New PDF release: Proceedings of the Conference on the Numerical Solution of

By Josef Stoer (auth.), Dale G. Bettis (eds.)

ISBN-10: 3540066020

ISBN-13: 9783540066026

ISBN-10: 3540379118

ISBN-13: 9783540379119

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Extra resources for Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations: 19,20 October 1972, The University of Texas at Austin

Sample text

E v e n w h e n the stepsize is changed on every step, the n e w algorithm c o m p u t e s the required coefficients m o r e efficiently than the algorithms in ~Z] - [7]. *This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract N A S 7 - 1 0 0 , sponsored by the National Aeronautics and Space Administration. 23 However, if there are m o r e than a p p r o x i m a t e l y 2 first order stiff equations a n d the stepsize is c h a n g e d on e v e r y step, then the overall a l g o r i t h m is slightly less efficient than a carefully organized L a g r a n g i a n formulation.

H(1) [9(I-2)(£) + i{(z)[~(I-l)(~)+ hZ(S)]]. y(i-z)(~)~ Q(i-z)(~)+ P6Z. If(i){9(l-i)(z)+ hZ(Z)}. y(I-1)(Z) ,- ~(I-1)(Jt) P63. P64. [Test if m o r e equations to be processed. 3 P65. [Set ~ to indicate that %0 is not updated. ] P66. [ E n d of predicting and updating diff. 5. + hE(l). ~- ~+1; if ~ ~ n , go to P5, e Exit. Interpolation to Off-Step Points A significant advantage of rnultistep methods over one-step methods is that a multistep m e t h o d has sufficient information stored to enable one to get the solution at any point passed during the integration: with the s a m e accuracy as is obtained at the end of the individual steps, without interfering with the integration process in any way, and without requiring any additional derivative 52 evaluations.

H n + z _ i then the m o d i f i e d divided difference q0i(n+l ) is equivalent to the i-th b a c k w a r d w at t = tn+ I. of q~i(n). T h e s e ~q ~ 0. L a t e r w e shall find it useful to estimate estimates difference q0i(n+l ) f r o m of the values are obtained using equation (Z. 6) i=q- I, q-Z ..... O. 26 Clearly ~i(n+l) = ~(ie)

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Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations: 19,20 October 1972, The University of Texas at Austin by Josef Stoer (auth.), Dale G. Bettis (eds.)


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