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ELSEVIER Applied Mathematics and Computation 96 (1998) 65 73

A numerical method for the integration of perturbed linear problems

David J. L6pez, Pablo Martin * Departamento de Matem4tica Aplicada a la lngenierla. E. T S. de Ingenieros Industriales.

Universidad de Vtdladolid. Paseo de Cauce s/n, 47011 Valladolid. Spain

Abstract

The purpose of this paper is the construction of a multistep numerical method to in- tegrate perturbed linear problems based on the expansion of the solution in terms of a family of functions. The method here presented is a generalization of the SMF method for perturbed oscillators. A numerical example is presented showing the advantages of the new method over the general purpose codes and over the SMF method when considering perturbed oscillators with damping. © 1998 Elsevier Science Inc. All rights reserved.

Keywords: Perturbed problems; Numerical integration; Multistep methods

i . Introduction

In this pape r we are interested in the numer ica l in tegra t ion o f p rob lems o f the fo rm

x (~) + 2 t x (~ ~) + . . . + 2 , x = eg (x , x ' , . . . , x (~-L), t),

x ( 0 ) = x l ° ~ , x ' ( 0 ) = - ~'~ - (~ '~ ( 1 ) ~0 , ' " , x/k I)(0) =-~0 ,

where the der ivat ives are with respect to the t ime and e is a small pa ramete r . F o r the pa r t i cu la r case o f k = 2 and 21 = 0, i.e. pe r tu rbed osci l lators , there

exist special numer ica l me thods for the in tegra t ion o f the p rob lem: Stiefel and Bettis [1], Bettis [2], Scheifele [3], or Mar t i n and Ferrf indiz [4,5]. The m e t h o d

* Corresponding author. E-mail: pabmar@wmatem.eis.uva.es.

0096-3003/98/$ - see front matter © 1998 Elsevier Science lnc. All rights reserved. PII: S0096-3003(97) 1 0121-7

66 D.J. L6pez, P. Martin / Appl. Math. Comput. 96 (1998) 65 73

presented in this work is a generalization of the method of Martin and Ferrfin- diz, the SMF method, based on the Scheifele G-functions.

Scheifele [3] suggested a systematic way to obtain the numerical solution of a perturbed oscillator by introducing a family of convenient functions, the G- functions (more details can be seen in [6] or in [7]). In Section 2, we generalize this technique for the problem (1) by defining, like Scheifele, a family of func- tions, and expressing the solution in terms of these functions. In Section 3, using this expression as the basis, we define a new numerical code in the same way as in [4,5]. The method defined in this form has the good property of ex- actly integrating the unperturbed problem (e = 0) and e, appears as a common factor of the truncation error when integrating the perturbed one. In Section 4, we obtain generating functions for the coefficients of the methods, and as a par- ticular case we give the generating functions for the coefficients of the SMF method, calculated in the work of Martin and Ferr~indiz by means of a product of matrices. In Section 5 a numerical example is presented. We show the ad- vantages of the new methods over a general purpose code for second-order dif- ferential equations and over the SMF method when considering perturbed oscillators with damping.

2. Solving the perturbed linear problem

In this section we will construct the solution of the initial value problem

xlk) + ;tlxlk l) + . . . + 2kx = ~ g ( x , x ' , . . . , X Ik J),t),

x (O) = x~O);xt(O) : X~I) , . . . ,x (k I)(0 ) : X0" (h-l). (2)

To this end we will consider the linear operator L defined by

Lx = x I*l + 21x Ik j) + . . . + ;tkx. (3)

In order to solve the homogeneous problem we will define a set of functions {H0, H i , . . . ,Hk-~ }. Each Hr will be the solution of the problem

L x : O , x ( j ) (O)=O, O < ~ j < ~ k - l , j C r , x (r)(O)= 1. (4)

The solution of the homogeneous problem will be a linear combination of the functions H~. For the complete solution of problem (2) we can define a family

G ~ of new functions { ,},=k, where Gn(t) is the solution of

t n k L x - (n - , ) ! ' x ( 0 ) = x ' ( 0 ) . . . . . = 0 ( n / > , ) . (5)

These new functions verify the relation

G',(t) = G,, ,(t), n > k, (6)

D.J. L6pez, P. Martin / Appl. Math. Comput. 96 (1998) 65-73 67

and from this result we can define G, for n = 0, 1 , . . . , k - 1 as follows:

t Gk-l(t) = Gk(),

Gk-2(t) = ' Gk l(t), (7)

c 0 ( t / = C', (t/.

In the particular case of k = 2, these G-functions are the same as those defined by Scheifele [3]. These functions are related to the H-functions defined in Eq. (4):

k 1 H,(t) = G,(t) + ~ 2~_nG~(t), n = 0, 1 . . . . , k - 1. (8)

j=n+l

Let x(t) be the solution of the problem (2) and let

f ( t ) = g(x(t), x' ( t ) , . . . , x (k- l)(t), t). (9)

With this notation the solution x(t) can be written as

k-1 x(t) -7- ~-~x~i)I-Ii(t) -4- ~__~f(n)(O)Gn+k(t)" (10)

i--O n--O By substituting Eq. (8) in Eq. (10) we obtain

k-I Q i-1 / X(I) = x~°)ao(t) -}- Z X~i) -]- Z ~'i-jx~) a i ( t ) + ~;~['(n)(O)Gn+k(t). ( l l )

i= 1 j=O /I n=O

In order to simplify this expression, we shall define a new set of coefficients as follows:

a~ ° i=x~ °t, a} ° / = x~ il + 2i_jx~ ) , l < ~ i < . k - 1 , (12)

and we can write Eq. (11) in the form

k-1 ~ f ( n ) x(t) = ~a}°)G~(t) + (0)G,+k(t). (13) i-0 n-0

We are also interested in the values ofx ' ( t ) ,x"( t ) , . . . ,x(k-lt(t). We can express these functions in terms of G-functions as follows:

k-I x(P)(t) = Zai(P)Gi(t) + E~_.f(")(0)G.+k p(t), 1 <<.p<<.k- 1. (14)

i=0 n 0

The expression of the coefficients a~ ) can be easily deduced from Eq. (7) and from the fact that

k 1 G0(t /= - (15)

i=0

68 D.J. L@ez, P. Martin / Appl. Math. Comput. 96 (1998) 65-73

This gives us the following recurrence relations:

ai (p+l) -(P)-)~i+io(~ ), O < ~ i < ~ k - 2 , a (p+l) -2ka~o ). • = ¢ / i + 1 k I = (16)

3. The numerical method

We can define a numerical method to integrate the problem (2) by truncat- ing the series of the solution obtained in the previous section, Eqs. (13) and (14). If we denote by Xm ~) the approximation to the pth derivative of the solution in t = mh, the numerical method will have the expression

k - 1 n

. (p) = Zai (P)Gi(h) + e ~ f ( i ) ( m h ) G j + k - p ( h ) , A m W l

i = 0 , / = 0

where

0~<p~<k- l, (17)

a~O) _ ~(o)

i 1

a(Ol = x~) + Z 2i_jx~l ' l ~ i ~ k - 1 , i

j o (18)

ai(P+'l = ai(i+l, - 2i+,a~o i, O <~ i <~ k - 2, p > 0 ,

a~_+, '1 = -~ka~o I, p > o.

The method defined in this way has the property (as that of Scheifele [3]) that e always appears as a common factor in the expression of the truncation error, so the method exactly integrates the unperturbed problem. In spite of this good behaviour, the method defined above has the great disadvantage that it can only be applied in very particular cases, due to the complexity of the prelimi- nary calculations that it demands. The calculation of the derivatives of the function f is usually very difficult and costly, and in fact rules out the applica- tion of the method when perturbations with slightly complicated expressions are considered.

This problem can be resolved, as in the works of Martin and Ferr~indiz [4,5], by converting this code into a multistep scheme without losing the above-men- tioned properties.

To this end, it is sufficient to approximate the derivatives of the function f for combinations of evaluations of the function at different points. For this purpose we will consider the following upper triangular matrices:

B~ = (b~) .+ l×~+ z,

D.J. L6pez, P. Martin I Appl. Math. Comput. 96 (1998) 65-73 69

where

1

bij = 0 ~j - - I bi Lk Z...~k= 1 j - k

D, = (dij),+,×,+x,

where

i f i = j = 1,

i f j > 1, i = 1 ,

i f / > 1.

1 i f / = j = 1,

-1 if i = 1,j = 2, dij= 0 i f / = 1,j > 2,

X-,j-1 a, L,k if i > 1. Z--~k=l j k

With these matrices we have the following relations between the derivatives of the function and the backward differences:

I f (mh)

hf'(mh) h2 f " } mh )

h"f(")(mh)

' f (mh) "~

hf'(mh)

hZf"(mh)

h.fl.I (mh)

= Bn

= Dn

im Vim V2fm

\ V"fm

( im+, Vim+l Vzim<

Wfm+l

+ O(h°+l),

+ O(hn+l).

These results let us substitute the derivative evaluations in Eq. (17) by back- ward differences and we obtain an explicit multistep method:

k-1

,=0

im Vfm

2 Vim

V"im

, O < ~ p < < . k - 1, (19)

70 D.J. L6pez, P. Martin / Appl. Math. Comput. 96 (1998) 65-73

and an implicit multistep method:

k - I

Xm+l , , i=0

f m + l

XT fm+l ~ 7 2 / m + 1

n V fm+l

, O<<.p<<.k- 1, (20)

where Vk,p,n(h ) = (Gk_p(h), Gl+k_p(h)/h,..., G,+k_p(h)/h"). Both methods have the property that e is a common factor of the truncation error and they exactly integrate polynomials of degree less than or equal to n.

4. Generating functions for the method

The multistep methods defined in the previous section can be written in the form

k - I

(P) = Za~)Gi(h) + g,(b~o)fm -'k b~l )Vfm + . . . --k b~)V"fm), Xm+l i=0

k !

(P) = Za}P) Gi(h ) + e(do(Pl fm+, + d~lV fm+, + . . . + dflV"f,,+, ). Xm+l i=0

In this section we will obtain generating functions which give us the coefficients for the methods. We will restrict ourselves to the explicit methods. The gener- ating functions for the implicit ones can be obtained in the same way.

Let Bp(s) be the function given by :N)

8~( , ) = ~ b . ~ s °, (21) n=0

and let P, (t) be the only polynomial of degree less than or equal to n which ver- ifies that

e . ( o ) = vP. (O) . . . . . v ° - ' e . ( 0 ) = o,

This polynomial is

V"P,(0) = 1. (22)

pn(t)=t( t+h). . . ( t+(n-1)h)h.n! _ ( - 1 ) . ( - ~ / ) h .

Let y.(t) be the solution to the problem

(23)

L(y . ) = P . , y . ( 0 ) ' y(k- l ) (0) (24) = yo(0) . . . . . . 0.

D.J. L@ez, P. Martin I Appl. Math. Comput. 96 (1998) 65 73 71

The numerical me thod is exact for polynomials o f degree less than or equal to n, then it will exactly integrate p rob l em (24). The solut ion which gives us the me thod in h is b(,°l,b~ 1), /,(k-l) then

.V~)(h) = b~ ), O<~p<.k - 1, (25)

and we can write Be(s ) as

Bp(s) = ~_~y~l (h)s n. (26) n=O

Let Y(t, s) be the funct ion

o@

r( t ,s) = ~_y~°l(t)s". (27) n=0

Lett ing the linear ope ra to r L act on this funct ion we have

LY = °)s" = s" = - s ) " = (1 - s) = at, n=0 n=0

where a = (1 - s) -1lb. Eq. (24) gives us initial condit ions for the funct ion Y:

Y(O,s) = Y'(O,s) . . . . . YIk-~)(O,s) = 0, (28)

where the derivatives are with respect to the first variable. F r o m these last re- sults it is easy to obtain the expression o f Y:

at k-1 Y(t, s) = - Y~'~j=0 ( log a)JHj(t)

o'(log a) ' (29)

where a(z) = z k + 212 a - l + - . . + 2k, and therefore,

- - ~-]~J--° ( l °g a)gHj(h)" (30) Bo(S) = Y(h,s) = (1 s ) - ' k-, a( log a)

The other generat ing funct ions can be easily obta ined by deriving the funct ion Y.

In the par t icular case o f k = 2 and 2~ = 0, the p rob lem (2) is a per tu rbed os- cillator, and the me thod here developed coincides with the S M F me thod de- scribed in [4,5]. In the work of Mar t in and Ferr~ndiz the coefficients o f the me thod are calculated by means of a p roduc t o f matr ices as in Eqs. (19) and (20). With the results obta ined in this section we can give the generat ing functions for the coefficients o f the S M F method . I f the p rob lem we are considering is

x" + = g(x, t), x(0 ) - - x0, x ' ( 0 ) = (31)

then, the generat ing funct ions for the S M F explicit me thod are

72 D.J . L 6 p e z , P. M a r t i n / A p p L M a t h . C o m p u t . 96 ( 1 9 9 8 ) 6 5 - 7 3

(1 - s) -1 - cos(09h) - log a (sin(09h)/09) no(s) = ( l o g a ) 2 q-- oJ 2 '

(1 - s ) l l og a + 0 9 sin(09h) - log acos(09h) n , ( s ) =

( log a) z + 09=

where a = (1 - s) -l/h.

5. A numerical example

In o rde r to check the p e r f o r m a n c e o f the m e t h o d s here defined we will inte- grate a simple p rob lem:

x" + Lr' + x = e cos 2t. (32)

Log(Error)

- 1 '.,.,V('~/';'/"i ['"/'"~/"\/'-'! ;"'k /"\ :'", ;", . . . . . . . - 2 ~ ", ~ J:" - ,L ' V " W v !~ ",d ",,;'",,:'" ,,',

/ \ / l i 'v'W~.7 " r t / ~ , r x d ~, .~ ,--..~ ~ ~ " tl : Stormer - ' ~ r V ~ v 7 " ! l ' ~ / ' ¢ V ~ v " ~ / f ' ~ , x v-sMs -4 i V I ,~' ~ v v V

-5 ~ / ~ ~ L L M -6

-7 -8

-9 -10

t 200 400 600 800 1000

Fig . 1. x " + 2x ' + x = e. cos 2t, P E m o d e , o r d e r 4, h = 0 . 5 , 2 = 10-2 ,8 = 10 -4.

Log(Error)

- 6 ,,., ,"!F",/'">,,{"'./"~ ['",,/" "",.,"',, ",. ,,-, ,-, . , "7 [:' ': i ~ ~ ~ il }~ " V v ',,! ~,: "4' ~/"<',1"",,:",, Stormer

-" I !'v'~i v~,' '~,~ 'h/V-V~, ,~, c.c. ~ ; " v "

- 8 mt "7 li ~ V v ~ ~ ~ ',, ;I' ',('~ "~ "~ ~ SMF i ~ " '~ " " " " " ' ~ " t i ' s i # "

- 9 ,' i

-i0 ~ L M t

2 0 0 400 6 0 0 8 0 0 1 0 0 0

Fig. 2. x" + 2x' + x = e co s 2t, P E m o d e , o r d e r 6, h = 0 . 1 , 2 = 10 -2, e = l0 4.

D.J. L6pez, P. Martin / Appl. Math. Comput. 96 (1998) 65-73 73

We will study the behaviour of the new code by comparing it with the SMF method and with the classical St6rmer code. In the figures we will denote the method constructed in this paper by LM. Fig. 1 shows the error when integrat- ing with a large step size (h = 0.5) and with a low order (4). The good perfor- mance of the method and the advantages of it over the other codes is clear. The same problem has been integrated in Fig. 2 with a lower step size (h = 0.1) and a higher order (6). The same relation between the methods has been obtained with better results due to the higher order and lower step size.

Acknowledgements

This work has been partially supported by Junta de Castilla y Le6n under project VA47/95 and by Spanish DGES under project PB95-696.

References

[1] E.L. Stiefel, D.G. Bettis, Stabilization of Cowell's method, Numer. Math. 13 (1969) 154-175. [2] D.G. Bettis, Numerical integration of products of Fourier and ordinary polynomials, Numer.

Math. 14 (1970) 421-434. [3] G. Scheifele, On numerical integration of perturbed linear oscillating systems, ZAMP 22 (1971)

186-210. [4] P. Martin, J.M. FerrS.ndiz, Behaviour of the SMF method for the numerical integration of

satellite orbits, Celestial Mech. Dyn. Astron. 63 (1995) 29-40. [5] P. Martin, J.M. Ferrfindiz, Multistep numerical methods based on the Scheifele G-functions

with application to satellite dynamics, SIAM J. Numer. Anal. 34 (1997) 359-375. [6] E.L. Stiefel, G. Scheifele, Linear and Regular Celestial Mechanics, Springer, New York, 1971. [7] V. Fairdn, P. Martin, J.M. Ferr~.ndiz, Numerical tracking of small deviations from analytically

known periodic orbits, Comp. Phys. 8 (1994) 455-461.