08

5.367E-09

Table 3.2. Maximum orbital errors (30) of the P-stable block methods. h = TT/10

h = TT/20

h = TT/40

h = TT/80

h = TT/160

4.317E-02

4.655E-04

6.613E-06

2.375E-07

1.348E-08

We can see from the tables that although the results of our P-stable block method are slightly less accurate for both problems, our P-stable method integrates the equa-

36

tier) stably for long range of intervals. 4. Local extrapolation for P-stable block method We have seen in the previous section that our P-stable block method does not necessary give an accurate result compared with that of Hairer's one. The reason for this is that the second method (7) of the block method has a large error constant (—15b — 1)/15; the value of the constant at b — —1/9 is greater by a factor of 64 compared with that of the first one for the test problem (4), In order to improve the accuracy of the second method we shall develop some extrapolation technique such as Milne's device, which is often used to enhance the order of convergence in conventional LM methods for the first order equations. In our extrapolation technique we need another approximation to y(x } of the same order. To get such an approximation we use the same method as (7) with different set of parameters. We attache the superscript * to all symbols relating to the second approximation. The method for the second approximation is 3

2

n+2

y:

+2

-

2

+ sk-a = h {b' {f; 2

+ u $ + Kif:

+2

+ 1 U ) + 2b / },

+l

ffi

0

s

£ M = "(fn+2 + fc-a) + 26*& -

+ ft {c'{/„

+ /„_ ) + 2d'f }.

+2

2

n

n

(31)

(32)

In the method above the free parameters b and d' should be chosen in the range which guarantees P-stability, The local truncation error of the method is given by 2

T- := y{x ) n+2

- y'

15

+

= ~^

n+2

1

e

y&(x )h n

+ 0(k").

(33)

Using two methods (7) and (31) we can easily estimate the local truncation error of (7). From (12) and (33) we have 6|

*V (z») = ^ ( y „

+

2

" C.J + 0(*ft

(34)

where -lMa + 1 15 '

-15ft;+ 1 15 '

and therefore we get Q

y{x ) n+2

=y

n+2

+ , _ c

c

(JW.2 - y* ) + 0(h'). n+2

(35)

37 Thus, we have the following modified block method: fcU = 2y - y _ + —{f^j n

n

+ (10 - ) / „ + /„_, + / }

1

7

= 2y - y _ + h?{b {} + n

n

2

2

y*+2

=

y*+2 +

c

,

_

2

+ /„_ ) + 6,(/

+3

+ A- ) + b;(/;

n

2

li+a = fa - y,-2 +

ft {6 (/; 2

(fn+2 -

c

7

2

m+1

2

+ !

n

+ f„_ ) + 26 /n} t

0

+ fn-i) + 2b- f„}

( 3 6 )

a

y' ) n+2

where the off-step values j j ^ j / and y^ are given by (6), (8) and (32), respectively. The locally extrapolated value £ in the above algorithm is expected to be more accurate than is y . Therefore, if we compute y and replace y „ by it before proceeding to the next block then we can improve the accuracy of our P-stable block method. Note that the calculations of the three values y„+i, t/„ and y^ can be performed in parallel using three processors, and that the calculation of j / requires no function evaluations. n + 1

+1

n + 2

n + 2

n + 2

+2

+2

+2

n + 2

Example 3 . Let us consider the same equation as that of Example I , The results by the modified algorithm are shown in Table 4. In this experiment we take b = -0.2 and d' = 0.0622, and take the same values for 6 and d as those in Example 1. 2

2

Table 4. Errors of modified block method. I = 5* X = IOTT X = 157T X = 20TT

h = TT/40

h = jr/80

k: = TT/160

h = TT/320

-1 8 5 E - 0 2 -7 4 0 E - 0 2 -1 6 5 E - 0 1 -2 87E-01

-4 5 1 E - 0 6 -1 81E-05 -4 0 8 E - 0 5 -7 2 6 E - 0 5

-1 1 3 E - 0 9

-2.99E-13 -i.UE-12 -2.51E-12 -4.46E-12

-4 54E-09

-1 0 2 E - 0 8 -1 8 2 E - 0 S

Example 4. Next we consider the same problem as that of Example 2. In this experiment we also take the same values for b , b' , d and d" as those in the previous examples. The maximum orbital errors are shown in Table 5. 2

2

Table 5. Maximum orbital errors (30) of modified P-stable block methods, ~~h = TT/10

5.661E-02

h = TT/20

k = TT/40

h = TT/80

h = TT/160

6.585E-04

5.387E-06

4.412E-08

6.529E-10

From the results above we can see a very considerable improvement over those shown in Table 1, 2 and 3, in particular for small h. 5. Stability analysis of modified method Next we study the stability of the modified method (36). If we solve the test

38 equation (4) by the algorithm, then j i

and y^

n + 2

2

y ~2S(H )y n+2

+s

2 S

{

H

+y-

a

n

2

!,;

must satisfy the recurrence relations

+2

- 2S-(H )y

_ > - M i w y

(37)

+ y _ = 0,

n

n

{

H

)

(38)

2

2

_

2 S

= 0,

2

R[(H ) m m

-

where Rg and R, are the polynomials defined by(18) and (19), and flj and R' are the polynomials defined by the same Eqs., but b and d are replaced by b and d' respectively. From Eqs.(36), (37) and (38) we can find the recurrence relation to be satisfied by y - The relation is 2

2

:

n+2

j}„ 2 +

2

2S(H )y„ + y n

= 0,

2

(39)

2

where S{H ) is given by

2

The modified method is stable if and only if the function S{H ), which is often called the stability function, is less than unity in modulus. A simple computation shows that the modified method (36) is P-stable , if b —• —oo and b remains finite, (if -oo and b remains finite), then S(H ) S'{H ) (S(H ) - S(H )). There may be some cases that the method is not P-stable for finite b and b . However, in these cases there exist the intervals (0, H ) and (H ,H ) (H < H\ < H ) in which the method is stable. A further study on the stability of the modified method will be needed. The graphs of the stability functions S(H ) of the modified algorithm for some b and b are shown in Fig.1-3. In these graphs we take d = d' = 0.062. 2

2

2

2

2

2

2

2

2

2

2

2

0

2

2

2

2

40

6. Concluding remark We have derived a certain type of P-stable block method for solving the second order IVP's on parallel computers, and developed the procedure to reduce the local truncation error of the method. The modified algorithm using this procedure

41 produces an excellent Tesult, and seems to be P-stable for almost parameters. Further consideration on implementation of the algorithm on parallel computers and its performance evaluation will be necessary. 7. References

1. 2. 3. 4. 5. 6. 7. 8.

K. Burrage, J. Comput. Applied Math. 45(1993) 139. J.R. Cash, Numer. Math. 37(1981) 355. P. Chartier, SIAM J. Numer. Anal. 31(1994) 552. M.M. Chawla, BIT 21(1981) 190. M.M. Chawla and P.S. Rao, IMA J. Num. Anal. 5(1985) 215. M.T. Chu and H. Hamilton, SIAM J. Sci. Stat. Comput. 3(1987) 342. G.G. Dahlquist, BIT 18(1978) 133. S.O. Fatunla, Numerical Methods for Initial Value Problems in Ordinary Differential Equations, (Academic Press, New York, 1987). 9. E. Hairer, S.P. Nsrsett and G. Wanner, Solving Ordinary Differential Equations I, (Springer, Berlin, 1987). 10. E. Hairer, Numer. Math. 32(1979) 373. 11. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, (John Wiley & Sons, New York, 1962). 12. M.S.H. Khiyal and R.M. Thomas, Computational Ordinary-Differential Equations ed. J.R. Cash and I . Gladviell, (Clarendon Press, Oxford, 1992). 13. J.D. Lambert and I.A. Watson, J. Inst. Maths. Applies. 18(1976) 189. 14. L. Lu, IMA J. Numer. Anal. 13(1993) 101. 15. Tie-Shan QI and T. Mitsui, JJAM 7(1990) 423. 16. L.F. Shampine and H.A. Watts, Math. Comp. 23(1969) 731. 17. T.E. Simos, JJIAM 10(1993) 289. 18. R.M. Thomas, BIT 27(1987) 599. 19. H.A. Watts and L.F. Shampine, BIT 12(1972) 252. 20. B. Zhou, J. of Comput. Math. 3(1985) 328.

43 TWO-POINT HERMITE-BIRKHOFF QUADRATURES AND ITS A P P L I C A T I O N S T O N U M E R I C A L S O L U T I O N OF ODE C H I S A T O SUZUKI

Department of Computer Science, Skizuoka Institute of Science and Technology, SSOO-g Toyosavia Fukvroi-shi Skizuoka 4SI, Japan E-mail: [email protected]

ABSTRACT In this paper a special class of Hermite-Birkhoff quadratures is investigated and applied to numerical method for solving directly any higher order ordinary differential equation. That is, if an incidence matrix B = (eij) denned by ey = e

e

7 1

i s

o i s e d

i n

t h e

0 or 1, (l
ft

/ P{x)dx = E Vii^ixi),

fa

= a and i = 6), a

to be exact for any polynomial P with degree at most n — I , where HUj'i are weight coefficients independent of P. In addition, a numerical method for solving directly the initial value problem of r-th order ordinary differential equations

f J/

LR,

(01

11

= /(i.J .---,!/'''- ),

(

1 » '>(*o) =

tf\

*>*»,

i = o,i,..., -i, r

is constructed from all the quadratures specified by £ - (e;>), (eio = ••• = = e , = •:-•- = ei,. = 1). with n = 2s (n = 1,2,....r). To verify efficiency of the method, a numerical example is included. a

0

1. I n t r o d u c t i o n Let 7 = [a, 6], (a < 6), X = {x^I \ a<0t<Xv<>• -<x 1). Let n _ ! be a space consisting of polynomials with degree s
(i)

Ci,=0orl,

pi)

E i=l j = 0

n

n

1 < i < M < j < n- 1,

E

This E is often called an incidence matrix of quadrature or interpolation. Let V„ denote all the set of n real numbers yi indexed by (£, j) such that eg = 1 for a given fc X n incidence matrix E = [ % ) , i.e., y„ = {yj | By = 1}. Given a it X n incidence matrix E and X„, we can define a quadrature formula in B

44 the form Jo

1

for any polynomial P e n _ i , where W^'B are the weight coefficients independent of P. This is called ' ' the Hermite-Birkhoff quadrature formula (hereafter, merely say HB-QF) for the incidence matrix E. In this paper, first we consider the question of existence and construction of HBQF specified by E with k = 2 and n > I . That is, this problem is of generalization of the Euler-Maclaurin quadrature. As an application of HB-QF to ordinary differential equations, next we develop a numerical integration method for the initial value problem of the r-th order ordinary differential equations n

1 2 3

j,M = f( ,

&%.,.,

x

(i)

y (xo)

= Vo\

i = 0.1

(1)

r - l ,

where r is any fixed positive integer greater than 1. Then the numerical integration formula in the form Vm+i = 1 5 * + <*fcF +

m = 0,1,..., M ,

m

is given for the initial value problem, where h is a step-size of integration, x XQ + mh,

m

tfm-H

and F

Ym+i —

m +

=

i = 0,1,

i = J*)

here for each s (0 < s < T) is an approximation of yW(x i), and at, /3|, are r-column vectors dependent upon h. To verify efficiency of this scheme, a numerical example is included. m+

2. Lacunary Interpolation Problem 4

According to the theory of lacunary interpolation , an incidence matrix E = ( v ) . 1 < » < f c , 0 < j < f l — 1, ia said to be unconditional-poised or, merely, poised if there exists a unique polynomial p€ll„^i such that for any F„ = {yl | ey = 1} and any X„ = {xi | 1 < i < n], e

£

W

P ( X . ) =~&,

1 < i < fc, 0 <

For a k x n incidence matrix E = (eg), let h

i=l

< n - 1.

45

and

<

= E < . 0
(2)

Then the following Pdlya's result is well-known. Theorem 1 (Polya's theorem ) A 2 x n incidence matrix E is unconditionalpoised if and only if M£ determined by E satisfies the inequality Af > p+ 1, 0
B

p

This inequality is called the Polya condition. For a 2 x n incidence matrix E = (ey), we define a 2 x n matrix G = (gij) as follows;

gij = 1 - e.,„_j_i,

1 < i < 2, 0 < j < n - 1.

Then G is also an incidence matrix since 2 * - l

E £ w i=l j=0

= *•

In this paper, the G is called a dual incidence matrix corresponding to E or, merely, a dual matrix of E. For example, for the incidence matrix F - (

E

~

1

°

°

{ 0 1 1

its dual matrix becomes G =

1 1 0 0 0 1

The following theorem give a relationship between a 2 x n incidence matrix E and its dual matrix G. Theorem 2 A 2 x n incidence matrix E = (ey) i s unconditional-poised if and only if its dual matrix G is unconditional-poised. Proof To prove this theorem, we use the following relations obtained immediately from the definition of the dual matrix G\ m

f

= 2 - E

t=i

e.^-j-i -

2 - m^,_

u

(3)

for each j , (0 < j < n- 1). On necessity: When p = 0 , 1 , . . . , n - 2, from Eq. (3) we have

M? - 2(p+l)-Em*_ _ , i=o

=

2(p+l)-^-

=

2{p + 1) - » +

i

1

E^-i-ij(4)

46

Therefore we obtain Mjf > p + 1,

0 < p < n - 2,

since by virtue of Theorem 1 for poisedness of B. When p = n — 1, on the other hand, we have n-i

Therefore G satisfies the Polya condition. On sufficiency: Substituting n — q — 2 for p in Eq. (4), we get M? = 2(q+l)-n

+ MZ_ _ , q

2

for q = 0 , 1 , . . . , n - 2. Thus we have > q - f 1 for each q (Qn — q — 1 by poisedness of G. When p = n-1, computing M ^ _ , from Eq.(3), we have Mf_, = n since = a. We have thus proved Theorem 2. q

3. Existence and Construction of H B - Q F 5

The following lemma can be shown by means of the Darboux formula . Lemma 1 Let X\ = a and x = b, (a < b). Let q be any monic polynomial of degree n. Then for every P e I I _ i , it holds that 2

n

/V(x)^ = E E ' i V F i +

J

"

J

,=o 1=1

0-6-a),

(5)

(
(6)

( i )

(^).

where Wti = ^ P l

| n

~'~

, )

(i ). i

2

Proof By using the Darboux formula it can be proved that, for any monic polynomial q of degree n and any (n + l)-th continuously differentiable function u on /, it follows that u(b) = u(a) + E ( - i y ^ y (y-'>(0)«W(a) - q^\l)u^(b))

+

ft,,

(7)

47

where =

*

( _ 1 )

>)(Oji

^

+

1

}

(

a

+

s

h

^ -

(»)

Now define u(x) = j * p{t)dt,

(Pen„_,), 1

then we have u«>(x) = P ^ ' f i ) for j = 1,2,• • • , n + 1, and substituting P ^ - ' for tt$ in Eqs. (7) and (8), we can obtain ft?

u(b) = u(a) + £ ( - i y J ^

(q^(Cj)pU-V(a)

- q^(l)P^{b))

,

(9)

and

w

However since P ( x ) = 0, we obtain R^-Q. u{a) = 0, and u(b) = £


Therefore by substituting g (0) = n!,

P{t)dt,

into Eq. (9), the proof of Lemma 1 is completed. A quadrature formula for any rc-th continuously differentiable function g defined on 7 is obtained as ;=0 t = l

by substituting g for P in Eq. (5). However, in this case, it should be noted that the remainder term is not necessary zero, that is, in general, jj(x)dx where

= h(g) + J t » ,

(10)

n + 1

R^ = ( - 1 ) " ^ - / ' q(x)g™(a + xh)dx, h = b - a. n\ Jo As an application of Lemma 1, we can use it for the proof of an existence theorem of HB-QF. In fact, in order to show the existence of HB-QF specified by a 2 x n incidence matrix E = (e^), now it is sufficient to prove that Wi, = 0 if e = 0. In other words, if there exists a monic polynomial q of degree n such that q ~'~ {ti) =0 for ejj = 0, then the existence of HB-QF becomes clear by Lemma 1. Formulating this assertion, we have the following lemma. Lemma 2 7Viere exists a HB-QF specified by a 2 x n incidence matrix E if and only if there exists a monic polynomial q of degree n such that
in

i}

2

l,

48 Proof Suppose the existence of HB-QF specified by E = [a,), then we have the expression in the form

E

?p{x)dx=

Pen..,.

m

mf W%

(ii)

On the other hand, by Lemma 1 we also have the expression in the form P(x)dx = f'* P{x)dx = E

+l

W)

fc' wyP (*i)+

E iPhnfltHpfi,.

_ _•

(12)

r*

for any P e n _ , . Therefore, comparing Eq. (11) with Eq. (12) we get the relation n

E » ^ % f % ) = o. ti;=0

Since this relation must be satisfied for all P e n „ _ i , it is necessary that tetj = 0 if ey = 0. This means by Lemma 1 that there must exist a monic polynomial q of degree n such that x

q^-i- \U)

= 0 if

= t,

because j)i,„-j-i = 1 — = 1. Conversely, if there exists a monic polynomial g of degree TI such that ?'*'((;) = 0 if = 1, then it is trivial by Lemma 1 that the HB-QH prescribed by E exists. We have completed the proof of Lemma 2. By virtue of Lemma 2, the following theorem is proved. Theorem 3 A 2 x n incidence matrix E is poised if and only if there exists a HB-QF specified by E. Proof Let G be the dual matrix of E and H = [hi, k ) with hi — 1 and h = 0, then we consider the horizontal sum G = (§y) consisting of G and H, i.e., T

2

9a =

f gtf, hf,

2

if 0 < j < j i - l , if j = n.

This horizontal sum is often denoted by G = (G\H), and we recall that the horizontal sum is poised if and only if both G and H are poised. Therefore if G is poised then the horizontal sum is also poised since H is evidently poised. Consequently since G is poised by Theorem 2 provided E is poised, there exists uniquely a polynomial q of degree n such that fiHti) = 0, if g - = 1, i}

( 0 < j < n - 1, i = 1,2),

(13)

= n\.

(14)

and
q (t ) 1

49 where (, = 0 and t = 1. Conversely, if E is not poised then G is also not poised. Therefore any polynomial q of degree n which satisfies the interpolation conditions in Eqs. (13) and (14) does not exist, and any HB-QF specified by E does also not exist by Lemma 2. We have completed the proof of Theorem 3. By Theorem 3 and Lemma 2, if a given incidence matrix E is poised then it is guaranteed that there exists a monic polynomial q of degree n such that q^{U) = 0 if = 1 for the dual matrix G of E. Especially it should be noted that finding such g is just equivalent to solve a homogeneous interpolation problem on G. Therefore if the homogeneous interpolation problem can be solved, we can then obtain the HB-QF specified by E from the solution, by computing the coefficient »);,• in the way shown in Lemma 1, 2

4. Several Examples of H B - Q F In this section, quadrature formulas of general Hermite and Euler-Maclaurin-like types are illustrated together with the Euler-Maclaurin quadrature formula as typical examples of HB-QF. In particular a feature of the Euler-Maclaurin-like type quadrature is to use even order derivatives in the formula, though the Euler-Maclaurin quadrature uses odd order derivatives.

4-3- Quadrature Formula of General Hermite Type Let ni and n (n. < 112) be non-negative integers and n = n. + n > 0. Define a 2 x n incidence matrix Ean(ni, »a) = (ey) by 2

2

e

*'

f 1,

if

I 0,

if

/=o,M,---.«i-i.

otherwise,

for i = 1 and 2, then the interpolation problem on Eofffni.ttl] is known as a general Hermite interpolation problem which always has a unique solution. Therefore the incidence matrix E (n n ) is poised. Then the dual matrix G = {gij) of Egsfa,fts) defied by GH

u

2

9ij m X - e,,,*-,-. = % , (1 < i < 2,0 < j < n - 1) is also poised by Lemma 2. The homogeneous interpolation problem on this G can be uniquely solved as q(x) = t y r - * $ *

a

i

+

k

'

50 without the factor of multiplier. Then the (n-j-

l)-th derivatives of q are given by

\(n + t - j - I ) !

BI

K

-SS*

0 < j < Bl - 1

By virtue of Lemma 1, therefore the quadrature formula on EGH{HI, f P{x)dx = " £ J

o

+1

tf »» P«>(a) li

+

7 i j ) is given as

fc'+Vw^H*).

j=0

;=0

where the weight coefficient uVy can be computed from Eq. (6) as follows

23

n\h

{

U +«i-J-l+9; J

'

+

(A + g i ) !

'

J+

here = min{tii, j } and q = max{0, j — r i i } . This quadrature formula is said to be a general Hermite type. We are especially interested in the class of quadratures with tii = n = r and n = 2r because there is an important application for solving directly higher order differential equations. 3

2

Example 1: (Case of n = 6; n\ = n = 3) (a) Incidence matrix : 2

.

,

/ 1 1 1 0 0 0 \

BMIMJ-^ i I I O O O J (b)

Dual incidence matrix G corresponding to EGH{3,3)

:

( 1 1 1 0 0 0 ^ 1 1 1 0 0 0 (c)

Solution of homogeneous interpolation problem on G: 6

5

4

3

q(x) = x - 3 i + 3z - x (d)

HB-QF of general Hermite type specified by Ec«(3,3): [P{x)dx

= ^P(a)

+

+ T^(P

P(b)) + ( 2

^(pW(a)-pM(b)) 2)

V ) + P' W).

51 4-2. Quadrature Formula of Euler-Maclaurin Type Since the Euler-Maclaurin quadrature formula is included as a special class of HB-QF stated in the previous section, accordingly the incidence matrix prescribing the formula is poised. In fact, the following theorem shows it. Theorem 4 Let r be a positive integer and n = 2r. Let E = (ey) be the 2 x n incidence matrix such that EM

f

1. = { i. [ 0,

if j = 0, if j = 2k-l and if otherwise,

l
for i = l and 2, then EEM is poised. Proof It is evident that E is satisfying the Polya condition. In fact, by computing M from Eq. (2) for E , we can obtain Mo = 2 for p = 0, Ma _. — 2p+2 and M = 2p + 2 for p = 1,2,... , r - 1, and Af _, = 2r + 2 for p = 2r- 1. Therefore since M >p + 1 for p = 0 , 1 , . . . , 2r - 1, E u satisfies Polya's condition. We have completed the proof of the theorem. By virtue of Lemma 1, the quadrature formula specified by EEM is given as E

T

M

EM

p

2 p

2p

p

E

j^P{x)dx

= h(w P(a) 10

+ w P(b)) i0

where the weight coefficient ttty can be computed from Eq. (6). This formula is well-known as the Euler-Maclaurin quadrature. Example 2: (Case of n = 6; r = 3) (a) Incidence matrix : EEM

(b)

10 10

0 0

Dual incidence matrix G corresponding to

G

(c)

1 1 0 1 1 0

-

=

E : EM

I 1 1 0 1 0 0 ^ 1 1 0 1 0 0

Solution of homogeneous interpolation problem on G: 6

s

g( )=X -3x + I

^

-

^

52 (d)

HB-QF of Euler-Maclaurin specified by EEM'. [P(x)dx

=

\{P{a)

- ^ ( P

4.3.

m

+ P(b)) + ^ ( ^ ' ( a ) i

3

)

( « ) - r

m

P (t>))

m

Quadrature Formula of Euler-Maclaurin-Like Type

Theorem 5 Let r be a positive integer and n = 2r. Let E and Q<j
E

M

L = (eij)> 1 < i < 2

j = 2k and l
is poised.

EML

Proof We shall show that E M L satisfies Polya's condition. Computing M from Eq.(2) for E ML< we obtain evaluations of M = 2 > 1 for p = 0, A/2,,-1 — 2p>2p for p = 1,2, . . . , r , and M = 2p + 2>2p + 1 for p = 1,2,..., r - 1. Therefore £ M I satisfies Pdlya's condition. We have completed the proof of Theorem 5. By virtue of Lemma 1, the quadrature formula specified by EEUL is given as E

T

E

0

B

2(!

where the weight coefficient

ijjj,-

1 (2> + l)!2(j + l )

can be computed as follows; Let an — 1/2, and

-E

(2j - 2 * + l ) ! '

then wift - Wisj = a, for j = 0 , 1 , . . . , r - 1. Example 3: (Case of J I = 6; r = 3) (a) Incidence matrix : 0 10 0 10 (b)

1 1

Dual incidence matrix G corresponding to E 10 10

10 10

10 10

(1 < 3 < r - 1),

S3 (c)

Solution of homogeneous interpolation problem on G: 6

a

3

q{x) = x - 3a: + 5a; - Zx (d)

HB-QF of Euler-Maclaurin-like type specified by

+

2^

( / > ( 4

'

( Q )

+

F ( 4 l ( 6 ) )

E : BML

"

Remarks-. In all the cases of examples illustrated above, it should be noted that each dual incidence matrix coincides with the corresponding incidence matrix. 5. Application to Numerical Solutions of Ordinary Differential Equations As an application of the quadrature formulas obtained in the previous section, first we consider a numerical method for the initial value problem of the ordinary differential equation,

Suppose that the solution y to the problem is sufficiently smooth, then we can obtain the following equation from the quadrature formula by substituting y' for g in Eq. (10) and computing the integration of the left hand side . y(b)

- y(a) = £ V

JLml-iyJL-

+

+

l

n+1)

[ (x)i (a+xk)dx, n! Jo g

E

+

h = b — a,

(16)

where is the error term of the quadrature. In this equation, truncating the term Rn, we have the numerical integration scheme y

m + 1

= y

m

=E A W P + t ,=i ;=i

(™ = o, l , . . . , M ) ,

(17)

(Qj = Wij-i and 0j = w - i ) , for the initial value problem (15), where x x + (m + i)h and 2 j

0

m+i

=

54 for i = 1 and 2. It follows from Eq. (16) that the order of local truncation error in this scheme is greater than or equal to n. In the scheme with n > 2, because some derivatives of higher order are used, we are usually a need of analytic computations to obtain the derivatives. Unfortunately it is, in general, not easy to do so, in particular, in the case where the differential equation is of a system, so that the scheme is not practical. But, as an application of the scheme, we can design a useful scheme for computing directly a numerical solution for the initial value problem of higher order differential equations. After some preliminaries, therefore we will give such a useful scheme in §5.2, 5.1. Numerical Integration Scheme of Quadrature of General Hermite Type In the numerical integration scheme (17), especially we are interested in the numerical integration scheme derived from the quadrature of general Hermite type with rii < "2- Then the scheme can be written as follows (18) where Cti

=

n, ni+k-j

Un + k-pjY. + qj) {k + q,)\ '

pj = min{fii,j) and q, = max{0,j — T I T } . A reason which we pay attention to this scheme is that it has a strong numerical stability. To see it, we will apply the scheme to the test equation y' =

(A€C, ReA > 0).

Then we can obtain the stability function S{z),

S(z) =

1!

P. [k + n.-j

+ qj)

(k

+

qj

)l

j

with z = Aft. For the stability function, we can easily shown that S{t) < 1 (t e (0,+co)) for any n and n with iii < n . In addition, we can also show that the scheme is A-stable if H i = nj. The latter property is proved in the following theorem. s

2

s

55 Theorem 6

Let p be a positive integer, then the scheme (19) 3=1

j= l

is A-stable, where 1 fp\

m = -±h-» ( k

p

Proof

'

p

' \p+k-j)

2

V

p

+

* - » k\

!

Let o^ = 1 and 01 = t. By using the relation of combination,

(A)=s-H w)rrO'

*

+

we can show that - $ ~ t - l M

(J = l , 2 , . . . , p ) .

From this relation, then we have

3=0

m = j=0

Evaluating this stability function at each point z = to/, (-co < u < oo), on the imaginary axis, we obtain S(iV) =

A+iB A — iS

where A=

£ ( - l y ^ w * . and S = £ (-#" j: even>o 3: odd>o

On the other hand, we obtain

5(1) =

j=0

w /

E(:)(2 -y)! P

j=o

< 1.

W , / a

0 VJ

56 at 2 = 1. Therefore by the principle of the maximum in the complex-valued function, it follows that S(z) < 1 for any zsC (Rez > 0). We have completed the proof of Theorem 6. It should be noted that the scheme (18) with m = 0 and n = n is equivalent to the Taylor expansion method. 2

5.2. Application to Higher Order Ordinary Differential Equation We construct a numerical integration scheme for solving an initial value problem of the r-th order ordinary differential equation given in Eq. (1). In Eq. (19), replacing y ^ with i £ f > and with yt~4 for J = 1,2, then we get +i)

m

= yir

r

)

+i j=i

+ i >=i

for p = 1 , 2 , . . . , r, and by setting t — p = s, we can moreover obtain

for s = 0 , 1 , . . . , r - 1. In addition, the right hand side of this equation can be represented, using vector notation, in the form L

+ (o.-.-.o,^-,...,^- ;;;_„/i'-' -;)f + ( o , . . . , o, ftiSp . . . , v—'ft:;.,, h'-'0;:;)Fm+l, a

Q

m

where (

1

wL ' \ ('+1)

, and F

m + I

=

Arranging j^mti i " a column and setting j , ^ = / scheme

m + i

,

f 3/L°> \

, (j = 0,1), we have the desired

l

/" fca[ . . . k - a' _ r

l

h'a

r

\

T

0

: 0

•• ...

ka\ 0

2

2

ha ha[ J

\

fm

}

Table 1: Absolute Errors in y ' and y ' with M = 5/ft M

Step-Size h 0.005 0.010 0.050 0.100 0.500

M

Absolute Error of y > This Method Euler Method 7.47 x 10^ 8.55 x lO"'* 2.99 x 10" 1.73 x 1 0 7.47 x 10" 9.56 x l O 2.58 x 10~ 2.17 x 10"' 8.66 x l f r 2.60 x 10+°

Absolute Error of jAV This Method Euler Method 4.52 x 10" 1.56 x 10" 1.81 x 10~ 3.13 x 10" 4.52 x 10" 1.62 x 10""' 3.36 x 10" 1.81 x 10~ 9.81 x 10" 1.57 x 10

M

6 s

-i

4

- 2

2

A'# \

2

3

1

3

+0

a)

u

Sm + 1

0

+

2

s

4

3

f Aft

6

0

0

(20)

J'-l) Sm+1

Aft

V /m+li }

1

1

for m = 0,1,..., M, where f = • • •, J/m+, ') for i = 0 and 1. Since this scheme is of an implicit form, we must solve Eq. (20) as a system of nonlinear equations with respect to variables of T / ^ . , . . . . j / m + i \ provided / in Eq. (1) is nonlinear with respect to j / ' , ; / ' , j ; ' ' y. However we can use this scheme as a corrector in a predictor-correct or method, and then the predictor can be easily obtained by replacing / i with / in the right hand side of Eq. (20) as a system of linear equations with an upper triangular coefficient matrix. m+i

+

0

m +

1

m

Numerical Example: In order to illustrate the efficiency of the numerical integration scheme described above, we consider the application of the scheme with r = 2 to solve the simple initial value problem y"(x) = y{x), 0
7m" 1)

+

0

¥2

IA - A * 0

+

,,(1) fm+1 m+1

where / = f [ x , a n d / + i = / ( W i , ! / ! i , £ l i ) - Then the numerical results is shown in Table 1, together with the results solved by the Euler method. m

1. 2. 3. 4. 5.

m

m

t

N. Dyn, J. App. Theory 31 (1981) 22-32. N. Dyn, Math. Comp. 43 (1984) 168. K. Jetter, SIAM J. Numer. Anal. 19 (1982) 1081-1089. A. Sharama, SIAM Rev. 14 (1972) 129-151. M . Mori, Numerical Analysis (in Japanese), (Kyoritsu-Shyuppan, 1972), p.280.

59 I M P R O V E D SOR-LDKE M E T H O D W I T H O R D E R I N G S FOR N O N - S Y M M E T R I C L I N E A R EQUATIONS D E R I V E D F R O M SINGULAR PERTURBATION PROBLEMS EMIKO ISHIWATA

Department of Mathematics, Waseda University Okkubo S-i-1 Sinjyuku-ku, Tokyo 169, JAPAN E-mail; [email protected] and YOSHIAKI MUROYA

Department of Mathematics, Waseda University Ohkubo S-4-1 Sinjyuku-ku, Tokyo 169, JAPAN ABSTRACT We consider the linear system Ax = b derivedfromsingular perturbation problems, l b solve such non-symmetric linear problems, we propose a generalised SOR method, which we have named the "improved SOR method with orderings", We use three ideas, that is, orderings,rariablerelaxation parameters and a not oecessarily strictly upper triangular splitting matrix U'vaA^D — L-U. The basic theorem, the selection of the relaxation parameters, orderings and several numerical experiments are also presented.

1. Introduction We will be concerned with the non-symmetric linear system of equations Ax = b. To solve such equations, we propose a generalized SOR method, which we have named the "improved SOR method with ordering^'. For an )i x n matrix A, we choose the proper permutation matrix P and put T

A = PAP ,

x=Px,

b = Pb.

Then the improved SOR method with orderings can be expressed as m+1

£( > = (D —

- $)£> + $[?}3:

(m)

+ 0 - *I)-'*b,

m = 0,1,2," • (1)

where for A = D — L — U, D is a diagonal matrix, U is an upper triangular matrix, L is a strictly lower triangular matrix and $ = diagfwi, • • • ,Q„) is a diagonal relaxation matrix. Let put ^"l^p^W,

r

* = F $P =

diag(>,,"-,ai„),

T

D = P DP,

T

L = P LP,

T

U = P UP.

Then Eq.(l) is expressed as follows. ^+V

= {D-*L)-\{I-*)D

+ W)aP*

l

+ {D-*L)- *b,

m = 0 , l , 2 , - - . (2)

60 For example, if P = I , then Eq.(l) is called the improved SOR method with natural orderings. In particular, if Q, = w, i = 1, • • • ,n, then we call Eq.(l) as the usual SOR method with natural ordering for U), If 0

1

1 '

0

i

P =

and Q{ = Ul, i = 1,• • • ,n,

then we call Eq.(l) as the SOR method with inverse ordering for UJ. Our method has the following three features compared with the usual SOR method. 1) We take orderings into account. This turns out to be very important for non-symmetric matrices. 2) We change the relaxation parameters G>i, i = 1, • • •, n of $ usefully. 3) U need not be a 'strictly' upper triangular matrix. Recently, H.Han et al. and H.C.Elman and M.P.Chernesky studied the effect of the partitioning and ordering of the unknowns on the convergence of the GaussSeidel iterations and gave a general procedure to automate the partitioning and ordering phase of the solution process, for not only one-dimensional problems but also two-dimensional problems of the discrete convection-diffusion equation. K.R.James expressed the iteration to vary all relaxation parameters as in Eq.(2) and derived a range of values of the relaxation parameters of the Gauss-Seidel and Jacobi type, together with the bounds of the spectral radii of the corresponding iteration matrices. P.H.Brazier , D.B.Russel , J.C.Strikwerda and L.W.Bhrlich respectively proposed special selections of the relaxation parameters for two-dimensional problems, but they have no analytic results. J.J.Buoni and R.S.Varga commented that the splitting matrices L , U need not be triangular matrices. 8

6

11

3

13

14

5

4

We use all three ideas at the same time to solve non-symmetric linear equations. As a result, we have obtained an effective SOR-like method which converges more rapidly and with fewer iterations than the usual SOR method. In this paper, we only prove the basic Theorem on the tridiagonal matrix case with constant coefficient and apply this theorem to several numerical examples of blocked systems, using the special relaxation parameter wt with proper orderings. Further results on the improved SOR method with orderings, that is, general convergence theorems, special selections of the relaxation parameter u),, i = 1, • - •, n and orderings in practical use and relationships between our method and the direct methods such as Gaussian Elimination, etc. will be published elsewhere (see E.Ishiwata and Y.Muroya ' ). 9 10

2. Difference Schemes Derived from Singular Perturbation Problems We first consider the singularly perturbed two-point boundary value problem

12,15

61

-«<"(*) " a(x)u'(x) + b(x)u(x) = f{x), u(0) = 7o,

=

x € (0,1)

7 l

where e is a parameter in (0,1] and the functions a, b,f lie in C?[0,1] and are independent of e. We first assume that there exist constants a, 8 such that 3

3

2

a{x) > o > 0, b(x) > /3,

a + 4e/3 > 0.

(4)

Under these hypotheses Eq.(3) has a unique solution u(x). This solution has, in genera], a boundary layer at x — 0 for e near 0. Now, as examples of non-symmetric difference equations, we show only two examples of difference equations derived from singular perturbation problems. Let n be a positive integer and h = l / ( n + 1) be the uniform mesh width. The nodes in (0,1] are Xi = ih, i = 0,1, • - • , n + 1. i) Upwind difference scheme — 2y + y h f

~

£

y ,~yi a . — +

< + t

i+

2

6 * = A, . = l , - , n

JAi+1 =

Vo — 7o>

{ 5 }

7i

where Oj = a(xi), = b(xi) and fi = f(xi). Then we obtain a difference equation -kVi-i + Vi- U i V , i = fc,, i = 1, • • •, n , where +

e

£ + a,h 2

2e + a ft + bih '

'

(

2

2£ + a,h +

b,h '

If bi = 0, then li + « i = 1 holds. ii) The El-Mistikawy and Werle difference scheme The piecewise constant approximation a(a:) of a(x) on [0,1] is defined by a o i,

x.£ [*(-l,as), x= 1

u

n +

where

hi = (a(z,_]) + a(xj))/2

for

i = 1,2,••-,«+ 1

i = 1,2, • • • , « + 1.

Piecewise constant approximations 6 and / of b(x) and f(x) respectively are defined analogously. The test functions [iik}t=] 8*8 defined by 1

-£^;' + av5' + oV* - 0, on (Xj,x ) k

j = 0,l,---,n

ju

(6) 5 , j = 0, l , - - . , n + 1. Under the assumption Eq.(4), the El-Mistikawy and Werle difference scheme is denoted by Aii — b where u = {^(xi),-• ,u (a; )) '. This matrix A = [flyr] is an irreducible tridiagonal matrix such that i>k(xj) =

11

ktj

f,

h

7

n

a

jJ-i
= - e ^ f o - i + ° ) < ° . %-irf = ^ l ( ^ - 0 ) < 0 -

b = (6i, - - . O n ) ' ,

+ °) + where 6,r

=

»(0t*» " 0) > 0,

(/,i/<j)

5

j = l,---,n,u+l j = V-1«

- fij,iai 7o - <5,>cm,n+i7i|0

62 This holds a > + If 6(x) = 0 hold, then = |0*j-il + Moreover, fljj-i — = " ( i j ) > a > 0. This difference scheme for solving Eq.(3) on a uniform mesh in (0,1] was proposed in T.M.El-Mistikawy and M.J.Werle . A.E.Berger et al. , and E.O'Riordan and M.Stynes ' gave independent proofs that the El-Mistikawy and Werle scheme is uniformly second-order accurate (that is, all nodal errors are bounded by Ch , where the constant C is independent of x, h, and e). If the function a(x) is allowed to have a zero in (0,1) and the zero of a(x) is assumed to be simple, and o(0)a(l) must not vanish, then R.B.Kellogg et al. proved that the modified El-Mistikawy and Werle scheme is uniformly first-order accurate in the case of having a turning point 27* such that fa < 0 or > 1 where 0% — b{xk)la'(x),). 3iJ

7

2

12 15

2

1

3. Basic Theorem for Simple Tridiagonal Matrices In this section, we show the error estimates for the SOR method with the special relaxation parameter w = Q . We use an n x n tridiagonal matrix expression as b

16

f h 02

0\

Cl

62

C2

A = [a ,b ,*\ = i

i

On-1

0„-l C„_i

In particular, let us consider a n n x n simple tridiagonal matrix A = [—1,1, — « ) , that is, cti = —I, bi = 1, Ci = — it, i = 1, • •• ,n. Then the eigenvalue u of the jacobi matrix of A is well known as }

(ij = 2Vl • ucos Assume 4lu < 1. We set u%~

n+V 2

(?)

j = 1,

Note that i+yr^ 1 < Wopi <•£%'= Jim u ^ < 2 if lu > 0 and 0 < £ 4 = lim < Wept < 1 if lu < 0 . Let us express x to be the unique vector solution and a r to be the m iteration vector in the SOR method for u = wj, and the p -th element of the error vector (m) (m) _ . > 0 be [e<*">] = e< >. Now we have the basic theorem for m = tit,. 1 + Vl - 4/u

and u t = ov

(m)

m

e

=

x

Theorem 1 ? =

0,1,2,---.

s

m

Assume lu / 0 and 4lu < 1. Then for w = Q , X = w - 1 and i / m = 9(71 + 1 ) , thenwehave ef = |A|«< >. e<°> and b

n+li

,1+1

b

63 if m = q(n + 1) + k, 1 < fe
mi

4 =

1

!

p = n - fe+ 1 JO)

<4°' | , n - Je + 2 < p < n

8

Proo/. Let the eigenvalues of the SOR matrix C be \ = (w — lje* ', j = 1,- -• ,n. Then , i can be expressed by %j and = cosfij + isindj, where i = —1. u

17

2

3

H

=

A,r+w - 1 ' .. = w(A )a

- 1)5 , i. (w - 1)5 ^ - ( 1 + e"-)e-i > = i ^-2cos ^ "j w 2

1

e

3

c o s

Since / i , - is already expressed by Eq.(7), we obtain cos^ = U\J-£ti ^i- Because of Cij = lit, we have u)^Jluj{us — 1) = 1 and fly = 2jw/{n+ 1), 1 < j < n. The eigenvector u , corresponding to A can be defined by }

pjTf ^ n + l " U

I — I A

,p5

e ' sinpfly

{isin2pfl +i(l-cos2pe? )| i

>

where tJj = Pj/2 = j j r / ( n + 1), j= l , - - , n . Then we can get the next relation n

2 £ < f «

ei (sin2A:e +tcos2*fl,-)0,

, where c[ —

Y"

i

(8)

The reason is as follows : £ 4 %

=

—*—• £ {sin2p^ + i ( l - cos2pfl,)} (sin 2/fcff,- + t cos 2*9^)

£

3=0

3=1

XA V'"

e[

*=1

0)

* y=o

We use the relations £ J sin 2pt?y - cos 2kS = 0 and £ " cos 2k&j = 0 . If p = A holds, then 5Z™ cos 2(p —fc)fl -= n + 1. Otherwise, if p / k and 1 < p, k < n, then E"= cos2(p - k)g~ = 0. Hence we obtain Eq.(8). = 0

=0

0

}

3

s

= 0

64 Therefore each error vector at m-th iteration is expressed as

F

"

„

j=l

\V

j=l

U

/

Substitute the above expression of cf^ to this formula, then the error vector is rewritten as -k v u j=0 where P

n

*(2m >-2fc)Jy

e

+J

.

a i n p 9 j

. _ {

c o s

(

+

1

2(m-fc))fl + £sin (p + 2(m - k))0j} • sinpfij

p +

>

= -r {sin2(p + m -fc)(7y- sin 2(m - k)S -} + 5 {-cos2(p + m }

+ cos2(m -

.

Since 8, = jv/(n + 1), then E"= sin 2 ( p + m - Jfc)# = £ J s i n 2 ( m - fc)0 = 0. The error vector is finally defined as 0

3

(0) 6

, « J 1

''

= 0

y

n

• J2 {cos2(m - k)9j - cos2(p + m -

+ i

,

from which each value of p, m determines the error vector explicitly and we can finally obtain the proof. • For practical purposes, we denote two error estimates more explicitly, one for \l\ > \u\ and the other for |f| < |uj (cf. H.C.Elman and M.P.Chernesky ). 8

Corollary 1

Assume \l\ > \u\ on Theorem I . Then for q — 0,1,2, • • •,

if m = q(n + 1), then we get simply e^

n+1))

=

• e<°>, and

if m = q{n + 1) + k, 1 < k < n, then

p = n —fc+ 1

\M «(«+!>

wfcere d, -

1

+

^

4

' " and |d,| -

~J\Xl/u\.

If\l + u\ = 1, then |d,| = 1. Otherwise, if\l + u\ < 1, men |A| < 1^| < 1, and if\l-rv\ > 1, then |A| < 1 < \di\.

(9)

Corollary 1 implies that if |/ + ti| < 1, then the convergence ratio per one iteration ° i ™ ^ J i ' ' ' ' l for 1 < fc < n is |A/di| which is greater than |A], but the number f

e

(n+1

+t>

65 of iterations (see section 5) is independent of n. On the other hand, if \l + u\ > 1, then |d] | > 1 holds and the number of iterations increases according to n, but if n is sufficiently large, then the number of iterations is not more than q(n + 1), where the integer q satisfies | A | * max |e< | < 6 for an admissible error bound 6. (n+11

0,

l
Corollary 2

"

Assume \l\ < \u\ on Theorem I . Then for q = 0,1,2, • • -, n+1

1

if m= q(n + 1), then we get simply ej?< » = jA-j*** * • e<°>, and if m = q(n+ 1) + k,

l
• 4 • U& H -

then

-

(f )"

• Jf ] ,

1< p
Iff ]

n+1



>• replaced di by

v

1 + Vl —4lu

Corollary 2 impfies that if |i + u\ < 1, then the convergence ratio per one iteration of max \e^ \ for 1 < k < n is |da| which is greater than Al, but in spite tn+1>+ki

l
*

of \t\ < \u\, the number of iterations is independent of n. On the other hand, if |( + u| > 1, then \di\ > 1 holds and the number of iterations should be theoretically almost q{n + 1) where the smallest integer q satisfies | A | ' " ' max |e{, '| < 6. But in practical computations, if n is large, then it may not be correct by the computational errors (see Example 2). Since (|A|/|di|)/|o2| — |A| < 1, the case of \l\ < |u| is not better than the case of |/j > |u|. Example 1 and 2 imply those results. That is, for p = 1, • • •, n — k, k — 1, • • • , n , the term d\ • efl or jA|" • 4 " • 4 ° ' Corollary 2 remains and e< > does not decrease until m becomes a multiple of (n + 1). On the other hand, we note in both cases of Corollary 1 and 2 that ,

P

k

e

m) p

o f

e

m )

+1

m

i n

m

p

m

= ]X\ ef\

m = (n + l), q

9 = 1,2,3,---

4. The Permutation Matrices and the Relaxation Parameters In this section, we mention how to determine the permutation matrices for good orderings (cf. H.Han et al. ) and the relaxation parameters. We first define the turning points of the matrix. 8

Definition 1 Let us consider an n x n tridiagonal system Ax = b, where A = [—li, 1, — i = 1, • - •, n. / / there is an integer k such that 3 < k < n — 2 and

66 - |«»-I|)
a n r f

(IM - l ) ( K I - 5) >

0

*

e

p°" 0

,

n t

8

In particular, if\l \,\u \ < 5, < |tt*-i| and | W l | > Nw-lli * * " *» a "stable" turning point and if \l \, |u*| > \, \l -\\ > \uk-i\ and ]l i\ < \u \, then vje coll x an "unstable" turning point. In this paper, we assume that I|,uj ^ 3, i = 1, • - - , n and if there are turning points x , k = l , - - , p where 1 < rj < r < • • • < r < n, then each x is only a stable turning point or an unstable turning point such that if p > 2, then <M-|)(l^-i)
k

k

k

k+

k+l

k

rt

2

p

•••]

Tk

oin))

corresponds to the permutation matrix P and a satisfies at least all conditions. 1) Suppose that x , 1 < k < n — 1 is not a turning point. Then if \l \ > \u \, then cr(k - 1) < cr(k) < e(k + 1) and if \l \ < |«*|, then a(k - 1) > a{k) > cr(k + 1). 2) Suppose that x is a turning point. Then if < ]u -i\ and > then a(k - l),£r(fc + 1) > a{k) and if > \u ^\ and \l \ < |u»+i|, then
k

k

k

k

k

k

k+l

t

(2 < 3

< r\) (l<j
lj = h = Hi

(r, + l < j < n )

h =h Uj = u\

(n<j
where

ii+wi = l ,

where

l + u^=l,

ii,fii>0

2

l ,Ui>0 2

Ci ~ |)(t/2 - I) > 0, then x , is a turning point. We call (r, - 1) x ( n - 1) submatrix Ai = [—li, 1, - U i ] and (n - ri) x (n - r,) submatrix A = |-i ,1, -uaj respectively the first block and the second block of A. As simple examples, the permutation matrix P is expressed as P, if 0 < Fi.fij < and P if (,,uj > | . | E

T

2

u

/ 0 ••• 0 1 0 ••• 0\ 1 0

0

0 01

0

r 1

0

1 0 0

0 1 ^0-010

0

2

67 Note that the turning points of matrices in Definition 1 are simiiar to the turning points of the singular perturbation problem, Eq.(3). Now let us consider a n n x n tridiagonal matrix A = (-i*, 1, — u,] with /, = 1 — Uj and tin = 1—1„ and assume 4i\iii < 1, t = 1, - • • , n . Assume this matrix A has several blocks as mentioned above. Then we select Q^i for the j - t h block [—h, 1, —u,] of A 2 1 as Qij j = —-—, Note that if L + u,- = 1, then wa ,• = ——.,, - ,. 1 + ^ 1 - 4/ max(|( |,K|) r

3llj

3

Finally, we propose the selection of u, for an n x n tridiagonal matrix A = [—It, 1, —Ui] (see Example 9). 2 i) If Xi is not a turning point, then Wi — - = = = = , If l - -iii = 1, then i r

_

1 +

1

i/l

ii) If there are turning points Xk, then we choose

+

u

k

AliiLi

as

if x is a stable turning point. k

l - ( i * + TiO' (lk

-

if Xk is an unstable turning point and Z*, u > 0. k

) - V

These selections are derived in consideration of a priori error estimates. 5. Numerical Experiments In this section, we show several numerical experiments satisfying previous conditions for the tridiagonal matrices with the turning points. For simplicity, in all examples we specify the vector solution x =• of Ax = b as x = 1, i — 1,' • • ,n and the starting vector x^ = [x' '] as x = 0, i = 1, - • - ,n. We iterate until we reach the first positive integer tn such that each component ej ' is less than the admissible error bound 6 — 10~ in magnitude. We call this fh as the number of iterations. We first apply the SOR method for w = u\ to a simple tridiagonal system Ax = b, where A — [—1,1, —u], I + u = 1. 0

t

t

ffl

8

Example 1. For A = [—i, 1, —v], I + u = 1, we apply u> = u\. Table 1. The number of iterations for ui = ui

b

71

10 50 100 200 500 1000

(=0.9 ( = 0.1 9 11 9 51 9 101 9 201 501 9 9 1001

(=0.75 i=0.25 17 22 17 51 17 101 17 201 17 501 17 1001

(=0.67 i=0.33 27 33 27 51 27 101 27 201 27 501 27 1001

;=o.e (=0.4 46 46 46 46 46 46

55 51 101 201 501 1001

68 If \l\ > \u\, then the number of iterations is small and independent of n. But if |/| < |«|, then that depends on n and equals to a multiple of ( n + 1). Example 2. For A = [ - / , 1 , -u], I + u ^ 1 and lu > 0, we apply w = w . 4

n 10 30 50 70 100 200 300

Table 2. The number of iterations for I + u ^ 1+u<1 i=0.8 0.1 0.6 0.1 0.6 0.3 1.0 0.1 1.0 u=0.1 0.8 0.1 0.6 0.3 0.6 0.1 9 11 9 11 9 11 17 22 31 9 31 9 31 20 31 10 9 42 20 51 10 51 9 51 71 9 71 9 42 20 71 13 9 101 9 42 20 76 14 101 9 141 9 42 20 76 20 204' 9 Ml 9 42 20 76 26 310*

1 and lu > 0 f+u >1 1.1 0.2 1.5 0.1 0.2 1.1 0.1 1.5 29 33 16 22 28 31 18 31 37 100* 24 57" 45 130" 30 109* 75 172* 49 147* 131 355* 86 276* 207 514" 126 411*

If |i + u\ < 1, then the numbers of iterations in both cases |/| > |u| and |/| < |u| are independent of n. On the other hand, if |f + u| > 1 and |/| < \u\, then the number of iterations must theoretically be a multiple of (n-r 1). But in these cases, if n becomes sufficiently large, then we may need more iterations than a multiple of (n +1) because of the computational errors. We represent such cases by the numbers of iterations with superscript * in Table 2. Hence in practical computations, we should transform the case of I + v ^ 1 into the case of i + u — 1, which can be done by using such d : dl + u/d = 1 as defined in Corollary 1 and 2. Example 3. For A = [—1,1, —u], I + u = 1, we change u>, 0 < u < 2. w 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Table 3. The ease of 0 < w < 2 and n = 100 i=0.1 (-0.9 m 1=0.25 i=0.75 m i=0.33 (=0.67 m 245 146 99 412 314 98 650 553 97 205 106 99 341 243 98 536 439 97 172 73 99 285 187 98 446 349 97 144 45 99 240 142 98 373 276 97 114 16 98 201 312 103 98 215 97 128 29 99 167 69 98 260 163 97 153 55 98 132 34 98 214 117 97 187 86 101 126 28 98 171 74 97 229 132 97 52 150 98 120 27 93 293 198 95 192 93 99 146 49 97 407 308 99 248 147 101 186 89 97 d

d

d

171,1 means the difference between the numbers of iterations in cases of |/| > and |(j < hu|. In Table 3, m is almost n — 1. H.C.Elman and M.P.Chernesky" considered the case of u> = 1 and obtained the similar results. d

69 Remark 1 For the n x rt tridiagonal matrix A = [—1,1, —It), let lu > 0 and wi, < Q < 2, then for any eigenvalue A of the fact \X\ = w —11 (8 fcrtowrc. Let m 6e trie num&er of iterations for w — Q. Let A = Qt — 1 and m 6e a constant suck that |A|* = 5. Traen anrfer l&e assumptions we can guess m& as m& = m + an, where a parameter a is determined by ^|A/A| = |A|" and does not depend on n.

Wojrt <

Q

For example in Table 3, if I = 0.9, w = 1.3 and u A = 0.3, a = 0.4124, rn = 15 and jrt, = 56.

= 1.111111111111111, them

b

Next we apply our method Eq.(l) to various tridiagonal matrices with the turning points. But before doing that, we shall explain some words used in all the following tables to describe the numbers of iterations for each condition. The parameter ujgp, in all examples is the calculated value for the nxn tridiagonal matrices A — [—li, 1, —ttj]. For UJ — uJopi, let m^i, m „ and m ^ , be the numbers of iterations for the SOR method with natural ordering, with inverse ordering and with good ordering for each block. Let m& be the number of iterations for the improved SOR method with good ordering and 4> = diagfwj,- - • ,u„) in each example. We actually compare the numerical experiments with the theoretical numbers which we guess by Remark 1 in the following tables. in

b

Example 4. We apply the nxn (j = 0.9, Uj = 0.1, !j = 0.9, u = 0.75, = 0.25, ty = Q.75, 3

tridiagonal matrix A = \—lj,

Wj = w = 1.111111111111111, wj= 1.538461538461539, ^ =1111,2 = 1.333333333333333, M

1, — Uj] such that

(1 < j < [ ^ ] - 1) (j = f ^ ] ) + 1 < j < n).

We denote that [x] is the maximal integer less than X. TaiAe 4. The case oF an unstable turning point at j = |(2n)/3] value of Uopt 1.320605864340204 1.329710139543094 1.330960730282924 1.332102986243628 1.332394148600067 1.332749378400198 1.332954574021790

60 120 150 210 240 300 360

"Lord

35 54 64 83 93 115 134

75 136 166 226 256 316 376

34 55 65 85 95 115 125

17 17 17 17 17 17 17

We apply good orderings and $ = diag(u>i, • • - , u ) denoted above to A. We guess by Remark 1 that the number of iterations equals to that for the second block Ai = [—0.25,1, —0.75] with good orderings because of Qhy. = maxuib,, =f Wopt. n

For iii = let m ,\ be the number of iterations for the first block A\ = [-0.9,1,-0.1] and m be that for the second block A = [-0.75,1,-0.25]. Then by Remark 1, we guess m = m& = rn and m^, = mt + n/3 where a = 0.5. Similarly, we guess m^i =• rftt + / 3 and = m + 2n/3 = mj, + n. We can see that the computational results satisfy these relations. b

b

2

t

b

n

t l

70 Example 5. We apply the n x n tridiagonal matrix A = 0.1, u, = 0.9, 0.1, uj = 0.25, 0.75, Uj = 0.25,

1, -Uj] such that

Wj = u = 1.111111111111111, ( 1 < J < [ ^ ] - 1 ) Wji = 1.538461538461539, u>j = = 1.333333333333333, (\^\ 4- 1 < j < n). M

Table 5, The cose of a stable turning point at j = [{2n)/3j n 60 120 150 210 240 300 360

value of Wopi 1.320605854262298 1.329710155009154 1.330960720871857 1.332102985834606 1.332394148552820 1.332749378434252 1.332954574049759

m„ 71 130 161 221 251 312 371 d

Ttinv

34 53 64 84 94 115 135

ma 18 18 18 18 18 18 18 t

34 55 64 83 95 115 132

Similarly, we guess the numbers of iterations by Remark 1 to be m ^ i — rhi + n/3, T/ionj = rfte + n and m™ = i h + rt/3. Practical computational results also satisfy these relations. Next we show the cases of having two turning points. b

Example 6. We apply the n x n tridiagonal matrix A = [—lj, 1,—Uj] such that l = 0.9, lj = 0.9, 1^ = 0.25, ij — 0.25, lj = 0.67, 3

u u u u u

= = = = =

0.1, 0.75, 0.75, 0.33, 0.33,

Table

n 60 120 180 240 300 360 420

= C7J = 1.111111111111111, (1 < 3 < [ ] " 1) uj = 1.538461538461539, Oy = i J , - 1.333333333333333, m\ + 1 < J < i f i - 1 ) u>j = 2.380952380952381, u>j ~ W6,3 = 1.492537313432836, (lf] + l<j
3

2

t

2

6. The case with an unstable and a stable turning point value of J.',,.,; 1.468250161411775 1.485146834836888 1.489059929933442 1.490556405149029 1.491295304620208 1.491724815483151 1.492005472669140

"lord "linn " W 48 75 40 69 135 61 97 144 83 122 185 105 148 226 125 173 268 147 201 309 170

ma* 26 27 27 27 27 27 27

We show another interesting example. The next table presents the results for a tridiagonal matrix which has a stable and an unstable turning point.

71

Example 7. We apply the nxn lj (, lj (j l }

= 0.012195, = 0.012195, = 0.67, = 0.67, = 0.1,

uj - 0.987805, uj = 0.33, uj = 0.33, U j = 0.9, uj = 0.9,

tridiagonal matrix A = [—lj, 1, — u \ such that }

Uj = u\i = 1.012345554031413, 1^ = 1.520207356283397, u = Stf, = 1.492537313432836, Uj = 1.754385964912281, uj = u , = 1.111111111111111,

(1 < j < [f | - 1) (j = Is!) (|§| + 1 < j < [ ^ ] - 1) (j = ( [ ^ | + 1 < j < n)

s

b 3

where Uj = Ugpt — 1.492537260055542, j = 1, • •• ,n for tno^,m „, and m in

u<J)1

.

Table 7. With turning points on j = [n/3], [(2n)/3] n 120 180 240 300 360 420 480 540 600 660 690 750 840 900

"lord "Km* 246 125 360 181 473 234 582 284 695 336 808 389 921 442 1035 496 1145 549 1255 601 845 616 917 668 1022 746 1095 799

"V... 167 241 314 386 456 529 602 676 746 816 617 669 747 800

27 27 27 27 27 27 27 27 27 27 27 27 27 27

We consider the numbers of iterations given by Remark 1. Now for w = Wopi = Qt,2, let fin,!, m ,2 and m ^ be the numbers of iterations for the first block, the second block and the third block of A. Then we guess these to be respectively mt,! = mi + 8n/9, m ,2 =? >n& and 771^3 = rftj, + n/3. Similarly we guess m d = m n + n / 3 = m i + l l n / 9 and m = t%, + n/3 < rh + 8n/9. For this tridiagonal matrix, the differences of the ui^j of two adjoining blocks are very large. In such a case, we usually get a greater number of iterations if n is not so large. Because we guess the number of iterations under the assumption that n is sufficiently large, we note that in Table 7 if n < 660, then m , ^ and are much more than our guess and monotone increasing according to n. But if n > 690, then we get the numbers of iterations which we guess by Remark I , The number of iterations m „ is monotone increasing according to n. But the number of iterations m ^ is much less than n and independent of n. 0

D

0

or

inv

h

in

Next, we consider the case of the tridiagonal matrix which has no turning points. We divide the matrix A into four blocks by |n/4], |n/2], and (3n/4] as indicated below. Since i , > u, > 0 holds, we can iterate with natural ordering which is good ordering.

72 Example 8. We apply the n x n tridiagonal matrix A = j i i j i s i

= = = = = = =

0.67, 0.67, 0.75, 0.75, 0.9, 0.9, 0.987805,

where

=

*** = *»i = «J = «* = Uj = i = *i = v

Wj) = Wf = Wj = Wj = Wj = i = J =

0.33, 0.25, 0.25, 0.1, 0.1, 0.012195, 0.012195,

w

W

1, —Uj] such that

1^,1 = 1.492537313432836, 1.492537313432836, (7J6, = 1.333333333333333, 1.333333333333333, iLt = 1.111111111111111, 1.111111111111111, w = 1.012345554031413 2

|3

M

:

(l<j
tm

(J =

([fi+i<j<[f]-i)

o=in) a?i+i<j<[^]-i)

(IfI

+ 1 < 3 < ")

+ w * ) / 2 , fc=l*i/4], |n/2], [3n/4]. +1

Table 8. The case with no turning points n 60 100 160 200 240

value of Wopt

raz

"lint,

1.450762203852421 1.474511840524642 1.484800375838160 1.487445518273925 1.488946709089202

132 212 330 412 491

68 129 196 243 290

71 132 199 246 293

b

25 24 24 24 24

We guess that the number of iterations is usually coincident with the number of iterations for the block with u = max LDJ,^, but in this case, rriQ is a little less than this. Finally, we consider the case of changing uj , i = 1, • • - ,n for each entry. What happens then? We consider the coefficient matrix which is derived from the upwind difference scheme and the lower elements lj, j = l , - - , n are monotone decreasing such that Ij + Uj = 1 and 0 < lj < j = 1, b

t

Example 9. We apply the n x n tridiagonal matrix A = [—lj, E + Ojft

2e + o /i' j

T>i=—-—r. it + mri

1

=—rr> n +1

h

£=

L

.,

2

ai-th,

1, —u,) such that 1

u =—, f

iij

i

= l , - -,n.

Table 9. The case of changing all uii n 80 120 160 200 240 280 320 360 400

value of uJopi 1.221880878114550 1.221880565772298 1.221880876205672 1.221880875020172 1.221880880390542 1.221880879444867 1.221880888127118 1.221880877534967 1.221880892204558

"lord

142 221 301 382 463 544 626 708 790

63 102 142 183 224 265 307 349 391

64 103 143 183 224 264 307 349 391

mi

mj

18 22 23 27 30 35 40 44 48

16 17 18 21 22 24 23 25 27

m

4Ji

6 5 5 5 5 5 5 5 5

73 Let m be the number of iterations to apply the above w j — 1, • - •, n. Moreover, mi is the number of iterations to apply each UJ — H\, corresponding to each i-th block which is split by the points [n/2), [3n/4], |9n/10] and m is the number of iterations to use each ui = H> corresponding to each block which is split by the points [n/2], [3n/4], [4n/5], [9n/10], [19n/30], where we choose u\j = & • min u< ulj

it

7

bii

and the non-diagonal entries of i-th block are l , Uj for r i < j < ri. It is clear that changing Wj, i = 1, • • •, n for each block is efficient from comparing with m i or m j . But the most efficient case is to change all w , i =!,••• ,n. Since the non-symmetry of this matrix is very strong, our method performs very efficiently for these types. }

t

All the above examples imply that the improved SOR method with orderings is more rapidly convergent than the usual SOR method. 6. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

A. E. Berger, H. Han, and R. B. Kellogg, Math. Comp. 42, (1984) 465-492. A. E. Berger, J. M. Solomon, and M. Cfment, Math. Comp. 37, (1981) 79-94. P. H. Brazier, Comput. Meth. Appl. Mech. Engrg. 3, (1974) 335-347. J. J. Buoni and R. S. Varga, in Numerical Mathematics, ed. R.Ansorge, K.Glashoff and B. Werner (Birkhauser, Basel, 1979) 65-75. L. W. Ehrlich, J. Comput Pkys. 44, (1981) 31-45. H. C. Ehnan and M. P. Chernesky, in Recent Advances in Iterative Methods, ed. G.Golub, A.Greenbaum and M.Luskin (1994) 45-57. T. M. El-Mistikawy, and M. J. Werle, AIAA J. 16, (1978) 749-751. H. Han, V. P. Il'in, W. Yuan and R. B. Kellogg, J. Comput Math. 10, (1992) 57-76. E. Ishiwata and Y. Muroya, Tech. Report 95-15, Advanced Research Center for Science and Engineering, Waseda University (1995). E. Ishiwata and Y. Muroya, Tech. Report 95-16, Advanced Research Center for Science and Engineering, Waseda University (1995), K. R. James, SIAM J. Numer. Anal. 10, (1973) 478-484. E. O'Riordan, and M. Stynes, Numer. Math. 50, (1986) 1-15. D. B. Russel, Ministry of Aviation, Aeronautical Research Council, Reports and Memoranda no.3331, (1963). J. C. Strikwerda, SIAM. J. Sci. Statist Comput. 1, (1980) 119-130. M. Stynes, and E. O'Riordan, Math. Comp. 46, (1986) 81-92. T. Torii, Information Processing in JapanG, (1965) 187-193. R. S. Varga, Mafrti Iterative Analysis, (Englewood Cliffs, New Jersey, 1962).

75 A N A L Y S I S OF T H E M I L N E D E V I C E F O R T H E F I N I T E C O R R E C T I O N M O D E OF T H E A D A M S PC M E T H O D S I Masatomo F U J J I Department of Mathematics, Fuktioka University of Education Miatakata, Pukuoka 811—it, Japan

E-mail: hijiini@lukuok&-edu ac.jp ABSTRACT The behavior of the difference between the values of the predictor and of the corrector for the Adams predictor-corrector method in the P{EC)"~ mode is analysed. This leads not only to an accurate estimation of local truncation errors but also to that of global truncation enors. 1. Introduction 1

In the previous paper , the author discussed an accurate method for estimating local truncation errors, and as its application, an accurate method for estimating global truncation errors. In that paper, he mentioned two theorems on the behavior of the difference between the values of the predictor and of the corrector for the Adams PC methods both in the P(EC) E mode and in the P(EC) mode. The proofs, however, were not given there. The purpose of this paper is to give the proof in the P(EC) mode. In Section 2, some preliminaries are given. In Section 3, the order of the error in the j^th correction (i = 0 , 1 , . . . , m) is investigated. In our discussion we need the asymptotic formula of the error in the (m — l)-st correction. In Section 4, the existence of the formula is shown. In Section 5, the behavior of the difference between the values of the predictor and of the corrector is analysed. m

m

m

2. Preliminaries We consider the initial value problem of the differential equation V

t

=

y { a ) ^ y

0

{ a < x < b ) ,

(1)

where we denote by y ( x ) the solution of this problem. The step points are given by x

n

= a + nh

(n

= 0,1,

...,N),

h = {b~

a)/N,

where N is the total number of the steps. Let p be the order of the Adams PC methods. Put v.—n+.p—X.

76 In what follows, we assume that f{x,y) in Eq.(l) is sufficiently smooth on the regions in question. We assume that the solution y(x) o/Eq,(l) exists. Let & fjt = 0 , l , . . . , p - l ) are p starting values and let Bp =

J/W) (pi = 0,1,..., J> - 1 ) ,

We also assume that yjs are chosen so that e„ = 0(tf)

(o->p+l;p. = 0 , l , . . . , p - l ) .

Let k)

9<») = / , ( * , » ( * ) ) ,

{k)

9l =9 M,

ff*=g{3v).

The formulae of the Adams predictor-corrector method of order p in the mode are given as follows:

P(EC)

m

;=' and

J=0

3=0 ]

where y$ is the i-th correction of yf , operator,

/£' = J{xk,y£)

, V is the backward difference

and ,

7 =^ (s-l)s---(s+j-2)ds/j!. J

For the solution

let p

Jpi(a:, •/(*); ft) = y{x) - y(x - ft) - ft | ] a / ( x - jft, y ( i - jfc)) H

3=1

and r ( z , j,(x); ft) = y(x) - y(x - ft) p2

ft£6 /(x w

3=0

- jft, y(x -

jh)).

77 For the formulae Eqs.(2) and (3), we define the local truncation errors at x by n

Tph, = T i{x , T

v

y(x );h) v

and

ft)

T 2n = T 2(xv,y(x ); respectively. For preparations of the succeeding discussion, we give three lemmas. Lemma 1 (M.Fujii ) For the Adams-Bashforth-MovXton pair of order p in the P(EC)*' mode, the identity P

P

a

1

3=0

holds, and for the exact solution y{x), the identity V - r

P

h =

- v i W V w

holds. The following lemma concerns the growth of solutions of the nonhomogeneous linear difference equation k Zn+* -

Zti+fc-i = h^Pj^+t-jZn+k-j

+ A

n

(n =

0 , l , . . . , N — k).

3=0 3

Lemma 2 (P.Henrici ) Let B', 0 and A be the constants such that El/M |A*|

<

B'>

\h»\
(» = 0 , l , . . . , i V ) ,

<

A (« = 0 , 1 , . . . , AT-A)

and let 0h < 1. TVien euen/ solution of Eq.(i) for which \z,\
\z \ < K'e" ' n

( = 0,l,... JV), n

1

where K'

= I - ( J V A + 2fcz),

t* = r - e * .

r * = 1/(1 -

ph).

The following lemma plays an important role in the proof of Theorem 3.

(4)

78 1

Lemma 3 {M.Fujii ) For any polynomial Pi(x) of degree i, the equality = 0

for 0 < i < p - 1

holds. 3. Order of errors i n the i - t h correction For the convenience of the later discussion, let us suppose that (ft = 0 , l , . . . , p - l ; i = 0 , l , . . and put 1

eg = $ - y(x )

(n = 0 , 1 , . . . , N; i = 0 , 1 , . . . , m ) .

n

Then we have the following theorem. Tn

Theorem 1 In the P(EC) mode, under the assumptions in Section 2, for a suitably chosen ft, there exists a positive constant K such that (n = d,\,...,N\

|egj
t = 0,l,...,ro).

(5)

Proof. Under the assumptions in Section 2, we may suppose that there exists the solution y(x) on the interval a < x < b and that for a small 6 > 0 the function f (x,y) is continuous on Vi = {{x,y)\ a<x
Let us put 0 < ft < 1,

I = \a,b],

A^I^T-^l,

ft where [ ] is Gaussian symbol. We may also suppose that C, (i = 0,1,2,3) be the constants such that \f (x,y)\
|e | < C , n

p + 1

pl

for

x e I, 0 < h < hn,

,+1

for

x e / , 0 < h < ho .

2

\T (x y{x) h)\
s

T+]

\T (x,y(x);h)\
[x,y)eV ,

for fte (O.ftr,] (u = 0 , 1 , . . . ,p - 1),

M

pl

for

0

3

Furthermore put

-4 = E M , i=i

B = El*wl. j=0

C = max(C ,C3), 2

79

J-IWft,

J

« = £ W

t*=o,i„4.

te = max (Idpil.loyl) (» = 1,2,... , p - 1 ) , fcp = |
ft- = ^(5(6 - )k" + 2pC,ft' , a

3=1

i=0

Let us choose ft, so that K' exp{(b — a)L'} < 6 for h satisfying 0 < ft < fti (fci < ho). Suppose that (XjJ^eVs

0 = 0 , 1 , . . . , N; 1 = 0 , 1 , . . . , l i t ) .

(6)

The validity of Eq.(6) will be shown later. Since 01

4 = e£f, +

1

ftE^.; '

- /(^->,y(x ))} w

r

p l B

(7)

3=1

and $

=

el^+ftVI/r

-IJ^

1 1

-/^,!/!^))}

(» = 1,2

m),

(8)

we obtain 01

|ei | < | e i

m|

m

1

1|

| - r f t C E | o | | e i 7 | +C7 h'' 0

P3

+1

2

(9)

3=1

and \e®\<s

i]

v

+ h3\et \

(i= 1,2,....m),

(10)

where s„ = n&i + ''Co E M k l V l + ^ f t

p+1

.

(ii)

J=I

From Eq.(10), we have 1

|eWf < rr^s,, + (flftJVl? ! (i = 1,2,... , m ) .

(12)

From Eqs.(9), (11) and (12), it follows that 1*1 < * < | e & l + ^ 0 E 3=1

+

(' = 0,1, - . . , m ) .

(13)

80 Put Then we obtain d
,+i

+ hC f^k d -

m

n

i

+ Ch> )

v j

(v=p,...,N).

(14)

In order to estimate the left-hand side of Eq.(14), let us consider the equation m-l

Zv

= 2 - i + ft|{<-„Cb*i + B E W K _ i v

i=0

+ £

CoffmfcjZ^j] +

(t) = R . . . , AT)

,=2

and let {z„} be a solution of this equation with the starting values Zj = |e | (j = 0 , 1 , . . .,p- 1). Then it follows that 3

U=

d,
0,l,...,N).

By Lemma 2, we see that 5, < <

hL

K-e" ' K'exp{(b-a)L'}

<S (n = 0 , 1 , . . . , JV).

Hence the validity of Eq.(6) is shown and Eq.(5) holds. 4. Asymptotic formula -1

For the asymptotic formula of ejj™ ', we have the following theorem. m

Theorem 2 In the P(EC)

mode, under the assumptions in Section 2, the relation

11

eJr = h"e(

Xn

+l

) + 0(h" )

(ft =0,1,...)

holds. Here e(x) is the magnified error function, which is the solution of the differential equation e' = g(x)e-Cy^(x),

e(x ) = 0,

(15)

o

uihere C is the error constant. Proof. First, we shall consider the case m > 2. For 0 < j < p — \, put y = Bp* " ', U =/j ', Wj = ur"~ and e, = ef' and also put 7

1

3

m _ 1

n

A

wf = £ f ( ,y(x ) y Xj

3

+ 9ef)de,

(IS)

81

1

5 W = n 7 (t,i + l ) J

3

Let us choose /12 (/12 < hi) so that 1

.

< 1

for

fi

g (O.A,].

(17)

-11

Now we make the difference equation on ej[" . Here we consider h which satisfies Eq.(17). From Eqs.(7) and (16), we have

and from Eq.(8) for i = m and Eq.(16), we have p-i J=0

Since

+ Tpi.z-p+i — 7Ja _ i, iZ

(19)

p+

p- I >v *

p t

\m—II \nt—11

j=0

+ 7J,i,3-p-)-i

L

\—»

fm—11 Im— 1

j=l —

T 2,z-p+i f

m

-hV7z(0,m)(4 l-ei?) 1

+ Wto7*(nt - l,m)(eW - e j - ' )

(20)

and e

M _ [m-H e

=

»J,( m

l)(ef - 4°1),

(21)

substituting Eq.(19) into Eq.(20), we have

j=0

+ r i , - p i -TpLz-p+i} P

I

+

+ hbpo^im - l , m ) ( H _ l — l ) e

e

{ 2 2

)

32 and substituting Eq.(22) into Eq.(21), we also have {1 - (ftSo)~7,(m - 1, m)SAm =

(hbto^SM

1

-

- 1)(1 -

e^ ) ^,{0, )){k±b wtfe£f m

pi

3=0

p - ftj^ flpjwi™J ei™7 + r i^-p+i - 7 p _ i } . 1|

11

p

2iI

p+

3=1

Put A= = (bpor-rAm - l,m)S«(m - 1) and 1

tT, - (fi )"- 5,(m - 1)(1 - /i6po7-(0. m ) ) . p0

Then we obtain

3=0

- hj2
+?P -p i - W P - H } V

+

(23)

3=1

Substituting Eq.(23) and the formula obtained by replacing z by z + 1 in Eq.(23) into Eq.(18), we have

j=0

- ft E 0pjwlT7-'' iTw} + TO/(1 - A - A . ) e

3=1

j'=l

j=0 p-1

+ A

E ^3 l+W L+W W

E

7

P2,!+p-2

3=0

X

+ W ®4{\

- frA:,)}{T ^v

p+1

- T , _p }. p2

I

+1

(24)

By Theorem 1, we may put e ! T - ft"e(^) + u , (n = 0,1,...).

(25)

Substituting Eq.(25) into Eq.(24) and using the relation 2

e ( i „ ) = e{x ) + he'(x ) + +1

n

0{h ),

n

we find the nonhomogeneous linear difference equation

3=0

p

E

lm-11

i

J=l ,

+

n

tr- u /(i-h' \ ) n

n

3=0

+ /t

p + 2

3=1

A _ n

Cn=j»,p+i ,,.,]¥-l),

p

1

and there exists a constant A' such that |A _„|
£» = p , p 4 - l , . . , , J ¥ - l ) .

n

Since 1 ^ 1 £<W**

U* = o,i

p-i),

then from Eq.(13), there exists a constant C such that |e^-«|
, + 1

and since e(^) = j / t ^ ' e'(io +

dd 0 = 0 , 1 , . . - , p ) ,

there exists a constant C such that M

= to - i f t ^ f f l ^ W

( J - o, i , . . . , ) , P

Put B" = BCb +

{2(5A2r-V(l -

WWHA

+ B)C . 0

Since the coefficient of /iMn+i in the right-hand side of Eq.(27) is given by l|

m

•\o*&7 (i - ' V - ' t W I i - A A i ) ) . n +

84 we may take

Put K'

= r*{(6 -

a)A' + 2(p +

.

L* =

T'T?*

Then, by Lemma 2, it follows that

< /fexpHft-aJL*}^ -

4 1

(R =

0,1,...,JV).

Second, let us consider the case m = 1. Let e(x) be the solution of Eq.(15). Put ]

ef =

tfeixi) + WP+V (t' = 0,l,...,A0.

(28)

Since e{x )=jk[

e'(x + 9jk}d6 JO there exists a constant A"o such that 3

(j = 0 , 1 , . . . , p ) ,

0

|%| < C + | j - y e'(«o + 8jh)d0 \
For brevity, let us put Since P

loi ep+k+i

=

101 . , n loi * E^J P+*+W p-U+i-.i +

V

/L

w

e

p—J P E V j.+*-j4 r*-> ~ £ p;Wp+*- ep _ j=0 j=l + Tpum - T, (A = 0,.. -, N - p - 1), +

/

l

1u

0

ft

a

3

J

rJ:

J

I J b + a

substituting Eq.(28) into Eq.(29) and using Eq.(26), we find r Wp+fc+i

=

^ % % i t i j=l

iip+fr +

j

P-i i

«

w

w - , '

P

w

<

+ ftE PJ i'+*-j™i>+*-j - * E - w * v + * - j % * * ^ J=0

+ A*

j=l

(* = 0 , l , . . . j V - p - l ) 1

(29)

and there exists a constant Kg such that \*k\
(k =

2

0,l,...,N-p-l).

By L^mma 2, there exists a constant such that \*\
(i =

0,h...,N)

and we obtain ]

ef = tVe^)

+

+

0(h* ').

Thus the proof is completed. 5. Behavior of y® - j , H At the beginning o f this section, we give the following proposition. Proposition 1 When z > p and m > 2, the equality eW-efl

=

{l-hVr.(0,">)} / { l - (hb^^im

- l,ra)S (m - 1)} ;

x {^V^f-'leL'"-

1 1

-Tpj,,.^}

(30)

/lo/ds.

Proo/. Prom Eqs.(2), (3), (16), (7), (8) and Lemma 1, it follows that ei

m|

01

p

- e' =

m ,|

ft7p- V i - ^1

W

,|

+ Tfi,*w

e

- W**,.

(31)

We denote both sides of the sign o f equality in Eq.(31) by D. Then we have the following equalitiy: e

m_ |o| e

/j_ft

=

0p07i

(o, )( M- loi) m

e

e

- 1. ™)(4

+ hbfnM

m|

- 4

m_11

)>

(32)

Substituting Bq.(31) into Eq.(32), we obtain e

IH_e|o]

=

{ l - f t 6

p 0 7 x

( 0 , m ) } D

+ Mtfh£« " l.m)( H - ef-'l). e

Substituting Eq.(21) into Eq.(33), we also obtain {1 -

- hm)S,(m =

-

{i-hb ^(Q,m)}D. p0

- e»)

(33)

Therefore we have

/ { I - ( A V r 7 * ( ™ - l,m)5,(m - 1)} This completes the proof. m

For j j f — j / [ ' , we have the following theorem, 1

Theorem 3 In the P(EC)" $

mode, under the assumptions in Section 2, the

-

=

- T„ + e ln

2

+1

+ 0(ft " )

ppv

holds,where p = p and P(

1

™

'

\v>t\j>-l)

+ l;

l2.

Proof. FVom Eqs.(2) and (3) by Lemma 1, we have e

1

- ^

-

-H-iVVJ -

=

- kfp-iVfixv,

y(x ))

- h^Vifl^

-

=

m

- r

p l n

l i

v

f(x ,y(x»))\ v

- ft -. V V " " ' ^ ! " - ' ! , 7p

First, we shall consider the case m = 1. For v > p, k > 0, it follows that fc-ip-i J

1=0 j=o

«+r-j P

— Tpi^+fc.p+i Hence, we see that

-rp^-p+j + o ^

1

) .

87 1

When we regard e„-i as ej, !,, the relation mentioned above holds for v = p. Since ^ - f f ^ + o f l ^

W=o,i,...),

for v > p, it follows that e

5+t

=

—

*1j8,v-p-ri — Tpi.u-^+i

+2

=

P,(fc) + 0(/i" ) + 0 ( f t

Ip+1

),

where Pi(k) is a polynomial in k of degree at most 1, Let P\(k) = «o + ajk. For v > 2p , k > 0, it follows that s

E

E *wl* + « " J'Jftffv + 0(ft )]|P,(« - j) + 0 ( ^ ) 1

*=0 J=0

r=o 3=0 =

+2

E k « o + « - kooAfc + « m ] +0(ft" )]

r=o

*

and E 3=1

+2

" w*^ - i * - ^ * - , = 9«"o + {* - ^)(c hg' 0

+ aig,) + 0 ( . V )

v

E Tpi,v+t-p+i + T i,v+t- +i r

+ *
P

P + 1

+0(O.

where I J ^ H = 7 ^ { i , v ( i „ ) ; n ) ( i = 1,2), Hence, u

e

«£U = ft(*> 4- 0 ( n

p + 3

) + 0(/i

i p + 1

lit'

)

By induction on j , if v > jp, we can show that 4 * = I J W + 0 ( f t ^ ) + Otft** ) + 1

1

s

expressed as follows:

(p > 2).

(p > j ) ,

88 where Pj{k) is a polynomial in k of degree at most j . For v > 0, we see that + ,

1

e™, = h'efx,) + 0 ( h ' ) = Fo(fc) + O ^ ) For v > p , by Lemma 3, it is seen that

=

D - l V ( J ) I * + 0(h)We(x )

=

0(A* ).

+1

+ 0(ft" ))

v

+I

Therefore, from Eq.(34), we have 0]

y[ - a!" =

- T

2

rln

+1

+ 0(h*+*) + 0(fc * )

(p > l ) .

For v > ip, it follows that

V ^ e ™ = E(-l) J (j)[ffv-jV ;4--l9

+ -

- l ) ' } * ? - " + W M I f l - i O j ) + OCA'*)]

0 ( 0 .

Therefore, we have p+

+l

2

1

vS" - J/!" = T , * . - 7 > , „ + 0 ( f t - ) + O f A ^ ) Second, we shall consider the case m > 2. For v > 0, 1

(p > i ) .

> 0, we see that + 1

- A"e(x„) + O f A ^ ) = P (k) + 0 ( A ' ) . 0

For ti > p,k > 0, by Proposition 1, we have * J"'-'\

Jm]

,

p-l

k

i

l

r=ij=o - (hj^r-'s^m

„,|m-l|

|m-l|

_v>r.

f=i - i)(«EU - «5L)

H

- 4 +AE E ^ « t £ i £ # - E (=lj=0

(=1

w

89 - (nVP

- 1

- D O - hV7«+fc(0, ™)}

/ { l - (nVJ'Vt-fcfm - l,m)S (Tn - 1)} v+t

6=1 j=0

*=1

- («*>"-W)"

_ 1

,,

(i -

ftv^+*)/{i

-

( / i v n ^ n

11

x {/t7 -iV '/ fcei,+t + Tpi.v+t-p+i - Tp2,„ - i} r

t+

+k

p+

2

-HOt^HOt/i '*™). FVom Theorem 2, for t> > p, we see that

+ 0(/i =

p + 2

) + 0(/t

P,(fc) ; - 0 ( h

p + 2

2p+1

)

)+0(A

2 p + l

)

(p>l).

By induction on j , for v > j(p — 1) + 1, it follows that e£*

f l

p+

+1

1

= fy*) + 0(fc > ) + © ( f t ^ )

(p > j).

In a similar way as in the case m = 1, for v > i(j> — 1) + 1, we have i $ - J/1" =

- r

+ o ( t f ^ ) + o(/t + 1

p l n

2p+1

) (p>»).

Thus the proof is completed. 0

m|

As is well known, the Milne device estimates a local truncation error by yj, ' — yj, multiplied by a suitably chosen constant. Theorem 3 shows that the estimation of Tptn ~ Tpin by yf® — j/j, "' is improved as the step proceeds. An application is shown in the paper . In general, the case m = 1 is often used. According to Theorem 3, however, the case m > 2 is superior to the former in accuracy. 1

1

6. Acknowledgements The author would like to express his gratitude to Professor Hisayoshi Shintani for his invaluable advice. 7. References 1. M. Fujii, An Extension of Milne's Device for the Adams Predictor-Corrector Methods, Japan J. Indust. Appt. Math. 8 (1991) 1-18. 2. P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, New York, (1962).

91 A N E W A L G O R I T H M FOR D I F F E R E N T I A L - A L G E B R A I C EQUATIONS BASED ON H I D M WATANABE Tsuguhiro National Institute for Fusion Science Ckikusaku, Nagoya, 464-01, Japan

E-mail: [email protected] Giovanni GNUDI National Institute for Fusion Science Chihisaku, Nagoya, 464-01, Japan

E-mail: [email protected] ABSTRACT A new algorithm is proposed to solve differential-algebraic equations. The algorithm is an extension of the algorithm of general purpose HIDM (higher order implicit difference method). A computer program named HDMTDV and based on the new algorithm is constructed and its high performance is proved numerically through several numerical computations, including index-2 problem of differential-algebraic equations and connected rigid pendulum equations. The new algorithm is also secular error free when applied to dissipationless dynamical systems. This nature is demonstrated numerically by computation of the Kepler motion. The new code can solve the initial value problem

where L and ip are vectors of length N. The values offirstor second derivatives of ip{x) are not always necessary in the equations. 1. I n t r o d u c t i o n Computer analysis is playing more and more important roles for the development of science and technology. High speed and large scale computers together with powerful algorithms are extending the field of activity of numerical computations. Many types of equations are waiting to be solved numerically in the course of research and development. There are many excellent algorithms to solve the initial value problems described by non-stiff ordinary differential equations. We can usually get good solutions for such problems by excellent ready-made computer programs. However, we encounter sometimes serious numerical difficulties if the problems are reduced to stiff ordinary differential equations, or to differential-algebraic equations. Differential-algebraic equations frequently arise in many physical problems, such as optimal control problems, dynamical systems with constrained conditions and so on. The present status of the research on differential-algebraic equations is described in references ' - . 1 2

3

92 4

In a previous paper we have constructed a new computer program named H I D M D V (HIDM with second derivative) to solve stiff ordinary differential equations or differential-algebraic equations, based on the algorithm H I D M (higher order implicit difference method) ' ' ' . The program H I D M D V can solve the equation 5 6 7 8

0 = L(
(1)

where L and

Sift

ft

sh 3

2ft

• 4 — ^ — I — $ — t — ^ — + p(0)

tp(h)

v(2ft)



Vf'(0)


Fig.l The difference scheme for H I D M D V . The values ip(Q) and
9 11

10

12

13

93 In this paper, we have extended the algorithm of H I D M D V in order to satisfy the time reversibility and to have a more easy-to-use nature. A computer program named H D M T D V (HIDM, time reversal with second derivative) is constructed based on this new algorithm. In section 2 we summarize the principle of the H D M T D V , Numerical examples of H D M T D V are shown in section 3. Section 4 is devoted to a short summary. 2.

Principle of H D M T D V 5

7

The principle of H I D M is shown in detail in references '^ . Here we summarize the principle of H D M T D V , which can solve differential-algebraic equations without the trouble accompanying non adaptive initial conditions. Furthermore, H D M T D V has a linearly symplectic nature, and guarantees absence of secular errors for recursive motions of dissipationless dynamical systems. There are 3 types of H D M T D V difference scheme, depending on the highest derivatives of each variable. These are discussed in the following subsections. 2.1. Difference Scheme for Variables with Second Derivative Here, we consider the difference scheme for variables which have second derivatives in Eq.(l). In this case, we use the difference scheme shown in Fig.2. S

0

Si

. ci) . T

S

S

.

ti) T

2

3

v(o)

S4

.

S5

S

S

- fa • t/h m

ff(-h) Fig.2 The difference scheme for variables having second derivatives in Eq.(l). The values "{sih) should be minimized (SQ = —1, S3 = 0, sg — 1). Expressions of the function and its derivatives at the points 1 = S(ft (t — 0, • • •, 6) are given by linear combinations of function values at grid points x = —h, 0, A as

94 follows k

E Q W w E W + l i Q&wm

+ E

• (2)

ip'(sh)

j=-l

jm-i

3=-l

The difference scheme for the second derivative
2

4

1 - 9s + 12s = 0 ,

(5)

that is 9l = -0.7838••• , 8 = -0.3682• • • , s = 0.3682«••-,.«$ = 0.7838• • • . 2

4

(6)

Then the values of (Pj(si), Qj($i), Rj{si)), {j = - 1 , 0 , 1 , i = 0,- •• ,6) are determined uniquely, and then truncation error of Eq.(2) becomes 0(h ). The difference scheme for tp', Eq.(3) has a total of 9 parameters (D,(s), Ej(s), Fj(s)). Then the truncation errors for Eq.(3) becomes 0(h}). This order is compatible with the one of Eq.(2). Then, the parameters (Dj(s), Ej(s), Fj(s)) are also determined uniquely. The difference scheme for tp, Eq.(4), has a total of 9 parameters (4,(s), Bj(s), Cj(s)). Then the truncation errors for Eq.(4) can be reduced to 0[h ). This order is one order higher than the one of Eq.(2) and Eq.(3). Then one parameters, for example Ci(s), becomes free if we are satisfied with the same order of accuracy of the discretization scheme for
s

1

<m

~

=-

Ci{8 )-C { ) 1

1

gt

=-

U

1 1 1

\ ^ s < ,

(7)

~ f*8 .

(8)

1

6

Two coefficients are still left undetermined for the parameters (-4j(s,), Bj(Si), Cfai)), (j = —1,0,1, i = 0, • - • ,6), if we request the compatibility for the truncation errors for representations Eqs.(2-3). We discuss about this points in some detail in the section 4. In the following, we determine the parameters in Eq.(4) in order to minimize the truncation error for (f(s). In this case the truncation error of Eq.(4) becomes C?(ft ), 9

95 which is one order higher than the one of ip' and (p" Coefficients for the discretization scheme of H D M T D V are reduced to the form ,

, ,

4278 + 5 9 8 A ± (7569 + 513A)s L

3^64

. . .

-

M

S

)

M

s

)

W

=

2469 - 299A =

.• . _ ( 2 9 3 7 + 361A) - (6372 + 164A)s ±

1

{

S

)

-

73728 _ C

,

(

^

_ E

,

(

p

i

(

)

)

S

)

s

i

(

=

_

-

Tro2

m

-

^

5

+

6

4

S

l

.

D

^8432

W

=

. _312

^

0

d

{

S

)

93A^

+

= J

o

!

s

J

i

^

)

^

,

m

261 + 53A (

1152—'

=

=

<">

^

=

= -

W

5

65AK(173

+

^

4 4 0 A T ( 9 4 - 735A),

+

^ 0 7 2 ~ '

„,..., £

'

-

( 1 0 )

'

283-21A C

'

351 - 5 2 9 A ± ( 2 0 8 8 - 600A)s

± ( 2 0 9

=

L

73728

^ ±

)

=

=

^

^

S

, ,

±

'

16H-17A±(444-4A)s

±

( 9 )

•

_ (405 - 59A)*

T

B

4608

«

,

3

)

(14)

35 + 55A _

^

_(^l_19A s ^ ]

117 + 3 7 A ± ( 9 9 + 51A)s

1

(

w

)

9 - 7A

where f V33 \ -V53

( for s = a, or s ) , ( for s = s or s ) . s

2

4

Time reversibility conditions given by Aj(»i) % (

S

i

= A-A'a-i), )

Bj{*i)

= - D - f a - i ) ,

P/{«3 = P-j(%- 3, i

Ej(Si)

C,( ) = e - j ^ j ,

=

(19)

S <

= B-j(st-i),

F (Si) s

=-Q-ifsg-i),

(j = - 1 , 0 , 1 ) ,

= -F-jise-i)

= *-,-(»«-<).

,

(20)

(21)

(i=l,2,4,5)

are completely satisfied. Next we consider the stability of the discretization scheme given by Eq.(9-17) by solving the characteristic frequency of the harmonic oscillator

ifi"(x) + uMx) where tu is some constant. discretization scheme by

= 0,

(22)

If we express the eigenfunction of Eq.(22) under the ip(nh) <x exp(i nhSl) ,

(23)

96 we get the dispersion relation cos(2hn) = 457228800 - 881118000fl + 239415750g - 20934585^ + 724410g - 9792g + 38g 457228800 + 33339600g + 1275750g + 33O750 + 540<> - 18<j + 2g (24) 2

4

2

3

4

5

5

6

6

2

where g = (kw) .

1.570792...

Fig. 3 The dispersion curve of the harmonic oscillator given by Eq.(22) under the discretization scheme of H D M T D V u is the frequency of the harmonic oscillator and ft is the eigenfrequency of the numerical solution, h is the step size of the numerical integration. When \u[h > 1.570792120-• -, ft becomes occasionally complex , and the periodic nature of the numerical solution begins to break down. When cos(2/iSl) is real and the condition |cos(2ftft)| < 1 ,

(25)

is satisfied, the numerical solution given by the above discretization scheme becomes periodic with the correct amplitude (= 1). The relation given by Eq.(24) is shown in Fig.3 when u)h is real. This figure shows that the largest step size h which guarantees the purely periodic solution of Eq.(22) is given by max

1.57079212078280060208152-8

| ( 1 - 2.67763---xlO ).

(26)

97 In other words, the largest step size which guarantees the periodic solution of Eq.(22) is 1/4 of the period of oscillation, and the relative error for the period is 2.67763- • • x I D - 6 The local error of the discretization scheme becomes cos(2 a -cos(2 f t

)

= - ^ | g 2

M

0

+

-

(27)

This analysis leads to the following conclusion. When we adopt the time step h as 1/20 of 1 period ( h = O.IJT/W), the local error for function value is of the order of 4.4 x 1 0 and the error for the period of the solution is of the order of 1.1 x 10~ Furthermore, this discretization scheme can be proved to be linearly symplectic. -14

13

2.2. Difference Scheme for Variables with First Derivatives In previous subsection, we have derived a discretization scheme for functions with second derivatives. Here, we consider the difference scheme for variables with no second order derivative, which appear in the equation like 0 = L(f(x),

#(4 0"(ar),

tp'(s), tf&f,

x) .

(28)

In this case, the value 1 • > ' J * W * K.

(TI

<$>

• fV • m Li * W

'

• f1 * t3 t/h

v'(-h)

m


Fig.4 The difference scheme for variables having first derivatives (without second derivative) in Eq.(l), which should be solved at 7 points (x = Sih , i = 0, 6), shown by • (unequally separated points). The value f(-h) is given as initial values for rank-1 ordinary differential equations. The remaining 7 function values at grid points shown by O obtained numerically. The values Si[i = 1,2,4,5) are uniquely determined by the difference scheme for the variables with second derivative as shown in Eq.(5). The additional embedded points x = ±h/V3 are determined from the condition that the truncation error for >p'(Sih) (i = 1,2,4,5) should be minimized. a r e

98 Expressions of the function and its derivatives at the points x = s h ( i = 0, - • *, 6) are given by linear combinations of function values at grid points x = ±ft, ±ft«, 0 as follows t

= I [ E ^(sMm-r

f.(sM-ku)

+ fAsMhu))

+

E (29)

i + h E o,isW{jh)

I


a

ft

i < * M J ) + c-UM-hu)

+ cAsMhu)

-

(30)

;=-l

The difference scheme for the first derivative iff, Eq.(29), has a total of 8 parameters (d,(s), e,(s), f±{s)). Then the truncation errors for Eq.(29) become 0{h ). To reduce this truncation error one more order, we choose the value of the embedded points ftu appropriately. It is slightly surprising that the value u = l / i / 3 guarantee the all truncation errors for <^(sjft) (t = 1,2,4,5) are reduced one order, and the truncations errors of above expressions become 0(ft ) which is just the same order for discretization scheme of functions with second derivative shown in previous subsections. Coefficients for this discretization are reduced to the form 7

8

2

M . ) -

9

3 +

8

5

a

* g

8

+

9 4 A ) s

.

6

" . W -

^ ,

(3D

. . , (25 + 9 A ) ( ^ - S) . , , (61 - 19A)s " Wi = 384 • , , . ±(1137+ 225A) + (1380 + 292A)s .. . ( 1 4 7 - 29A)s d±M = , d (s) = — , ±

l

W

=

i

b o { s )

( 3 3 )

.... (34)

0

. UM ,

T(39 + 55A)3\/3 + (216-168A)35 =~ ,

(35)

m

- 1 4 3 - 3 1 A ( 1 6 8 + 40A)s . . ( - 6 1 + 19A) 3072 > *<«>384 where the value A is given by Eq.(18). We study the stability of the discretization scheme given by Eq.(31 -36) using the equation
T

=

L

( 3 6 )

where A is a some constant. The difference scheme Eq.(31-36) gives the following solution V {

..., '

2

=

3

5

6

7560 + 7560Aft + 3465(Aft) + 945(Afe) + 165(Aft)< + I8(Aft) + (Aft) 7560 - 7560Aft + 3465(Aft) - 945(Aft) + 165(Aft) - 18(Aft)* + (Aft)« 2

3

4

1

(38)

99 This solution (the amplification factor of the difference scheme) has the following characteristics \f{2h)\ < 1 |(^(2/i)| = 1

when3tAft<0, when Xh is pure imaginary.

(39) (40)

The relation (39) shows that the difference scheme is A-stable and Eq.(40) guarantees purely periodic numerical solutions independent of the step size h when A is pure imaginary. The linear symplectic nature is also verified for this discretization scheme. The local error of the difference scheme is given by the difference between the discretized solution Eq.(38) and the analytic solution exp(2Aft) « -

e

^ 2 A /

1

)

=

-

J

I

^

....

+

(41)

These results show the excellent nature of the difference scheme of Eq.(31-36). 2.3. Difference Scheme for Variables with no Derivatives Here, we consider the difference scheme for the variables with no derivatives, which appear in the equation like 0 = lfax%

ib(x), ^(4

&x% f ( s ) , ?(?), x)

(42)

In this case the difference scheme is very simple, as shown in Fig.5, and and no truncation errors are included. -1

Si

y(sih) ifii-h)

s

0

@

1 — @

2


s

4

f{sih)

2


s

s

©

i m

®


t/h

Fig.5 The difference scheme for variables having no derivatives in Eq.(I), which should be solved at a total of 7 points (x = Sjft, i = 0,---,6), shown by • (unequally separated points). The values y>(sih) (i = 0,1, • - •, 6) shown by O obtained numerically. The values S{(i = 1,2,4,5) are uniquely determined by the difference scheme for the variables with second derivative as shown in Eq.(5). (s = - 1 , s = 0, s = 1) a r e

0

3

6

2-4- Remarks on the coding of the program In previous sections, we have considered the truncation error of the discretization method. In actual numerical computations the roundoff errors deteriorate the accuracy of derivatives. To reduce these effects, increments of variables are treated as

100

practical unknown quantities in the actual program coding. For example, the quantities Uh) = v(-h)

+

V'(jh)

+

l ) V ( - f c ) + ip{jh),

=
(43) («)

(J =0,1) . If it is possible to determine all the highest derivatives using the given system of equation (1), we can get solutions directly by the above mentioned algorithm. There are, however, problems in which we cannot determine the highest derivatives only by the given system of equations. An example is given by G(,tM)=0,

(45)

/r(vMfr,j)=0.

(46)

In this case, both variables J/>(I) have second derivatives, so the discretization scheme given by Eq.(9 -17) is applied. This discretization scheme assumes that y>(-h), if/(—h), il>{—h) and ^(—h) are given as initial conditions. In this example, however, we have only two true initial conditions, for example ip{-h) and f'{-h). The other quantities ip(—h) and ift{—h) are not initial conditions, and should be determined consistently from Eq.(46). In this case, two additional equations are necessary to determine the values i>{—h) and i)'(-k). An example of a set of additional equations is ^ff(,M,x)=0,

(47)

£pH( ib,x)=0.

(48)

Vl

Since the program should be informed of these facts, a index for each variables is prepared in the program. Example of the index is rank-2 array variable named JVR as shown in the following. JVR{ 1, n) = JVR(2

}

highest derivatives of n-th variable.

n) — number of additional equations for n-th variable.

When equation is one of standard form as iF--/fy,r,*),

(49)

and high speed computations is requested, separate program should be prepared which does not treat the second variables as unknown functions because this value is given

101

in Eq.(49) . In this case the computation speed can exceed the speed of standard Runge-Kutta method program. 3. Numerical Examples of H D M T D V In this section we show several numerical examples of H D M T D V Computation was carried out on a Fujitsu M-1800 with double precision (1 word is 64 bits). The 'exact value' is calculated by long double accuracy (1 word is 128 bits). First we show the accuracy of the discretization scheme of H D M T D V . Giving the 'exact values' of ip{nh),
Fig.6 An example of numerical error of the discretization scheme given by Eqs.(9-17). First and second derivatives and ip") have almost the same order of accuracy and the accuracy for ip is one order higher compared to them. Next, we have calculated the numerical error for the variables with first order derivative (without second order derivative). The discretization scheme is given by Eqs.(31-36). In this case, we give the 'exact values' of
102

Fig.7 An example of numerical error of the discretization scheme given by Eqs.(31-36). p and iff have almost the same order of accuracy.

3.1. Kepler Motion As a example of dissipationless dynamical systems, we have integrated the equation for the Kepler motion 2

dx

_

dt* ~ 2

d y ~dtJ

=

x _

A

V +

?/)V2 '

y ~^{x + y )^ (I = 7T /16 , 2

2

2

2

'

(

5

0

)

( 5 1 )

103 where the constant u and the initial conditions axe chosen such that the analytic solution has period T = 64 and a relatively large value for the eccentricity. In this system energy and angular momentum are conserved and it is possible to check the accuracy of the numerical computations. For this system the index of the variables is shown in Table 1. No additional equations are necessary. x(t) y(t) variables 1 2 n JVBfl.nl 2 2 JVR{2, n) 0 0 Table 1 Index JVR of each variables to solve the Kepler motion given by Eqs.(50-51) by H D M T D V . For the numerical computations by H D M T D V , we have used step size h = 0.25(= T/256) and total time step number 10 (0 < t < 5 X 10 ). Plots of the orbit [x(t), y(t)) and of the error for energy and angular momentum are shown in Fig.8 and Fig.9. 6

u,-nVl6, J

5

s

h = 0.25, 0 < ; t < ; 5 x l 0 , (Tanai^c-64)

1 i I r i i J i f i i J

i | i i r i i i j i i | i i i i T' i T i r | r i i i i T j

Fig.8 Plot of the orbit (x(t),y(t)) of the Kepler motion Eq.(50-51). Because the period for the numerical solution is not the same of the analytical one ( = 64), the phase of the motion gradually shifts from the analytical position. The discrete plots of (x(t), y(t)} appears like a continuous line. But the motion is guaranteed to come back to the initial state. No secular error are present in the numerical results. The center of the force is marked by ' + ' .

104

Fig.9 Plot of error for energy and angular momentum of the numerical solution of Eq.(50-51) (Kepler Motion). Because the period of the numerical solution is very close to the analytical value ( = 64), the recursion time of the numerical solution is very long. This figure shows the secular error free computation characteristics of H D M T D V .

3.2. Numerical Solution of a Connected Rigid Pendulum In this section we solve the motion of a connected rigid body pendulum as an example of differential-algebraic equations. The equations are

mi

- 9j = -21-an +%• ( « - i i } .

m>

m

=

7t^

~

T l

'

y i

+

T

i

* '

^ ( ^ - s ) = - T i - ( x t

m

a

^*r fi.

=

m

X

~ ' l

) ,

T

- '"<»-»»>•

(52)

( 5 3 )

(54)

5 5

( ) (56)

+ Vl = ti , 1

^ ( a i - ^ + dfi-Ift) = 4 .

(57)

105 where £, and l represent the length of mass-less rigid rods. T and T correspond to to the tensions of the rigid rods, 1%, m , g are constants. The unknown variables are the position of the tip of each rod ( i , , jft, x , y ) and the tension of each rod, (T and T ). The former group of variables has second derivatives, but last group of variables has no derivatives. In this case, the the index JVR is given in Table 2. 2

t

2

2

2

3

2

2

variables n JVR(l,n) JVR(2 n) l

Xi 1 2 0

Pi 2 2 2

%2 3 2 0

V2 r i 4 5 2 0 2 0

T 6 0 0 2

Table 2 Index JVR of each variables to solve the differential-algebraic equations given by Eqs.(52- 57) by H D M T D V . The system of Eqs.(52-57) has energy conservation law given by — niigxi =

constant,

which is used to check the accuracy of the numerical results. Numerical example are shown in Figs.10 and 11.

Fig.10 Numerical solution of Eq.(52-57) by H D M T D V with step size h = 0.001. i i and £ 2 are plotted as a function of time, mi = 65, mi = 35, t\ = 10, i = 5, g = 9.8. Initial angle of rod-l= 175 (deg), rod-2= 187 (deg). Initial velocities of rods are assumed to be 0. 2

(58)

106

Fig.ll Numerical solution of Eq.(52-57) by H D M T D V with step size ft = 0.001. yi and the error for the energy are plotted as a function of time. Parameters and initial conditions are the same of those of Fig.10. This system is dissipationless, but numerical results show that numerical error suddenly increase at special points. This will break the time reversal nature of the motion. The reason of this phenomena will be discussed in the next section. 3.3. Differential-Algebraic Equation of Index 2 As an example of differential-algebraic equation of index 2, we have integrated the following equations by H D M T D V , 0=

Lifolf'.M)

0=

t (v,() 2

2

2

=^- cos(t)z -8exp(-t)y ,

(59)

= i-(i- m{t)

(60)

a

Qii

+ 3exp(-tj)-y,

where a and 8 are constants. The analytical solution of this system is given by

rtt)

t)B

^ l-a^W

( 6 1 )

In this case the variable y{t) contains the first order derivative, but it is determined by the algebraic Eq.(60). Then, an additional equation is necessary to determine the value j/(()- So, the index of the variables y{t) and z(t) becomes as shown in Table 3.

107

variables n JVR(l,n) JVR{2,n)

y(t) 1 1 1

z(t) 2 0 0

Table 3 Index JV/fofeachvariablestosolvethedifferential-algebraic equations of index-2 given by Eqs.(59-60) by H D M T D V . We adopted as additional equation

=

o,

n

: positive integer.

(62)

Numerical results of H D M T D V for this differential-algebraic equations are shown in Fig. 12.

a = 0.9, p = 0.2, h=0.01 IIIII[II[|IITIIIII1|M[II[1II|IHIII11I|H

Time

a = 0.9, (J-0.2, h = 0.01 U II11II1111111111II1111111IIIJII11III I f n

Time

Fig. 12 Plot of numerical solution and its relative error for differentialalgebraic equations of index-2, Eq.(59-60). Since the variables z(t) is solved by the algebraic equation, Eq.(60), the error is only due to roundoff error, i.e., order of 10~ 15

4. Summary and Discussion We have developed a new integration method with high accuracy and high applicability. A new program named H D M T D V can solve dissipationless dynamical

108

systems without secular error. Stiff ordinary equations or differential-algebraic equations can also be solved by the same program. These properties are demonstrated by several numerical examples.

t

t

Fig.13 Numerical solution of Eq.(52-57) by H D M T D V under automatic change of the value of JVR guided by the Table 4. yi and the error for the energy are plotted as a function of time. Parameters and initial conditions are the same of those of Fig.10. In subsection 3.2, we have observed a sudden increase of numerical errors. Let us consider the reason of this phenomena. When we treat the constraint given by Eq.(56), we use two additional equations, SiSH'teKS = 0 ,

(63)

x x'l + x[x[ + fctf + y[y{ = 0 .

(64)

L

These equations are expected to work to determine the values y'(t) and y"{t). But, when the rod is nearly vertical (xi n £\ and y\ ^ 0), the left hand side of Eqs.(63¬ 64) becomes very close to zero independently of the values of j/'ff) and y"(t), and it becomes difficult to determine the accurate value of y'(t) and y"(t). This will be the reason of the sudden increase of numerical errors shown in Fig.l 1. A quick treatment for this problem is provided by the replacement of the value of the index JVR according to the relation > |xi| or \yi\ < \xi\, i.e., in the former case, we treat yi{t) as a rank-2 variable, on the other hand, in the latter case, we treat x,(t) as a rank-2 variable as shown in the Table 4.

109 !/i < 1*1

yi

n 1 2

yi > l^i 1 JVR(l,n) JVR(2,n) 2 0 2 2

;VR(l,n) 2 2

variables x V2

n 3 4

\Ui-yi\ JVR(l,n) 2 2

lift - yi < x - I i | J V R ( l , n ) JVR(2,n) 2 2 2 0

variables

2

>

x -xi JVn*(2,n) 0 2 2

JVr?(2,n) 2 0

2

Table 4 An improved index J V i i to solve the differential-algebraic equations given by Eqs.(52- 57) by H D M T D V . The index JVR for the values Ti and T are same of those of Table 2. 2

The physical meaning of this process is the following. We treat j/i(/) as a rank-2 variable when |j/i(t)| > | Z ] ( f ) | . In this case, y\{—h) is treated as a initial condition and ft) is determined by the equation of motions. The x\(-h) and x"(—h) are determined by the additional equations Eqs.(63-64). When |j/i(t)| > |xi(t)|, %i(t) is treated as a rank-2 variables. We show a numerical example in Fig.13, using this choice of the value of the index JVR.

Fig.14 Numerical solution of Eq.(52-57) by H D M T D V introducing the polar angles $ and 0 defined in Eqs.(65-66). j/i and the error for the energy are plotted as a function of time. Parameters and initial conditions are the same of those of Fig.10. t

2

no A more fundamental treatment for this problems is the introduction of polar angles, 8\ and 82 as unknown variables instead of Xi, j / i , x and 3/2, 2

H = 3% — x

l

=

4 C O S 0 ,

,

jft

g COsS , 2

V2-yi

2

=f infl l S

,

1

(65)

=f2Sinfl .

(66)

2

In this case, Eqs.(56-57) are automatically satisfied. Highly accurate numerical solution is obtained as shown in Fig.14. In subsection 2.1, we found that two of the coefficients (Aj(si), B,{s ), C,(s,)), (j = —1,0,1, i = 0 , - - ,6) are undetermined (for example the values of Ci(s and Ci(s ), if we are satisfied with the compatibility for the truncation errors for representations Eqs.(2-3). In this case, we can change the dispersion relation Eq.(24) so as to satisfy t

4

5

cos(2fcf2) < 1,

for ft|w| £ y

,

(67)

by appropriate choice of the values of Ci(s ] and Ci(s ). In this case, the largest step size h which guarantees the purely periodic solution of Eq.(22) is given by 3/4 of the period of oscillation. It will be easy to extend H D M T D V to solve boundary value and eigenvalue problems. This will be published elsewhere. The next big task is the construction of a general purpose computer program to solve time evolution of multi-dimension boundary value problems described by partial differential equations. When the space dimension is 1-D, we have already constructed such a general purpose computer program based on H I D M . The present work represents also an important contribution to accomplish this task. 4

5

max

1 4 , 1 5

5. Acknowledgements G. Gnudi acknowledges the Japan Society for the Promotion of Science for the financial support. 6. References

1. 2. 3. 4. 5.

K. E. Brenan, Annals of Numerical Mathematics, 1 (1994) 247. S. L. Campbell, Annals of Numerical Mathematics, 1 (1994) 265. R. Marz, Annals of Numerical Mathematics, 1 (1994) 279. T. Watanabe, Annals of Numerical Mathematics, 1 (1994) 293. K. Abe, A. Ishida, T. Watanabe, Y. Kanada and K. Nishikawa, Kakuyugo Kenkyu (In Japanese), 57 (1987) 85. 6. T. Watanabe and M. Takagi, Trans. JPN Soc. Ind. and Appl. Mat. (In Japanese), 1 (1991) 135.

111 7. T. Watanabe, K. Abe, A. Ishida, Y. Kanada and K. Nishikawa, Kakuyugo Kenkyu (In Japanese), 58 (1987) 265-278. 8. T. Watanabe, RIMS koukyuuroku (Research Institute for Mathematical Science, Kyoto University) (In Japanese), 841 (1993) 43. 9. J. M . Sanz-Serna, BIT, 28 (1988) 877. 10. S. Saito, H. Sugiura and T, Mitsui, BIT, 345 (1992) 345. 11. H. Yoshida, Cel. Meek, and Dyn. Astr., 56 (1993) 27. 12. G. Gnudi and T. Watanabe, J. Phys. Soc. JPN, 62 (1993) 3492. 13. T. Watanabe and G. Gnudi, ISM Cooperative Research Report (In Japanese), 55 (1994) 211. 14. T. Watanabe, Trans. JPN Soc. Ind. and Appl. Mat. (In Japanese), 2 (1992) 93. 15. T. Watanabe, GAKUTO International Series, Mathematical Science and Applications, 1 (1993) 189.

113 S e m i - e x p l i c i t M e t h o d s for D i f f e r e n t i a l - A l g e b r a i c Systems of Index 1 and Index 2

Hisayoshi SHINTANI Department of Mathematics, Facility of School Education Hiroshima University, Higashi-Hiroshima

739,Japan

Abstract A-stable semi-explicit methods are constructed for differential-algebraic systems of index 1 and for those of index 2 and their convergence is shown. 1. Introduction Consider the differential-algebraic system of index 1 y' = f(y,z),

S(v,i) = 0

(1)

J - m &

9(ff) = 0,

(2)

and that of index 2 with the initial condition yfro) = yo> z(x ) - zo, 0

where y and /, z and g are vectors of the same dimension respectively, / and g are sufficiently smooth and for (1) gi{y,z) has a bounded inverse in the convex closed domain D and so does g (y)f,ly,z) for (2). The initial value (yo>zo) is saied to be consistent if g{yo,zo) = 0 for (1) and if g(yo) — 0 and 9Ayo)flyo, zo) = 0 for (2). Let y

x = x + 'ft (0 < h, t

0

0 < t,

th < C).

We are concerned with the case where the approximations (y„, z„) to (y(x„), z(x )) (TI = 1,2,...) are obtained by semi-explicit methods. Rosenbrock-type methods are convenient because they are non-iterative,but they are liable to be less accurate because they do not solve the equation g — 0 directly, so that at the n-th step of integration we cannot always expect (y -i> n - i ) to be a consistent initial value. Rosenbrock methods for (1) with inconsistent initial values are described in the literature . n

z

n

1

The object of this paper is to construct A-stable semi-explicit methods for (1) and (2) that correct the errors of the initial values and to show their convergence. We also construct interpolation formulas for approximating y(xi) and z(x ) (t ^ 0,1,...) and obtain the formulas useful for stepsize control. t

114

2 . Methods for Systems of Index 1 Let do — — (ff7'ff)(!/o>zo)> and assume that ||doll is so small that there exists a unique zrj such that g(ya, ZQ) — 0 and put ZQ - ZQ+CIQ. Thenifj = zo-t-0(|[do|| ). Let (yo(x)«2o(x)) be the solution of (1) satisfying i/o(xo) = yo,zo(xo) = ZQ- Put 2

-1

G = (- )-\

K = Gg , L - / , G , T = f + KU

F = f-rfzdn,

where all function values are evaluated at (yo.^o)- Then we have

Sl

y

2

yr>o) = F + O(||do|| ),

4 = (I-

s

2D

aftT") (a > 0),

( x ) = JrF + 0(||doH), yo(xo) = TF + O(||do||), 0

We construct A-stable semi-explicit methods for approximating (j/o{xi),£o(xi)) of the form m n yi =yo + E £ p i j o ' ( ) i=ij=i m n m z\ = ao + m i + E E P o ' o + E ' ' ( ) i=lj=l i=2 f c

3

r

m

4

where hi = {A- ly-'Aklfi

+ Lgt), Ut = Kkj,

m, = Ggu f, = f(v.i,Vi),

(i = l,2,...,m;j - I,2,..,n),

gi = S(tfc,%J, iii = yo, «i = a , 0

n i - l TI i—1 ^ = W + E E ' ' ^ * j ' ' ' = - M ~ m i + E E M > * E ' > j (* = 2,3,...,m), i—l

c

v

i

;=1*=1

+

JJ=1fc=1 +1

e

m

j=2

2

2

yi - yo(xi) = 0(ftP + A ||doll + ft Noll ), si - s (xi) = 0 ( f t 0

, + l

+ ft lldol + lldoll ) (P > 9 > 0).

We also construct interpolation formulas of the form m n yi =!W + E E P O ' ' . J i + hvt{f(yuz\) + Lgim,zi)} i=ij=i C

*,=* + m i + E E « ^ i=l i=l and the formula 771 e

(5)

2

+

h

K

?' f(yi

(6)

(0 < ( < l ) ,

p

, n

.*l)+E * * uc2

TI fc

= E E Py y

such that JI, - vo(xi) and z, - zo{xt) are of the order (5) and (6) respectively and e — 0(ft ). The quantity e is used for stepsize control. p

115 Let z = k(y) be the branch of the implicit function defined by g{y, z) = 0 such that io = k(yn) and put F(y) = f(y,k(y)). Then (1) is equivalent to the system so that the A-stability of the method (3)-(4) can be verified by the test equation y = >~y We have 1

Theorem 1 For m = 1 A-stable methods, interpolation formulas and the formulas with (p, q) — (2,1) exist. For m — 2 those with (p, q) — (3,1) exist. For m — 3 those with (p, 9) = (4,2) exist. For m = 4 fftose wtffi (p, q) — (4,3) exist but methods with p = 5 do not exist. Examples of the formulas Case m = 1 " = g, 0=7.

P11

4 Case m = 2 1 3

= li

— 1, Pin =

P11

P12 = 1,

3

P12

put = fc Pia = 3

— 1;

2< - 1 , P12 2

11

1.

—

11

7

16

-

—, Pll - ^f, P12 - - g , P l 3 - —g", P21 16 16 , , 16, , 16, 16 , TI = y Pn. - t--t , p, , = - t + 2 t — f , p i = t - 4 t + y ( , P21, 0 = 7 . Can = - , C212 - - , C13 = 2

3

2

3

a

2

3

3 1

16 32 ra - y i ,pn = - , Pis = 2

E

32

~

32 Pi3 = ^ . P21

w

32 - gj-.

Case m = 3 1 0

=

3 =

7>

5'

3 0 2 1 2

=

^

~ 25'

33 ~ 125'

1 C 3 U

=

0 3 , 2

=

l f

C 3 1 3 =

C 3 H

~3'

=

50 8 8 10 4 125 1 C32i = 0, e 2 = j , P11 = 0, P12 - - g . Pis = — j p P14 = g, P21 = P3i - g> 3

3

2

r = B, r = 1, put = ^ t f O f - 49f + 27), p 2

3

3

2

Put - |(12( +5t -36t+9), pi

it

3

3

3

l a

2

3

m ^(24( -65t +54t+9), n = ^ t ( 4 - 3 t ) , P2

4

2

P3i, = | i ( 4 - 3 t ) , p, - t - i , ra - | j t ( 5 t - l ) , r 20 P i 3 - -

Case m — 4 1 1 a = 2'

4 : 2 1 1

=

T

,

2

2

= ^(24t - 41t + 18t - 9),

5

2

31

« i/. (5t-3), p

125 _

P l 4 = g , P2i = - ^ .

11

2

C

3 ' 212 = - g . C213 = — , C

3U

n

== y , P12 = | ,

5

P3i-g.

8

26

- - , C312 - - —, C i3 3

- — ,

116

5

C321 -

1030 - "jjy, C

CHI = 1,

27 9 621 c«i = ~ , CJ2 = 5, <42 = 131 PM -

4

4

p

189 13 *43 = -fc. Pi] - ^

P U (

121

3

- 2,

151 463 - -go". P13 -

P12

9 119 9 13 Tq, P22 = - T J - P3t = 55, P4! = 720-

P21 =

r = I£ 15

1331 560 - - 3 ^ - , C421 = gg-,

413

239 69 = -20' ' " 15-

r a

3

2

= -4r(378t - 1647t + 2255S - 1080t + 120), 120 3

= l->-76Uf yu

2

+ H685i - 6920t + 90( - 90), 4

3

2

Pi3< = TT^I-27621t +52440f -23345( - 1080t + 540), 540 p

4

l4[

3

2

= _ L { i o i 2 5 ( - 19140( +8105i + S72t-324},

2

2

3

2

PS1I = ^* (36t - 75f + 40), P22, = Y^( (16119t - 30810t + 14215), P3i- 3

+

1

5

(

"

5

|

p l

'

3

(

2

J

r

( 5 ( _

3 )

P l 1

=

m

=

5 4

" • T^* '" *

p, - jt (27t - 501 + 23), rj, = -^-( (65t 4 40 13 , ... 25 . 13 . * - 3o i 2 ' y = 16 . 3 P22 = - y , We have shown the following 2

p

4t3(-9(2

P

l

3

2 +

1

0

S

(

"

5 0 )

-

2

+ 93), r , = Jrt (85t - 39), 20 43 . 65 . 15 is = M' ™ = - y . 13 #M-jr; 3

m

-

2

,+1

Theorem 2 Suppose that do - 0(ft ) (s > 0) ,0 < h < ho , nA < C. Then for sufficiently small ho V„ - yoM = O(ft'), in - z„{x ) - O(ft'), z - zo(x„) = 0{A") (n = 1,2,...), n

n

( = min(p,7 + 2,s + 3), u = min(p,7 + l,s + 2).

3. Methods for Systems of Index 2 3.1. Construction of the Methods -1

Let G[y,z) = - ( f r f o ) / ^ , * ) ) , ff(!/,z} = /«(»,a)G(tf,«), let M (j = 0,1,2,3) be the constants such that f

\\G(y,z)\\ < Mo, \\H{y,z)\\ < M,, \\g {y, z)/2\\ < M , \\g iy)f Ay,z)f2\\ vs

2

s

1

< M inD 3

117 and put D = M\M , £>i — M M , 0

2

0

m

CQ = H(y , z )g(y ),

3

a

0

So = Vo + co, y

a

= yo-

Then we have 1

Proposition 1 Suppose that r = A>IMI < L. Then the sequence { j / " } defined by the iteration . „C*) + W H{g

i

o, i,...)

(fc =

(7)

converges to yo,u)kich satisfies the equation g(y) = 0 and the estimate IISo-wll < Do Hcoll / ( l - T-2) 2

(8)

holds. +1

h

Proof. Put c = y<* ' - s/W (fc - 0,1,...). Since g(y< )) + j „ ( # > ) c i - 0, we have k

<7(i/

fc+1)

) =jf (1-*)frf

W

+ Oc )(Ck, c )d&, k

k

(9)

which yields the estimate 2

\\c i\\ < Do\\ck\\ , k+

so that 2

1

llcfcll < r ' - Hcoll (* = 0,1,...).

(10)

For any positive integer p we have U*-H» _3,W|| < £

h

k

+

. _

< 2 ' - l ,| ||

A

C0

r

/ ( 1

_ ,2^

which impUes that tip"*} ia a Cauchy sequence . Taking k — 1 and letting p —> o o in (11), we obtain (8). From (9) and (10) it follows that |j(»

( t + I )

ffe-OO),

2

)|<«2llCfc|| -*0

which shows g(yo) — 0 by the continuity of g. This completes the proof. Let eo = G{yo,zn)gyly )f(yn,zo),z

m

= zo-

0

Then by the same reasoning we have Proposition 2 Suppose that s — D\ ||eo|| < 1. Then the sequence )z'"'} defined by the iteration z

[k + l)

=

z

{k)

+

z

W)g (y )f'y^ y

a

Z

W)

(fc = 0, 1, ...)

converges to zo,uihich satisfies the equation gy(yo)f(yo,z} — O and the estimate Wzo-zWjKD! holds.

2

2

||e || /(l- ) 0

s

118 Corollary 1 Let d = G{yo,z )g (yn)\f(y ,z ) Q

0

y

0

+ fy(.yo, Z O ) C Q } >

0

zo + dn.

ZO =

Then e ^d 0

+ 0{\\dof + \\cf),

n

(12)

2

Z

||io-5o|| = 0(||(iol| + l|col| )-

(13)

Proof. From eo = {G(iiD,^H0(IMI)}{ftr(»^ 3

- * + 0 ( 1 * 1 11*11+INI ) the estimate (12) follows. The inequality \\ZO-ZQ\\

<

W

jzQ-

z \\

+

W

\\z

yields (13) and the proof is completed. We assume that ||
v

L

G = -(S ,A)" , R = Gg , K = Rf , L — /,G, A — (I — a f i T ) ' (a > 0 ) , 1

-

S

s

y

T = Pfv, P = ' -Q< Q = -Lg», F — f + f c y

0

+/ d , E= ;

0

9v!

,(F,F),

where all function values are evaluated at (yo,zo). Then 2

2

Bfifjee) = F + OIHcoll 4- ||do|| ), z (z ) = KF + G £ + 0( | | | | + ||do||), 0

0

Co

y '(x ) = TF + LE + O(||co|| + ||d ||),... . 0

0

D

We construct semi-explicit methods for approximating (yo(xi), form m

ZQ(X,))

n

yi = vo + co + ^ ^ ( p i j * i . f + rymiji, <=U=i m

of the

n

(14)

m I

+r

+

s

81 =zo + do + E Z ^ ( P y ' J ' j " y ' 5 I ( ' P ' 1=1j=i

1=2

+

t

''9')'

15

( >

where J

1

fc = (A-7) '- ,4/iF, li^iEferjjj 0 = 1.2,...,n), u

l

k - {A - iy- Ah{fi tj

- F), kj = Kkij

rm = {A- iy-'ALgi, }

(i = 2,3,...,m;j = 1,2, ...,rt),

rty = Km,, Pi = R(fi - F),

119

5i = Ggt/h, fi - f(ui,vi), i-l n

gi — g{wi),

Ui = yo + c + E ^2(djkkjk

+ dij*mjfc)|

a

y=i*=i Vi = z

i-l n i-l c + d + J2 E( y**jfc + <*y*«i*) + £ ( e i j p j j=ifc-J j=2 i-l n Wi = jo + co + + u* j*)' j=ifc=i

+

0

0

d

1

m

2

2

2

ft = S1 - So£*i) = Of**" + A M l + 0 ||doll + Hcoll + h ||d || ),

(16)

0

+1

r , - z i - z i ( x ) = O(A« +fc||^||+ft||d || <

1

0

2

2

|| || + l|d || ) ( p > > 0 ) . (17)

+

Ca

o

7

We also construct interpolation formulas of the form in n yt = yo + co +

m ,

i

EEj 'i ^'+5] i=lj=l

2, = ZO +

rrtmil

+ rirh,

i=2

m n TIT + 7 7til do +EEPy'''J 5Z' "" + iiPi 17=1 j=l i=2 s

+'"3i)

and the formula m n

m

i=l j=l

i=2

where m n m m - ALg(w),w = yo + c + E H + ,=1 j=l i=2 The equation (2) is equivalent to the system j / = F(y), z = &(!/), so that the A-stability of the method is verified by the test equation j/ = \y . We have Theorem 3 For m — 1 X-itaWe methods, interpolation formulas and the formulas with {p,q) = (1,0) exist. For m — 2 (Aose (p,g) = (2,1) exist and for m = 3 (ftose wiift (p, 5) = (3,1) exist. For m = 4 (ftose with (p, g) — (3,2) eiis( methods with p = 4 do not eiis( , Examples of the formulas 0

Case m = 1

^ o=

2>

Pn

=

1

' P"i

=

(

' Pn

=

1

-

Case m — 1 1 O=

1 Cn 2

- , Cai2 -

1 -yjj,

,

.

1

C2)i - 1, C212 = - , pn = 1, P12 -

5 —j

120 33 = 3, r = 1, s = 9, t = 2, pn, = i , p i = K - y £ + » ? , P211 = -6f+18( -9( , 2

p

2l

2

2

2

2

2

r , = i , s , = 9t , t , = 2(, pn = 0, pu = -5, 2

3

2 (

2

P21

2

= 6 , f = ^. 2

Case m = 3 1

3

a = j , O i l = 1,

9

2

1

2

«32 •

C11

3

= - - , GJI, =

9

0321 = y y

dMl = 1.

C

, C3i

21

- — , c

<WS -

, ,

pn = 1, p i = - - ,

3

7

P13 = - , P21

2

— ,

.

C313 = y y

3

(f =

69 =

321

1 .

&2U = C212 = C n = 1, C312 = - g ,

2

=

99

= -jjg.

2

3 ,

Yg. ff3 =

=

7

= 0,

1 =

3'

P22 = - - ,

r

P31 - ^ y 21 = 1, T - = - 1 , r = s = 1, S3 m —, 1 — —, t = 3, •> 16 , 16 , 8 o 19 3 1

2 2

2

3

2

2

2

2 2 [

= -t , r

=

C2U

P 2

3

2

3

P21 = 0,

f31 = 1. 1 "9' 207

0 2 1 3

=

3

C 3 n

=

?' 33

C312 =

39 "250' 3

cm = - £ g , * M = — , < * » = — ,C322 = j g . A u - 1.4M = J. . 64 , 128 , 64 . 686 81 dill = 2 2 5 . ^ - Tl25 ST'^ 243' 25' ' 50 128 1664 686 9 45 9 —

4 2 1=

31

=

632

=

e 4 2

=

9

643 = y J32 = y-jy ?42 = — = JjJ.ejl, = ^Cm = 32,C3ii = , 99 . „, , . 41 . 7 . 200 ; 64 C312 = j y ^ i =o,c4ii - i , c i 2 = - y - 4 2 i = - j . « m = y f . ^ i = T_-, , 64 64 343 8 1 257 *2l - ^.^422 - 2 ^ 3 , ^ 3 , = - — , P U = 1.P12 = - g , P , 3 = 3^,P.4 = g j , 41 59 125 1 64 64 ?

c

4

P2, - 0,P22 = "^,P23 = 77T.P31 = ^ P" = g, r , = g j . f ^ = - - . 349 50 2048 1372 3 r « = - ^ , r « = 1,82 = 9 = y , S = 1,(2 = . t 3 = , t 4 = " y , 2

J

> S J

3

4

W

W

z

3

2

PlU = *,Pia = ^ ( 2 2 5 f - 6 4 i + 36t-18),p22i = ^ ' ( 1 5 t + 64t - 72i + 18), P3U =

2

- i),pm = a

- 4),r , = g t V t t , = - ^ t , 2I

2

2

2

2

r i i . = f | | ' ( 8 * - 9), r « , = - | t ( 7 t - 9), , = 9t , * = ^ ( , s , = ( , S 2

4

T

put - —( ,

22

2

C 2 1 2

16 J

„ - 0, p , - --t\

- ( , r, = 8 ( t - ( ) , p,i = 0, p , = - y P13 = y

P31 = Casern = 4 1 1 1 4' " 3' = "5' 337 37 a

2

pu, = y t - - t + ( , 3

3 1 1

2

3

Pill = t. PlM - 2t -t—-t , f i. = t , r

2

121

t 2

'

=

128 243

t ( 1 3 ( +

3 )

t 3

' '

686 243

=

3

i(5f

P 4 1

~ 2 . -21'

3 >

r 3 1

'

( 4 !

=

,

8

t ( 7 (

" "2 784 . 729' ' 4l

=

~ -

6 )

'^

2 =

~

P

u

=

21'

L

3.2. Convergence Let (yj, zj) (j = 1,2,...) be the approximations to (j/ofijl.zofxj)) obtained by one of our methods and put y = y + c , z = Zj + dj (j = 0,1,2,...), j

j

j

3

where Cj = H (yj),

dj = Gjg {yj)(f(yj,Zj)

jg

+ f (yj,

y

H = H{y ,z ), i

i

z,)cj},

v

Gj = G{yj,Zj).

j

Then we have Theorem 4 Suppose r+1

1+l

c = O{k ),

d = O(h )

0

{r,a > 0), 0 < f t < f c ,

0

0

nh
Then for sufficiently small ho fe-»0(*i)

= O(A'), % - * o ( x ) = 0{ft')

(J = l,2,...),

(18)

ft "

= O(A'), z - z (x )

= 0(A") (* = 2,3,...),

(19)

i

k

0

k

m

ft - W N - O(fe'), n - a f n j = 0 ( / i ) ,

(20)

tuftere

( - min(p,g + 2, r + 3,2r + 2, s + 3), u = min(j), g + 1,r + 3,2r + 2, s + 3), I = min(p + l,f- + 3,2r + 2,s + 3), m = rnin(g + l , r +2,s + 2). Proof. Let £y be the solution of g — 0 obtained by the iteration (7) with the starting value (j(j,«f) and let Zj = fc(jy). Let (tfj-(z),Zj(x)) be the solution of (2) such that yj(xj) — yj, Zj(ij) — ij and put z

Sj+1 = Vj+i - l o f o + i ) . Tj+i = i+\ ~ Zj(xj+i). Then for some constants oo, &o, * (j = 1,2,3) and (16) and (17) 1

| | S j i | | < aoh^ +

| | T j l | < 6oA +1

(fc = 1,2) we have by

2

2

+ UihHWcjW + \\dj\\) + 02 IICJH + ftaj ||d,|| , ,+1

+b h(\\c \\ + \\dj\\) + b2(\\cjf 4- \\djf). 1

j

(21) (22)

122 Let Q} = -Myi>zj)Gi9(yj), Aj« =

Pj = i-Q,,

Ri^GmfaffidViiZi)'

life - yo(xj)\\, A«j = \\z, - ao(xj)|| 0' = 0.I. ••).

and let La be the constant such that ||F(u) — F(v)\\ < La ||u — «||. Since 90J

~ » ( * ) = Wte) - » ( * / ) + / {

i?

^

i - (fo(())}d( ( I > * ; ) ,

we have b j d ) - j/ofaOII < Ay, + £ f

\\y it) - y (t)\\ dt,

0

3

so that by Gronwall's lemma \\y (xj+i) - So(sj+i)|| <

0

e^Ayj.

3

Prom Sfetzj+l)) -

" S i) j+

=

-

2

ft,(tt+i)S i

+ 0 { | | S | | ) - 0,

i+

3+l

it follows that c i =

fff+ij^+l)

i +

a

= -Qi+iS i

+0(|%iS ),

j+

(23)

and so a

fc'+l - ft&fcl] - Fj+iSj + i + 0(||Sj+i|| ), because -fe+l=^

a

+0(||c i|| ).

+ J

i+

Hence for some constants fa and fcj < e"" A

Vj

CL

Setting d = kie °

2

+ ft, HSf+,11 + k

\\S \\ .

2

j+1

(i = 1,2) , we have

Ay. < C , £ | % ] + C £ HSjll J=I j=i

2

2

( = 1,2,...), n

(24)

because Ay = 0 , and there exist constants C, (i = 3,4,5) such that 0

2

\\y^-yo(x )\\
+ C \\S \\

n

3

( r i - 1,2,...),

n

\\yn ~ Vo(x )ll < Ay„ + C ||S || + C ||S„|| B

4

n

h

Ilia •-•••»B.(*i)ll = l i f t ! .

2

(n - 2,3,...),

(25) (26) (27)

123 Let k be tbe constant such that ||fc(«) - k(v)\\
0

Az„ = \\k(y„) - k(yo(x ))\\ < k Ay n

0

n

(n = 0,l,..),

so that for some constants kj (j = 3,4,5) 2

2

P „ - io(*»)ll
h i - tt(xt)l < koAy + fc IIrfjt|| + k (\\c \\ k

4

5

+ \\d f)

k

(28)

lit = 2,3,...),

k

l k i - ^ ( n ) l l = llr,||.

(29) (30)

Since 2

Vj(Xj+i) = Vj+i ~ Fj+iSj+i +0(||S - i|| ), zjixj+i) = J

+

-Tj+i,

we have Ss(w(^+i))/(y,(^+i),z,(xj i)) +

= Ss(W+i)(/ + V j + O - S j t / j P j + i S j + i + ATj+O - ^ ( P j + i S j + i , / ) +o(||s || + ||r || } 2

2

J+1

i + 1

= 0, which yields (31) where function values without arguments are evaluated at j/j+i or at ( i / j i , z i ) . Prom (21), (22) and the assumption it follows that for some constants A\ and Bi liftII
J+

m

By (21), (22), (23) and (31) there exist constants a such that

t

,

2

\\S 4< h^ + n {\\S \\ j+

ao

| | T i | | < 6oft i+

ai

,+1

+ \\T \\)+

j

j

(j - 1,2,3), h and b

2

Wf-H&pyf,

a2

2

+ M d l f t l l + 113)11) + W l l S i f + P J | )

(j = lj2,...).

We shall show that for sufficiently small ho inequalities +

llftll < Ajh" \

\\Tj\\ < Bjh

w+i

are vahd with bounded coefficients for j = 2,3,... , where v = min(p,g + 2,r + 3, s + 3), w — min(g, T + 2, s + 2).

(32)

124

Put 1

A = a^-"

+1

+ aiiAih? -'

2

21

+ B,h? -')

+ a A\h -"-

1

-

4- ^ ^ M * * ,

2

Then (32) holds for j = 2. Suppose that (32) holds for j = 2,3,.... k and put A

k+1

= h' -» ao

+

0

+2

v

(A hl + B ^ ~ )

a i

k

+ aiAlh°

k

+l

2

2

+2

+ a B h r -\

0

3

+1

Bfc+i = 6ohT" + 6i(iltfcS -" + B^fto) + f ^ f t o

4 1

(33)

k

1

" " + BgftJ?* ).

(34)

Then (32) is valid for j = ft + 1. Setting j l t = 0 + i f c , B* = c+feoB (fc = 3,4,...), t

2

2w+2

v

where a = ao^"" + MNf**~" + a b h - , 6 6 /to . we have

b = bohf", c = b + Wifto +

3

2

+l

2

=

«4

3

AB^ai*. + «^Ao^*H-

a T ^ * ^ ^ == 4>C^^ ^fc^

where Oj (t — 4,5,6,7) , bj (j — 3,4,5) , A and B are polynomials in fto with nonnegative coefficients. From (33) and (34) it follows that A > 0 and B > 0 (ft = 3,4,...) , because -4 > 0 and B > 0. Put 3

3

k

fc

3

(A ,B f k

3

= U , (
k

=T{U)

k

and choose ha small so that |T(10|| < r < \

for 11(7 - t/ || < 2 ||(/ - t/ ||.

0

3

4

3

If for j = 4,5,...,fc \\Uj-V \\<2\\Ui-U \\, 3

(35)

3

then since \\u -u \\ k+l


k

Q

k

k

we have \\u -u \\ k+l

3

< <

Eiall^+i-^n ( l + r + ... + i £ - ) | | E J - 7 i | | < 2 | l ( / - f / | | , s

0

,

4

l

4

3

and (35) holds for j — k 4-1. Hence j i \ and B^ are bounded and there exist constants A and B such that Aj < A, Bj
(j= 1,2,...).

125 By (24) we have AiM < <

v+l

2

2

a

+2

CiA{h! + (n-l)h } + C A {h ' + (n - l ) A * } Ci>l(A + Cft'')+C2vl (/t ' + C A " ) < D / i ( n = l , 2 , . . . ) . 2

i

2

2

2

+1

!

Estimates (18), (19) and (20) follow from (25), (28), (26), (29), (27) and (30) respectively

4. R e f e r e n c e s

1. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, (Springer-Verlag, Berlin, 1991), 453 - 454. 2. H. Shintani, Semiexplicit Methods for Differential-Algebraic Systems of Index 1, Bull. Fac. Sch. Educ. Hiroshima Univ., Part II 1 7 (1995), 23 32.

127 COMPUTATIONAL C H A L L E N G E S IN T H E SOLUTION OF NONLINEAR OSCILLATORY MULTIBODY DYNAMICS SYSTEMS" JENG Y E N Army High Performance Computing Research Center, University of Minnesota Minneapolis, MN 55415, USA

E-mail: [email protected]

and LINDA PETZOLD Department of Computer Science, University of Minnesota Minneapolis, MN 55455, USA

E-mail: [email protected] ABSTRACT One of the outstanding problems in the numerical simulation of mechanical systems is the development of efficient methods for dealing with highly oscillatory systems. These types of systems arise for example in vehicle simulation in modelling the suspension system or tires, in some models for contact and impact, in flexible body simulation from vibrations in the structural model, and in molecular dynamics. Simulations involving high frequency vibration can take a huge number of time steps, often as a consequence of oscillations which are not physically important. On the other hand, the components causing the oscillations cannot usually be eliminated from the model because in some situations they are critical to the simulation. The equations of motion of a rnultibody mechanical system are described by a system of differential-algebraic equations (DAEs). In this paper, we will explore two types of methods. The first class of methods damps out the oscillation via highly stable implicit methods. Even in this relatively simple approach, unforseen problems may arise for Newton iteration convergence, due to the nonlinearities. The second class of methods involves formulating thernultibodysystem in such a way that the oscillations are determined by a linear subsystem, which can potentially be solved rapidly via a number of different methods. 1. Introduction Much recent work has been focused on the development of numerical methods and underlying theory for the solution of rnultibody dynamic systems (MBS) consisting of fast and slow subsystems ' , These types of systems occur frequently as initial value problems in the computer-aided design and modeling of constrained mechanical systems, molecular dynamics, and in many other applications It is well-known that the characteristics of fast or slow solution is determined not only by the modeling as28

20

1 , 4 2

"This work was partially supported by ARO contract numbers DAAL03-92-G-0247 and DAAH04-94G-0409 and by ARO contract number DAAL03-89-C-0038 with the University of Minnesota Army High Performance Computing Center, and by the Minnesota Supercomputer Institute.

128 pects, e.g., the coefficients of stiffness and damping, but also via the initial conditions and events that may excite stiff components in the system during the simulation. As an example, the governing equations of motion of a mechanical system of stiff or highly oscillatory force devices may be written as differential-algebraic equations (DAE) : 6

-£/*(«.?. 0 + G ( « ) * r

=

0

l a

< )


(lb)

T

where 5 = \qi,...,q„] is the generalized coordinate, q = ^ is the generalized velocity, q = ^ is the acceleration and A = [Ai, . . . , A ] is the Lagrange multiplier. There are rif stiff or oscillatory forces /;, Q includes all the field forces and the external forces which are non-stiff compared to the stiff components, g is the kinematic constraints, G^A = I * A represents the internal constraint reaction forces, and M is the massinertia matrix. The stiff force components in (la) are usually expressed by T

m

A

/, = T,( , )(KMw.) qil qil

(2)

+ Ci^)

K

1

where the i' force, i € 1,.... ny, is a bilateral force between the pair if and i f bodies, 75, which may be the orientational transformation matrix of a local coordinate system, is a nonlinear transformation of configuration spaces, which may represent relative distance or angles between adjacent bodies, is a function of the generalized coordinates qi and q,,, and finally Ki, C, are the associated spring and damping factors > ^ For some generalized coordinate sets, such as optimal relative coordinates of mechanical systems, the function IJ may be linear or even identities, e.g., for instance TJ; = q^ for some t, ij, the nonlinearity of / j with respect to the generalized coordinate q is due to the spatial transformation T; When the components of the coefficient matrices Ki and Ci become large, these force components may cause rapid decay or high frequency oscillation in the solution of (1). The purpose of this article is to study these systems and their numerical solution. 33

29

7

3 3

To demostrate the problem of oscillation and the recent developments in this area, we present two examples: a stiff pendulum and a 2D bushing problem. The former is a very simple example of a type of system often seen in modeling molecular dynamic systems, and the latter is a general form of modebng force devices in rnultibody mechanical systems. Stiff pendulum In Cartesian coordinates, a simple stiff pendulum model, with unit mass and gravity, may be expressed as 0 = i-u

(3a)

129 0 = 0 =

y - V

0 =

ii + yX-

ii +

eA =

1.0

+

y/x*

2

(3b) (3c) (3d)

xX

2

y

-

1.0

(3e)

where the stiff spring of natural length 1.0 and stiffness j , is attached to the center of mass of the pendulum. Preloading the spring by using e = \ / l 0 ~ , the initial condition (0.9,0.1) of (x, y) and the zero initial velocity of (u, v), the results of the states (x,y,u,v) in the 0 to 10 second simulation are shown in Fig. 1. The corresponding eigenvalues of the uderlying ODE of (3), i.e., substituting (3e) into (3c, 3d), are illustrated in Fig. 2, where the 3D figures contain all the eigenvalues on the complex plane drawn along the time-axis. The dominant pair of eigenvalues in the example are i-i, as shown in Fig. 2. As e —* 0, the pair of eigenvalues approaches ±oo along the imaginary axis. The other pair of eigenvalues oscillates on the complex plane with the amplitude and frequecny approaching ±co. Decreasing e to V l O , the eigenvalues of the uderlying ODE of (3) are shown in Fig. 3. Comparing to those in Fig. 2, two pairs of eigenvalues in Fig. 3 are 10 times the magnitude of those in Fig. 2, and the oscillating pair increases its frequency proportional to the size of e. 3

-5

Stiff

I

~

^

3

I

I

?

in

3

*

4

n

•

5

i

••

*

6

T

7

G

*

Q

S

1

U

D

Figure 1: Stiff Pendulum in Cartesian Coordinates 2 0

Lubich shows that numerical solution of a class of Runge-Kutta methods for stiff mechanical systems of a strong potential energy, e.g., stiff spring force such as the stiff pendulum (3), converges to the slowly varying part of the solution, with the stepsize independent of the parameter e in (3). Reich extends the principle of slow manifold to DAE of MBS with highly oscillatory force terms. Algebraic 2 8

1 0 , i ?

130

" 20

·20

... 20

•• real-axis

Tlm.

Figure 2: Eigenvalues of Stiff Pendulum in Cartesian Coordinates, epsilon

= 10e-1.5

eigenvalues olllllrl pendulum n cartealan ooontinale (epsIIon-1D"-2.S)


200

!~ • ·2. .

-4. .

200 .

... ....

Figure 3: Eigenvalues of Stiff Pendulum in Cartesian Coordinate, epsilon

= l Oe-2.5

131 constraints corresponding to the slow motion were introduced with a relaxation parameter to preserve the slow solution while adding flexibility to it in the slow manifold approach. It is not clear that a slow solution appears in the above example. In fact, we can only identify the slow solution of (3) using a proper nonlinear coordinate transformation. In polar coordinates (r, 6), we obtain the equations of motion of (3): 0 = r-z 0 = 0-u 0 =

(4a) (4b)

z + r<J + ~(r-

0 = u --{2zw

1) - s i n 0

(4c)

- cosd)

(4d)

T

where (z,u>) is the velocity. In the 0 to 10 second simulation, using the same initial conditions as the Cartesian coordinates, we obtain the solution in Fig. 4, where the fast solution is (r, z) and the slow solution is (0,w), The eigenvalues along the solution trajectory are presented in Fig. 5. Note the dominate eigenvalues are of the same as those in the Cartesian coordinate formulation. This is because the coordinate transformation, x — rcosS, y = rsinS, is linear with respect to the fast moving r. The eigenvalues of (4) with e = V l O are presented in Fig. 6, Similar to the comparison in the Cartesian formulation, we obtain the eigenvalues of 10 times magnification. However, the eigenvalues corresponding to the slow motion have near zero imaginary parts, therefore, the oscillations along the imaginary axis of eigenvalues 3,4 in Fig. 6 remain insignificant. - 5

--' D

1

I

3

in Pom 4

6

B

?

s

i

n

Figure 4: Stiff Pendulum in Polar Coordinates

132

Figure 6: Eigenvalues of Stiff Pendulum in Polar Coordinates, epsilon = 10e-2.5

133

Although there are ongoing developments to extend the results of Lubieh to multistage multistep methods , and impressive application of the slow manifold technique in some molecular dynamic models, it is not clear that these results may apply directly to all the types of oscillatory components in MBS. As indicated in , the representation of stiff or oscillatory components in an appropriate coordinate system of MBS is not always possible, i.e., the constraints associated with the stiff or oscillatory potential force can be difficult to obtain in general. Moreover, convergence of Newton's method may be an obstacle in obtaining efficient numerical solution of an oscillatory MBS in either of the above-mentioned approaches 3 1

2 0

2 0 , 2 8

Bushing force We have been studying more general MBS of nonlinear oscillatory components such as a bushing force, which is often used in modelling vehicle suspension systems. Different from the linear spring, this element is usually an anisotropic force, i.e., it has different spring coefficients along the principle axes of the bushing local coordinate frame. The bushing force between body-i and body-j may be defined using the relative displacement dy, its time derivative djj, and the relative angle and its time derivative 9ij of two body-fixed local coordinate frames at the bushing location on two bodies. Using the vectors s[ and representing the bushing location in the body-Vs and body-j's centroid local coordinate systems, respectively, we have Xj

d

ij

+ Ai4 - Ajs'j

=

(5)

where the orientation transformation matrices Ai and Aj are A, = A(f>,) =

cosf, — sinf, sin 8; cos &\-

A, = A{8j) =

COS0-; — sin 8j sinf?j cosdj

and [xi, yt, 9<] and [xj, yj, 9j\ are Cartesian coordinates at body-fixed frames. The bushing force / j can then be written as A =

'A

k

=Ai

n

0

&

Ajdij+Ai

. Jb J and the applied torque is n = k%j + c V 1

v

3

e

0

o Ajd\j C*

(6) (7)

a

where lOij = k , k , and k are spring coefficients associated with x, y, and 9 coordinates, and
A simple example may be obtained from this model using unit mass-inertia and gravity, grounding the first body, and setting the bushing location on the second body

134 as s' = [-|, 0]. A bushing element with no damping attached at the global position of [ i , 0 ] yields 0 = »-*-(|-. 2»£j

(8a)

0 = if+ * » ( » - 2 ^ ) + 1

(8b)

+

-i

,.„

f l

sin , ,l T

cost?,

cosf

sinf?

It is easy to see from (8) that the local eigenstructure of the system may change rapidly, depending on the size of the stiffness coefficients. I

Usingtheinitialvaluesof(i, ,,fl) = (l.l,0.1.0.0)with(fc ,fcv,fc'') = (lOMoUO*), the solution of (8) exhibits high frequency oscillation for all coordinates, as shown in Fig. 7. Solving the eigenvalue problem of (8) at each time step yields three pairs as illustrated in Fig. 8, !

•vWWVW BJ

S

«

ts

A/WWW

o*

a< 04

M

04

Ql

Figure 7: Bushing Problem in Cartesian Coordinates The bushing example represents a different type oscillatory forcing function than the stiff pendulum. For the bushing element, the coupling of translational and rotational coordinates yields varying local eigenvectors in Cartesian coordinate space, and two or more fast pairs of eigenvalues, see Fig. 8. The bushing force cannot be represented as a simple potential force as well as the constraint reaction force of simple constraints, for which we may directly apply either the results of Lubich or the slow manifold approach by Reich to the numerical solution. Many methods for efficient solution of oscillatory dynamic systems are predicated on a nearly linear form of the equation. For example, the method of averaging requires the linear part of the oscillation equations of motion to be dominant, and 5

135 r . « * : o r o l ^ . l I J I..: J

|^|ttM

Figure 8: Eigenvalues of Bushing Problem 3 5

the mode-acceferation method for structural dynamics, which eliminates higher modes in the computation of the mode-dispmcemerrf solution , is based on the time-invariant eigenvalues of the structual dynamic equations. Our aim is to treat the class of general nonlinear stiff and oscillatory forces represented in the MBS of (1), One approach is based on the study of a class of MBS DAE solvers and the energy dissipative method > , which may damp out the oscillation that is not important. The other approach is to localize the oscillatory components, and then apply fast numerical solution techniques to approximate the oscillation. 4 0

3 9

a3

11

2. Damping the oscillation Given the possibility of a rapidly changing local eigenvalue structure, perhaps the simplest strategy is to consider damping the oscillation whenever it is not important via highly stable implicit numerical methods. Since the amount of damping is controlled by the time-step, and automatic stepsize selection increases the time-step whenever the solution is slowly-varying (i.e. if the amplitude of the oscillation is small in comparison with the local error tolerances), the stepsize is increased when the oscillation is no longer important. 3 9

In recent work we have considered the solution of mechanical systems with high frequency vibrations via this type of technique. In our initial experiments with the bushing problem (8) solved directly by low-order BDF methods, we found that the methods experienced severe problems with Newton convergence. To overcome these problems, we proposed a coordinate-split (CS) formulation of the equations of motion, and a Newton-type iteration for solving the coordinate-split equations at each time step. The coordinate-split formulation eliminates problems due to obtaining an

136 accurate predictor for the Lagrange multiplier variables because these variables are no longer present in the computation. We found that the coordinate-split formulation worked well foT several test problems involving mechanical systems with high frequency oscillations. However, for problems with very high-frequency oscillations, there are still difficulties with Newton convergence. The Jacohian matrix for solving the nonlinear equations of the coord in ate-split formulation at each time step involves several terms which are complicated to compute and which are small at the solution of the nonlinear system. These are terms of second-order which correspond to the derivative of the projection operator onto the constraints- Away from the solution, these terms are highly oscillatory. By neglecting these terms, we found that the resulting Newton-type method converged much faster for oscillating test problems like the bushing problem. We called the resulting method the modified coordinate-split, or CM method. The modified coordinate-split (CM) method performed extremely well in numerical experiments described in and in other experiments we performed subsequently. The constraints g(p) = [ p i , g s ] of a two-body pendulum may be written as 3 9

T

Si =

3-1

(9a)

Vi 0, g* = (x -x f 92

=

S3

=

t

Ss =

2

(9b) (9c)

+ (y,

(9d)

(9e)

&2

where X\ and tft, i = 1,2 are Cartesian coordinates of the center of mass of body i , and &i is the orientation coordinate of the body centroid reference coordinate system, and the length of the pendulum is 1. Applying the bushing force (6) with [it , k", k ] = [1000,1000,1000] and [c\cV,c°] = [10,10,10] to the pendulum, small oscillations of the numerical solution appear. Using the initial values q [0,0,0,9.9989e-l,-1.4852e-2,0] and v - [0,0,0,-6.75e-5,-4.5444e-3], numerical results from the BDF code DASSL are contained in Table 1, in which error test failures (etf — s) and convergence test failures {ctf - s) are listed. We denoted by CS the coordinate-splitting formulation, LG the stabilized index-2 formulation proposed by Gear , CM the coordinatesplit form using a modified iteration matrix with the second-ordered derivative terms omitted, and LM the modified LG using the new predictor of the multipliers by the CS method. Using simplified Newton iterations and the corresponding modified local error estimate, CS, CM, LG and LM obtain consistent results. 1

e

2 7

l 3

To see the effect of more severe oscillation, we increased the spring constant of the bushing to 10 . Time steps of these methods selected by DASSL are shown in 5

137

Method CS CM LM LG CS CM LG LM CS CM LG LM CS CM LG LM

TOL 10"

3

10-

3

lO" 10" ID" 10" lO" 10" 10" 10" irr 10" 10" 10" 10"

3

4 1

4 1

E 5 5 3

6 6 6 6

step S-s J ~ * et} - s ct] - s 62 156 48 0 13 62 156 48 0 13 62 154 48 0 13 59 141 46 12 0 77 193 65 1 16 77 193 1 65 16 61 136 27 1 5 77 193 65 1 16 87 215 54 0 13 87 54 215 0 13 108 259 77 1 21 87 215 54 0 13 138 343 97 1 25 138 343 1 97 25 131 308 65 0 16 138 343 97 1 7b 3

Table 1: Simple Pendulum with a Bushing Force, Spring Constant — 1Q ' Method cs, CM n

m

ToL lO lfj10" io- 4 4

4

LM,„

4

time step 0-0-1 2252 0-0-1 20 0-0.1 5267 0-0.1 2251

etf 3-8 3361 40 7 10536 7899 4900 3360

-s 1 0 0 1

/-* 4901

ctf-s 1120 0 2633 1120

6

Table 2: Results of Bushing, Spring Constant = 10 , Damping = 10 Figure 9. Clearly, CM took much larger steps than the other methods. Moreover, if the spring constant is increased to 10 , we found severe convergence problems for LG, CS and LM; the results are contained in Table 2. 6

The amount of damping in the CM method is controlled via automatic timestep selection. In real-time simulation, it is not always practical to vary the timestep. Hence we have also been investigating the possibility of damping the oscillation via changes to the method parameters. This idea has often been considered in structural analysis, where the Newmark /?-method is commonly used to selectively damp high frequency oscillations. The /?-method bears some similarity to the fl-method of numerical ODEs, however it applies directly to second-order ODE systems such as the equations of motion. The parameters in these methods are used to vary the damping (stability) properties. Usually the methods are first order, however for the critically damped parameters they are second-order. Hence when any significant amount of damping is added into the method it becomes first-order. This deficiency has been overcome via the a-modification of the ^-methods. By adding one additional parameter, the a-methods are able to achieve second order and selectively damp high2 3

11

138 •

B.OS

0.1

DIG

- ID.

0J

0 2

03

T m I.- •

OK

OA

0.5

0*$

5

Figure 9; Time Steps Used in Solving the Bushing Problem, Spring Constant — 10 8

frequency components. Recently Cardona and Geradin have considered extending the a-methods to second-order DAE systems from rnultibody dynamics, however the new methods are plagued by oscillations which we believe are largely unphysical. We can extend the a-methods to DAEs in a way which does not introduce additional oscillations. Given a second-order ODE, y = f(t,y,y)

(10)

the o-method for this system is given by a , n+

= (1 + a)f(t„ ,d ,v ) +l

n+l

- af(t ,d ,v )

n+1

n

2

d

n+l

= d + hv„ + h [(^-8)a n

v„+i =

n

(11a)

a

+ 0a \

n

" + h[(l - 7 K

n

(lib)

B+l

+70n+i]

(He)

2

where 7 = 1/2 — or, 8 = (1 — a) /r, for a e [-1/3,0], and d, v, and a are the position, velocity and acceleration, respectively. To extend this method to the DAEs which describe rnultibody systems, M m

= /(«,y,!/) + G (y)A 0 = 9(g) T

(12a) (12b)

7

where G = dg/dy, we consider a class of methods of the following type, M(d i)a n +

n + 1

= (l+c.)/(l i,d n +

n + 1

,u„ )-a/(t ,d ,j) ) + 1

n

n

n

(13 ) a

139

d

n+l

n

+

v

a+1

1

= d + hv + ^

h' [(^-0)a +0a ]

n

G

n

r ^±i_rA f

= v + h[(l - ) a „ + a

7

K

7

+

n+l

i


( 1 3 b )

T dn+l

hG ( +

d

")A

n + 1

9(<Wi) = 0 G (d )v = 0

(13d) (13e)

T

nirl

(13c)

n+1

The new method projects the solution at the internal stages onto the constraints similarly to the Projected Implicit Runge-Kutta methods introduced in . We note that a in (13) is not the acceleration. However, the acuta] accelerations can be computed via a post-processing step if they are actually needed. Using the concept of essential underlying ODE introduced in , we can show that for linear model problems, this method corresponds to discretizing the essential underlying ODE directly by the a-method, up to terms of order 0(h }, and is second order for the position variables. Further analysis of the method, testing and extension of these ideas to higher order methods remains to be done. 2

n+1

3

3

3. Localizing the Oscillation Often in rnultibody systems the components exhibiting high frequency oscillation are modeled using very stiff linear springs or nonlinear springs of the dominant first order term. In the case of rnultibody mechanical systems, there are usually nonlinear transformations applying to the spring forces to obtain equations of motion in the generalized coordinate space > . The resulting system of equations contains nonlinear high frequency oscillatory forces. 33,M

37

One approach to solving the high frequency oscillation problem is to carry out modal analysis and then eliminate the higher modes, since lower modes may preserve the slowly varying part of the solution For example, the extreme high modes of a structure are often rejected in modelingflexibleeffects of mehanisms, since the details of the oscillating solution are not so important as the long-term solution behavior. A similar approach has been developed in recent work on molecular dynamics simulation However, due to nonlinear oscillatory forces in the rnultibody formulation, the modal analysis needs to be carried out at each time step to resolve the rapidly varying local eigenvalue structure of the system, resulting in very costy computations. 9

4 2 , 4 3

Another approach is to resolve the oscillation efficiently via look-up tables. This idea is frequently used in a real-time simulation environment One such example 1 2 , 3 4

140 is the modeling of contact compliance in rigid body simulation, where the localized oscillation may be of interest. Applying linear constitutive laws to modeling the contact compliance, e.g., the elastic half space theory, Boussinesq's influence functions and Hertz' contact model leads to linear spring forces between contact bodies 38,H,2J