MOHAMED SULEIMAN Mathematics Department, Faculty of Science and Environmental Studies, Universiti Pertanian Malaysia, Serdang, Selangor, Malaysia.

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Pertanika 8(1), 59-66(1985) Convergence of the Variable Order and Variable Stepsize Direct Integration Methods for the Solution of the Higher Order Ordinary Differential Equations MOHAMED SULEIMAN

1 Pertanika 8(1), 59-66(1985) Convergence of the Variable Order and Variable Stepsize Direct Integration Methods for the Solution of the Higher Order Ordinary Differential Equations MOHAMED SULEIMAN Mathematics Department, Faculty of Science and Environmental Studies, Universiti Pertanian Malaysia, Serdang, Selangor, Malaysia. Key words: Convergence; higher order ordinary differential equations (ODEs); Direct Integration (DI) methods; predicted and corrected values; integration coefficients; variable order and stepsize. ABSTRAK Kaedah Pengkamilan Terus bagi penyelesaian terus satu sistem persamaan pembezaan biasa peringkat tinggi dibincangkan. Penumpuan bagi kaedah Pengkamilan Terus dengan langkah dan peringkat yang tetap dibuktikan dahulu, sebelum penumpuan bagi kaedah tersebut dengan langkah dan peringkat yang berubah dibuktikan. ABSTRACT The DI methods for directly solving a system of a general higher order ODEs are discussed. The convergence of the constant stepsize and constant order formulation of the DI methods is proven first before the convergence for the variable order and stepsize case. 1. INTRODUCTION Many problems in engineering and science can be formulated in terms of such a system. The general system of higher order initial Some of the higher order ODEs problems found values ODEs is given by in'the literature are on the symmetrical bending of a laterally loaded circular plate, the bending Y ^ 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 with uniform sunction; given in Serth (1975), control in the interval a < x < b, the i-th equa- theory in Enright et al (1974) copi anar c i rcu lar tionisoforderdjand two body motion in shampine and Gordon /ft *\ ^ _ j\ (1975), gravitational n-body problem in Dol et = ( yi n ->y y/~ )> ^ W~0?i, O>» T h> dj 1, *> 7?s>o>» ^s* d _i) # a/(1972) ' The current technique of solving the problem in (1) is by reducing it to a system of first Key to author's name: M.B. Suleiman.

2 M.B. SULEIMAN order equations. Let Integrating (3) d - r times, r = 0, 1,..., d 1 leads to the identity and =y.. (d 2 - (d s - x n + d r times d-l-r i = 0 r y (r+i) (* ) + (4) dx... dx z = and [ z ll> Z 12> "' z l dj' '"> Z sl > z s2>-> z sd s 1 F (x, z) - [z 12, z 13,..., fj,..., z s2, z s3,..,, f s J then (1) reduces to the equivalent first order problem, z'=f(x, z), (2) with z(a) =1? The system in (2) is then solved using either the Adams or the Runge-Kutta class of methods. But the work of Krogh (1969) - and Suleiman (1979) have shown that for many problems, it is more efficient to solve (1) by integrating directly, rather than reducing it to (2). This DI method is in fact a generalization of the Adams-type of methods to higher order ODEs. Hall and Suleiman (1081) and Gear (1978) have studied the stability of the method for solving second order ODEs. Gear (1971) proves the convergence for certain methods which solved (1), those which can be transformed into the Nordsieck's arrays, while Shampine and Gordon (1975) prove the case for the first order problem. 2 THE DI METHODS IN THE PREDICTOR-CORRECTOR MODE For simplicity of description and without loss of generality, the discussion which follows will be for the single equation y< d > = f(x, Y), Y(a) = A (S) Y T = (y, y',..., y ( d ->> ) and A* =(,,, ',...,nm-»>). f is then approximated with an interpolation polynomial P k n (x) of degree k - 1 which interpolates f at the k preceding points (f n, jc n ), (f n _j * n _! ),.», (f n -k + i.*n-k+i ) Thus expressed in terms of divided differences, ( x " ^n-., n-i] (X - X n )... In, n-1,..., n-k+1 ] x [n,n- 1,... n-i+1 1 -f, Mn- 1, n-2,..., n-i] *n -*n-i Hence a discrete replacement for (4) is i = o i! P k> Jx) dx... dx (5) p n^+r J are predicted values associated with totally know back values. Define, the integration coefficients g to be the t fold integral r*n+i r x r x git = - J x n Jx n J x n *- t times -* (x - x n _ i+1 ) dx dx... dx (6) 60 PERTANIKA VOL. 8 NO. 1, 1985

CONVERGENCE OF THE DI METHODS Then the predicted values in (5) can be written as >v t i N. 0 1 2... k + d + I k-1-0 0 1 2 goo Bio 20 goi gll fel S02 8ta 22... O,k + d + l.

3 CONVERGENCE OF THE DI METHODS Then the predicted values in (5) can be written as >v t i N k + d + I k goo Bio 20 goi gll fel S02 8ta O,k + d + l... 8i,k + d 82, k+d-1 gi,d-r f [n, n-1,..., n-i]. (7 a ) The corrector formula is obtained by approximating f in (3) by the k degree polynomial P k+1 n + 1 (JC) interpolating the (k + 1) points v n+l> X n+1^' ^n> *n)>» ^n-k+j» ^n-k+l)' ^*n+i - f(*n+i> ^*n+r' In terms of P n > Pn> Pn n k + 1 8k+i, D From (8) and (9) it is observed that g Qt = i t- i ^se t0 trie relation (x - x n _ t )... (x - * n _ k+1 ) f*. n + 1 n n_k+1 ] > the last divided difference is obtained using the f evaluated by the predicted values P* n+1. Replacing P k+1> n + 1 (jc)in (4) and integrating, we have the corrected values Cd-r) _ (d-r) *n+l " Pn+1 + 8k,r l [n+1, n,..., n-k+l]> r = 1, 2,..., d (7b) Formulae (7a, b) represent the k-step predictor-corrector mode of the DI methods. In the variable order implementation of the method, k r if changed, is usually either decreasing or increasing by one from the value at the previous step. From the definition of g it in (6) it can be shown by integrating by parts that 8i,t = Sot For the case i = 0, l-l, t dx dx...dx = h n+1 = (x n + l - x n ). (9) These integration coefficients are generated in a triangular array of the form shown below starting with the first row and from left to right with k + d + 2 coefficients along the first row and (k + 2) along the first column. and hence the computation of the first row is avoided. Constant stepsize will further account for the saving of computation, since if the stepsize remains constant for R steps, and the coefficients of the R-th row depend only on the current stepsize, therefore the coefficients the first R rows need not be recomputed and computation need only start at row R + 1. A change in order corresponds to adding or dropping an interpolation point. If at the current point to be computed, the step is not changed for R + 1 steps and the order is reduced, no special action is required. If however, the order is raised, one additional term needs to be added to the end table for each of the first R rows of the intergration coefficients, before commencing with computation of the row R + 1. Despite the simplicity and advantages in speed, storage and accuracy of the DI methods, they are yet to be popularised as a general purpose code for the solution of the general order ODEs. 3. CONVERGENCE OF THE PREDICTOR-CORRECTOR MODE OF THE DI METHODS Before the proof of convergence of the variable stepsize and order method in (7a, b) is given, the proof for the constant stepsize h and order k formulation of the method is first given. PERTANIKA VOL. 8 NO. 1,

$MB. SULEIMAN Definition polynomial interpolating the set of values A linear k-step method is said to be conver- This form Q{ ^ interpolation p O l ynom ial is used gent if for any function f(x, Y),$

4 MB. SULEIMAN Definition polynomial interpolating the set of values A linear k-step method is said to be conver- This form Q{ ^ interpolation p O l ynom ial is used gent if for any function f(x, Y), for which the here for convenience of proving the convergence constants Tj-Cm), o m = 0, 1,... d-- h-0 nh=jc-a s = 0, 1,..., d - 1; n = (x - a)/h for any xe.[a, b], with starting values y^m^ = r/ m \h) satisfying the con- of our theorem. The constant stepsize and order formulation of the predictor in (7 a) is (d-r) (d-r). 1 (d-r+l) (d-r+1). h*" 1 (<J-1) Pn + 1 = Vn +h y n + - +^I^yn + h r 2 7i, d-r fn+l-i. r = 1,..., d (10) the coefficients dition lim = 17 0 (m) M = 0, 1,... k-1; m = 0, 1,..., d - 1. r times i (x, 1) dx... dbc Let e n be the global error between the true solution Y(x n ) and the computed solution Y n, i.e., T T T e n = Y (x n ) - Y n are constants independent of h. To see this, put x = sh + x n then s k-1 C, n (s + j) ds...ds o j=o (d-l) C) (d-1) We now quote the following theorem. Theorem 1 If f(x, Y) is a function such that the solution to (3) exists and the starting errors are such that E o max I e, I < ch, c > 0, then the i constant stepsize h and constant order k formulation of 7 (a, b) converges. and c x are constants. The constant stepsize, Lagrangian form of the corrector polynomial is given by P k+1.n,lw = Proof The Lagrangian form of the polynomial P k, n W which interpolates the points, { (x n+ 1 _ {9 f n+1 _.)i = 1 5 im k } associated with the predictor is k P (r} = T T (Y 1^f 62 i-l k Li(x, r) \ n j=r ^ is the Lagrangian an d tne corrected values satisfy (d-r) r-1 Z h'y^ 0 t] PERTANIKA VOL. 8 NO i=l

5 CONVERGENCE OF THE DI METHODS Jo J o r times Jo + sh, 0) ds... ds. Subtracting (15) from (13), the global error e satisfied n+i Let H T - (h d, h d ->,..., h), ctj a dx d diagonal matrix, = diagonal (7jO» Y t n -> 7 ii( j-i)> a j* a dx d diagonal matrix, a!" = diagonal (y* o> y* lt..., 7* d _!)> A a d x d matrix, A = +r n+i i.e. e n+1 =Ae n + 3Y' 1 h h 2 2! 0 1 h (d - 1)! h d - 2 (d - 2)! J n+1 (Ae nt - 0 Then using the above notation, (10) and (11) can be written in the matrix form as P n+1 =AY n+ I a. (12) Y n+1 = AY n a* Hf* n+1 (13) 5 n+1 = ( <** Q T +1 + r n+1 ) is the local truncation error for the combined predictor-corrector method and df G(^+I _p, n+i_ i = f _ j. i : is a vector between " Y(x _ ; )and Y n+1 _ i( 1 Y(x n+1 )anc andp n+,,i = 0. Therefore, Since Y(JC n+l) is the true solution at x then AY(x n )+ 2 «, Hf(Y(* n+,_,)) +r* n+1 n+ {, e n+1 IK I A I lle n» + h*l 2 h*la* I A I I e n I + h* 2 L 2 a* (14) k ( 2»e n+1 _ijiii IIa, II)+6 It or* I le n+1 _jl (16) Y(x n+1 ) = AY(x n ) + 2 a*] h* = I H II, L = max I G n+, _ { II, + <*$ Hf(Y(x n+1 )) + r n+1 (15) r* n+1 and r n+l are vectors for the local truncation errors of the explicit and implicit methods respectively. 5 = max II 5 m II, a* = 2 I h, * I, m i=0 k a= 2 Hag II. PERTANIKA VOL. 8 NO. 1,

$M.B. SULEIMAN Here, we introduce {E.}a sequence such 4. CONVERGENCE FOR VARIABLE that I ej II^Ejforj =0, 1,..., i and for all i = o, 1,.$

6 M.B. SULEIMAN Here, we introduce {E.}a sequence such 4. CONVERGENCE FOR VARIABLE that I ej II^Ejforj =0, 1,..., i and for all i = o, 1,...,n(n>k - 1) and E o is given, E o = max STEP, VARIABLE ORDER METHOD I y (* r ) - y r II, r = 0, 1,... k - 1. Noting that I A I < 1 + Cjh and h* < c 2 h, for some The proof is very similar to that of constant stepsize and order. In this case the variable step constants c x > 0, c 2 > 0 and sufficiently small h, size is first bounded and then a lemma is stated. then from (17), we have Let the variable stepsize hj and order kj, be used on the j - thstep, Xj^ to x- y j = 1, 2,... (17) for some constant c 3 > 0, c 3 > (c x c 2 La* i c 2 L 2 a*a) ). Define R= 1 + Cjh + 2hc 2 L/** + C 3 h 2 thenlle n + 1 IKE n+1 and 0,1 a <E n <E n+1 (18) The equation in (18), for R and 5 positive gives 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 c n+l IKR n+1 R n+1-1)<(e R n+1 R , since R 1 )exp((n+l)x), N, such that b - a = 2 hj..let H* = max hj, D*. K are positive constants such that NH* < D* and k < K for all j. Let the ratio of successive stepsize be bounded, h. j-i < 7? for all j and kj > 1. For k = 1, (19) the above restriction is not necessary. Lemma 0 -* 0 JO is bounded, 0 < S.< 1. 7ud_ r 6.. (s) = L i (x n + sh n+1,l)ds...ds M r s,s k... n lfl j8 (8)ldi 0 J 0 ) Oj=l sh n + l - x = (c x h + 2hc 2 La* + c 3 h 2 ). Noting that(n +l)h = x n a, the above inequality becomes e n+1 U< o h(c! +2c 2 La* + c 3 exp ( (x n+1 - a) (c, + 2c 2 La* + c 3 h) ) Since 6 = 0(h k + 2 ) and since E o < ch, therefore I e n+ x I -> 0 as h Therefore the constant stepsize and order formulation of (7a, b) converges. If E Q is 0(h k+1 ), then I e n+1 I converges with order k+ 1. and L 4 (x n + sh n+1, 1) = n 6 yi (s); 1*1 We have only to show that each term 0.. (s) I is bounded in order for y { d _ r to bounded. This is done by showing that and i+l 64 PERTANIKA VOL. 8 NO. 1, 1985

CONVERGENCE OF THE DI METHODS are bounded. this leads to n +1 -x n+l -j h 1+1 IKUV H*( Cl +2c 2 La* + c 3 H* < 1+ V + rj 2 +.

7 CONVERGENCE OF THE DI METHODS are bounded. this leads to n +1 -x n+l -j h 1+1 IKUV H*( Cl +2c 2 La* + c 3 H* < 1+ V + rj />-» = (rji for all j, since for all m = 0, 1, - 1) exp (D* (c, + 2c 2 L a* + c 3 H*) ) Hence the convergence of the variable stepsize, variable order method (7a, b). REFERENCES In a similar manner, the denominator of 8 (s)l is bounded below, the bound depending on the relative sizes of i and j. For j > i, Similar bounds exist for j < i. "n+l Hence I c^ r i [ s bounded for all r. Therefore I tt I is bounded. Similarly I a* I is bounded. Theorem 2 If f(x, y) is a function such that the solution to (3) exists and the starting errors are such that E o = max le, l< c* H*, i = 0, 1,..., i k ] - 1, then the variable stepsize, variable order method (7a, b) converges. Proof The remaining argument in proving convergence for the variable stepsize, variable order case 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 I a*ii, then we find that IIe n+1 < (1 + ClH* + 2H*c 2 L* + c 3 H* 2 + 5, ) E n DOLD, A. and ECKMAN, B. (1972): Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations October 1972, The University of Texas at Austin. Lecture Notes No Springer-Varlag. ENRIGHT.W.H., HULL.T.E. and LlNDBERG.B. (1974): Comparing Numerical Methods for Systems of Ordinary Differential Equations. Technical Report No. 69, Department of Computer Science, University of Toronto. GEAR, C.W. (1971): Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall. GEAR, C.W. (1978): The Stability of Numerical Methods for Second Order Ordinary Differential Equations. J. SIAM Numer. Anal. 15(1): 18S-197. HALL, G and SULEIMAN, M.B. (1981): Stability of Adams-Type Formulae for Second Order Ordinary Differential Equations. IMA J. of Numer. Anal. 1: KROGH, F.T. (1969): A Variable Step Variable Order Multistep Method for the Numerical Solution of Ordinary Differential Equation. Information Processing 68 North Holland Publishing Company Amsterdam, RUSSELL, R.D. and SHAMPINE, L.F. (1972): A Collocation Method for Boundary Value Problems. Num. Math. 19: SHAMPINE, L.F. and GORDON, M.K. (1975): Computer Solution of Ordinary Differential Equations. Freeman. PERTANIKA VOL. 8 NO. 1,

SERTH, R.W. (1975): Solution of Stiff Boundary Value Problems by Orthogonal Collocation Int. J. for Num. Meth. ineng. 9: 691-199. M.B. SULEIMAN Methods for the Solution of Stiff and Non- Stiff ODEs.

8 SERTH, R.W. (1975): Solution of Stiff Boundary Value Problems by Orthogonal Collocation Int. J. for Num. Meth. ineng. 9: M.B. SULEIMAN Methods for the Solution of Stiff and Non- Stiff ODEs. Ph. D. Thesis, University of Manchester. SULEIMAN, M.B. (1979): Generalised Multistep Adams and Backward Differentiation Received 24 August, 1984) 66 PERTANIKA VOL. 8 NO. 1, 1985

Research Article Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations

International Mathematics and Mathematical Sciences Volume 212, Article ID 767328, 8 pages doi:1.1155/212/767328 Research Article Diagonally Implicit Block Backward Differentiation Formulas for Solving