Pertanika 8(1), 59-66(1985)

Convergence of the Variable Orderand Variable Stepsize Direct Integration Methods

for the Solution of the HigherOrder Ordinary Differential Equations

MOHAMED SULEIMANMathematics Department,

Faculty of Science and Environmental Studies,Universiti Pertanian Malaysia,Serdang, Selangor, Malaysia.

Key words: Convergence; higher order ordinary differential equations (ODEs); Direct Integra-tion (DI) methods; predicted and corrected values; integration coefficients; variableorder and stepsize.

ABSTRAK

Kaedah Pengkamilan Terus bagi penyelesaian terus satu sistem persamaan pembezaan biasaperingkat tinggi dibincangkan. Penumpuan bagi kaedah Pengkamilan Terus dengan langkah danperingkat yang tetap dibuktikan dahulu, sebelum penumpuan bagi kaedah tersebut dengan langkahdan peringkat yang berubah dibuktikan.

ABSTRACT

The DI methods for directly solving a system of a general higher order ODEs are discussed. Theconvergence of the constant stepsize and constant order formulation of the DI methods is proven firstbefore the convergence for the variable order and stepsize case.

1. INTRODUCTION Many problems in engineering and sciencecan be formulated in terms of such a system.

The general system of higher order initial Some of the higher order ODEs problems foundvalues ODEs is given by in'the literature are on the symmetrical bending

of a laterally loaded circular plate, the bendingY i' = f / Y ) ' = 1 ? s (1) of a thin beam clamped at both ends. Both nrob-

' ' ' lems are given in Russell and Shampine (1972).with initial conditions Others are on the steady flow of a viscoelastic

Y(a\ = v fluid parallel to an infinite plane surface withuniform sunction; given in Serth (1975), control

in the interval a < x < b, where the i-th equa- t h e o r y i n E n r i g h t et al ( 1 9 7 4 ) c o p i a n a r circulartionisoforderdjand t w o b o d y m o t i o n i n shampine and Gordon

/ft *\ ^ _ j \ (1975), gravitational n-body problem in Dol et

= (yi n ->y y / ~ )> a / ( 1 9 7 2 ) 'The current technique of solving the prob-

^ W ~ 0 ? i , O> •••»Th> dj — 1, ••*>7?s>o> •••» ^s* d _ i ) # lem in (1) is by reducing it to a system of first

Key to author's name: M.B. Suleiman.

M.B. SULEIMAN

order equations. Let

( d 2 -

(ds -=y . .

and

z = [ z l l > Z 1 2 > "' z l d j ' '"> Z s l > z s 2 > - > z s d s 1

and

F (x, z) - [z12, z13, ..., fj, ..., zs2, zs3,..,, fs J

then (1) reduces to the equivalent first orderproblem,

z '=F(x , z), (2)

with

z(a) = 1?

The system in (2) is then solved using eitherthe Adams or the Runge-Kutta class of methods.But the work of Krogh (1969)-and Suleiman(1979) have shown that for many problems, it ismore efficient to solve (1) by integrating directly,rather than reducing it to (2). This DI method isin fact a generalization of the Adams-type ofmethods to higher order ODEs. Hall and Sulei-man (1081) and Gear (1978) have studied thestability of the method for solving second orderODEs. Gear (1971) proves the convergence forcertain methods which solved (1), those whichcan be transformed into the Nordsieck's arrays,while Shampine and Gordon (1975) prove thecase for the first order problem.

2 THE DI METHODS IN THEPREDICTOR-CORRECTOR MODE

For simplicity of description and withoutloss of generality, the discussion which followswill be for the single equation

y<d> = f(x, Y), Y(a) = A ( S )

where YT = (y, y',..., y (d->> )

and A* =(,,,„',...,nM-»>).

60 PERTANIKA

Integrating (3) d - r times, r = 0, 1, ..., d— 1 leads to the identity

x n +

d — r times

d-l-r

i = 0

ry ( r + i ) (*„) + (4)

dx ... dx

f is then approximated with an interpolationpolynomial Pk n(x) of degree k - 1 which inter-polates f at the k preceding points (fn, jcn), (fn_j* n _ ! ), .», (fn-k + i .*n-k+i )• Thus expressedin terms of divided differences,

(x " ^ n -

(X - Xn ) ...

In , n - 1 , . . . , n - k + 1 ]

where

., n - i ]

- f ,x [ n , n - 1, .. . . n - i + 1 1 M n - 1 , n - 2 , . . . , n - i ]

* n - * n - i

Hence a discrete replacement for (4) is

i = o i!

Pk> Jx) dx ... dx (5)

where p n +rJ are predicted values associated with

totally know back values.

Define, the integration coefficients g tobe the t fold integral

r*n+i rx rx

git = -J xn J x n J xn

*- t times -*

(x - x n _ i + 1 ) dx dx ... dx (6)

VOL. 8 NO. 1, 1985

CONVERGENCE OF THE DI METHODS

Then the predicted values in (5) can bewritten as

- 0

k - 1

g i , d - r f[n, n - 1 , ..., n - i ] . (7a)

The corrector formula is obtained by ap-proximating f in (3) by the k degree polynomialPk+1 n + 1 (JC) interpolating the (k + 1) points

v n+l> Xn+1^' ^n> *n)> •••» ^n -k+J» ^ n - k + l ) '

where ^*n+i - f(*n+i> ^*n+r' n

Pn> Pn> Pn

In terms of

(x - xn_t) ... (x - *n_k + 1)f*. n + 1 n n_k+1 ] > where the last divided diffe-rence is obtained using the f evaluated by the pre-dicted values P* n + 1 . Replacing Pk + 1 > n + 1(jc)in(4) and integrating, we have the corrected values

C d - r ) _ ( d - r )

* n + l " P n + 1 + 8 k , r l [ n + 1 , n , . . . , n - k + l ] >

r = 1, 2, ..., d (7b)

Formulae (7a, b) represent the k-step pre-dictor-corrector mode of the DI methods. In thevariable order implementation of the method, kr

if changed, is usually either decreasing or in-creasing by one from the value at the previousstep.

From the definition of g i t in (6) it can beshown by integrating by parts that

8i,t = l - l , t

For the case i = 0,

Sot dx dx...dx =

where h n + 1 = (xn + l - xn). (9)

These integration coefficients are generatedin a triangular array of the form shown belowstarting with the first row and from left to rightwith k + d + 2 coefficients along the first rowand (k + 2) along the first column.

>v t

i N.

0

1

2

k + 1

0

goo

Bio

§20

8k+i,

1

goi

gl l

fel

D

2

S02

8ta§22

•

. . . k

. . .

. . .

• • • 82,

+ d + I

§O,k + d + l

8i ,k + d

k + d - 1

i t -

From (8) and (9) it is observed that gQt =i ^se t 0 t r i e relation

and hence the computation of the first row isavoided. Constant stepsize will further accountfor the saving of computation, since if the step-size remains constant for R steps, and the co-efficients of the R-th row depend only on thecurrent stepsize, therefore the coefficients thefirst R rows need not be recomputed and com-putation need only start at row R + 1. A changein order corresponds to adding or dropping aninterpolation point. If at the current point to becomputed, the step is not changed for R + 1steps and the order is reduced, no special actionis required. If however, the order is raised, oneadditional term needs to be added to the endtable for each of the first R rows of the intergra-tion coefficients, before commencing with com-putation of the row R + 1.

Despite the simplicity and advantages inspeed, storage and accuracy of the DI methods,they are yet to be popularised as a general pur-pose code for the solution of the general orderODEs.

3. CONVERGENCE OF THEPREDICTOR-CORRECTOR MODE OF

THE DI METHODS

Before the proof of convergence of thevariable stepsize and order method in (7a, b) isgiven, the proof for the constant stepsize h andorder k formulation of the method is first given.

PERTANIKA VOL. 8 NO. 1, 1985 61

Definition

MB. SULEIMAN

polynomial interpolating the set of values

A linear k-step method is said to be conver- T h i s f o r m Q{ ^ i n t e r p o l a t i o n pOlynomial is usedgent if for any function f(x, Y), for which the h e r e for c o n v e n i e n c e o f p r o v i n g t h e c o n v e rgence

of our theorem. The constant stepsize and orderformulation of the predictor in (7 a) is

(d-r) . 1 (d-r+l)

constants Tj-Cm),o

m = 0, 1, .... d- -

h-0nh=jc-a

s = 0, 1, ..., d - 1;

where n = (x - a)/h for any xe.[a, b], withstarting values y^m^ = r / m \ h ) satisfying the con-

(d-r) (d-r) (d-r+1) . h*"1 (<J-1)Pn + 1 =Vn + h y n

+-+^I^yn

+ hr 2 7i, d-r fn+l-i. r = 1, ..., d (10)

where the coefficients

dition lim = 170

(m)M = 0, 1, .... k - 1 ;

m = 0, 1, ..., d - 1.r times •

i (x , 1) dx ... dbc

Let enbe the global error between the true are constants independent of h. To see this, putsolution Y(xn) and the computed solution Yn, x = sh + xn theni.e.,

T T Ten = Y (xn) - Yn

s k - 1

C, n (s + j) ds...dso j = o

and cx are constants.

The constant stepsize, Lagrangian form ofthe corrector polynomial is given by

P k + 1 . n , l W =

( d - l ) ( d - 1 )

C)We now quote the following theorem.

Theorem 1 If f(x, Y) is a function such that thesolution to (3) exists and the starting errors aresuch that Eo • max II e, II < ch, c > 0, then the

iconstant stepsize h and constant order k formu-lation of 7 (a, b) converges.

Proof The Lagrangian form of the polynomialP k , n W which interpolates the points, { (xn+ 1_{9 a n d t n e corrected values satisfyf n + 1_.) i = 15 iM k } associated with the pre-dictor is

kP (r} = T T (Y 1 f

i = l

(d-r)

where

62

i - l

Li(x, r)k

\ • nj=r ^

is the LagrangianPERTANIKA VOL.

r - 1

Z h'y^

8 NO.

0 t]

1. 1985

where

CONVERGENCE OF THE DI METHODS

Subtracting (15) from (13), the global error

Jo J o Jo+ sh, 0)

ds ... ds.r times

A a d x d matrix, A =

1 h h2

0 1

2!

h

(d - 1)!

h d - 2

(d - 2)!

- 0

Then using the above notation, (10) and(11) can be written in the matrix form as

Pn+1 = A Y n + I a. (12)

Yn+1 = AYn a* Hf*n + 1

(13)

Since Y(JC n+l) is the t rue solution at xthen

AY(xn)+ 2 «, Hf(Y(*n+,_,)) + r *

n+ {,

n+1

Y(xn+1) = AY(xn) + 2 a*]

+ <*$ Hf(Y(xn+1)) + rn+1 (15)

where r * n + 1 a n d rn+l are vectors for the localtruncation errors of the explicit and implicitmethods respectively.

e satisfiedn+i

Let HT - (hd, hd->, ..., h), ctj a dx ddiagonal matrix, = diagonal (7jO» Y|tn ••-> + r n + i7ii(j-i)>

a j * a d x d diagonal matrix, a!" =diagonal (y*o> y*lt ..., 7*d_!)>

3Y'

i.e.

en+1=Aen+

(Aent

Jn+1

where 5 n + 1 = ( <**Q T + 1 + r n + 1 )is the local truncation error for the combinedpredictor-corrector method and

dfG ( ^ + I _ p , where n + I _ i = —

f _ j. i : is a vector between

_;)and Y n + 1 _ i (

andP n + , , i = 0.

Therefore,

" Y(x )a

1 Y(xn+1)anc

e n + 1 I K II A II l len» + h*L 2 It or* II l e n + 1 _ j l

h * L a * II A II II en II + h* 2 L 2 a*

(14) k( 2 » e n + 1 _ i J I I I I I a , I I ) + 6

where h* = II H II, L = max II G n + , _{ II,

(16)

5 = max II 5 m II, a* = 2 II h, * I ,

m i=0

k

a = 2 Hag II.

PERTANIKA VOL. 8 NO. 1, 1985 63

M.B. SULEIMAN

Here, we introduce {E.}a sequence such 4. CONVERGENCE FOR VARIABLEthat II ej II^Ejforj = 0 , 1, ..., i and for all i =o, 1, . . . , n ( n > k - 1) and Eois given, E o = maxII y (*r) - yr II, r = 0, 1, .... k - 1. Noting

that II A II < 1 + Cjh and h* < c2 h, for some stepsize and order. In this case the variable stepconstants cx > 0, c2 > 0 and sufficiently small h, size is first bounded and then a lemma is stated.

STEP, VARIABLE ORDER METHOD

The proof is very similar to that of constant

then from (17), we have

(17)

Let the variable stepsize hj and order kj, beused on the j - thstep, X j ^ to x-y j = 1, 2, ....

for some constant c3 > 0, where c3 > (cx c2La* i N , such that b - a = 2 hj..Let H* = max hj,c2L2a*a) ).

Define

where R= 1 + Cjh + 2hc2L/** + C3h2

thenllen + 1 IKEn+1and

0,1 a

(18)

<En <E n + 1

The equation in (18), for R and 5 positivegives rise to the relationship

Rn+1 _ ]

R - 1,n = 0, 1,... (20)

Further, since (1 + x)n '< exp (nx), x > 0,using (18), (19) and (20) then (17) becomes

Rn+1

cn+l IKR n + 1

Rn+1-1)<(E

R - 1

5

5, since

R — 1) e x p ( ( n + l ) x ) ,

where x = (cxh + 2hc2La* + c3h2). Notingthat(n + l ) h = x n + 1 - a, the above inequalitybbecomes

e n + 1 U<o

h(c! +2c2La* + c3

exp ( (xn+1 - a) (c, + 2c2La* + c3h) )

Since 6 = 0(hk + 2) and since Eo < ch,therefore II en+ x II -> 0 as h -+ 0.

Therefore the constant stepsize and orderformulation of (7a, b) converges. If EQ is0(hk + 1 ) , then II en+1 II converges with orderk + 1.

D*. K are positive constants such that NH* < D*and k < K for all j . Let the ratio of successivestepsize be bounded,

h.j - i

< 7? for all j and kj > 1. For k = 1,

(19) the above restriction is not necessary.

Lemma

L i ( x n + s h n + 1 , l ) d s . . . d s

0 -* 0 JO

is bounded, where 0 < S.< 1.

M r s ,s k

... n lfl j8(8)ldi

0 J 0 ) Oj=l

s h n + l -

7 u d _ r

where 6.. (s) =

and L4 (xn + shn + 1 , 1) = n 6yi (s);1*1

We have only to show that each term

|0.. (s) I is bounded in order for y{ d _ r t o

bounded. This is done by showing

that

andi + l

64 PERTANIKA VOL. 8 NO. 1, 1985

CONVERGENCE OF THE DI METHODS

are bounded.

n +1

h

-x n + l - j

this leads to

1+1

< 1 + V + rj2 + ... + 7/>-» = (rji

for all j , since for all m = 0, 1,

- 1)

In a similar manner, the denominator of| 8 (s)l is bounded below, the bound depend-

ing on the relative sizes of i and j . For j > i,

"n+l

Similar bounds exist for j < i.

Hence I c r i [s bounded for all r. There-fore II tt| II is bounded. Similarly II a* II isbounded.

Theorem 2 If f(x, y) is a function such that thesolution to (3) exists and the starting errors aresuch that Eo = max le , l < c* H*, i = 0, 1, ...,

i

k ] - 1, then the variable stepsize, variable ordermethod (7a, b) converges.

Proof The remaining argument in proving con-vergence for the variable stepsize, variable ordercase is the same as in Theorem 1. Equation (17)remains true except that H is replaced by H n + x

Let a> 2 llttj II and Ct* > 2 II a*II ,

then we find that

IIen+1 | < (1 + C l H* + 2H*c2L* + c3H*2 ) En

+ 5 ,

IKUVH*(Cl +2c 2 La* + c3H*

exp (D* (c, + 2c2L a* + c3H*) )

Hence the convergence of the variable step-size, variable order method (7a, b).

REFERENCES

DOLD, A. and ECKMAN, B. (1972): Proceedingsof the Conference on the Numerical Solu-tion of Ordinary Differential Equations.19-20 October 1972, The University ofTexas at Austin. Lecture Notes No. 362.Springer-Varlag.

ENRIGHT.W.H., HULL.T.E. and LlNDBERG.B.(1974): Comparing Numerical Methods forSystems of Ordinary Differential Equations.Technical Report No. 69, Department ofComputer Science, University of Toronto.

GEAR, C.W. (1971): Numerical Initial ValueProblems in Ordinary Differential Equa-tions. Prentice Hall.

GEAR, C.W. (1978): The Stability of NumericalMethods for Second Order Ordinary Diffe-rential Equations. J. SIAM Numer. Anal.15(1): 18S-197.

HALL, G and SULEIMAN, M.B. (1981): Stabilityof Adams-Type Formulae for Second OrderOrdinary Differential Equations. IMA J. ofNumer. Anal. 1:427-428.

KROGH, F.T. (1969): A Variable Step VariableOrder Multistep Method for the NumericalSolution of Ordinary Differential Equation.Information Processing 68 — North Hol-land Publishing Company Amsterdam,194-199.

RUSSELL, R.D. and SHAMPINE, L.F. (1972): ACollocation Method for Boundary ValueProblems. Num. Math. 19: 1-28 .

SHAMPINE, L.F. and GORDON, M.K. (1975):Computer Solution of Ordinary DifferentialEquations. Freeman.

PERTANIKA VOL. 8 NO. 1, 1985 65

M.B. S U L E I M A N

SERTH, R.W. (1975): Solution of Stiff Boundary Methods for the Solution of Stiff and Non-Value Problems by Orthogonal Collocation Stiff ODEs. Ph. D. Thesis, University ofInt. J. for Num. Meth. inEng. 9: 691 - 199. Manchester.

SULEIMAN, M.B. (1979): Generalised MultistepAdams and Backward Differentiation Received 24 August, 1984)

66 PERTANIKA VOL. 8 NO. 1, 1985