Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Exponential fitting of matricial multistep methods for ordinary differential equations

Authors: E. F. Sarkany and W. Liniger
Journal: Math. Comp. 28 (1974), 1035-1052
MSC: Primary 65L05
MathSciNet review: 0368441
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Abstract: We study a class of explicit or implicit multistep integration formulas for solving $ N \times N$ systems of ordinary differential equations. The coefficients of these formulas are diagonal matrices of order N, depending on a diagonal matrix of parameters Q of the same order. By definition, the formulas considered here are exact with respect to $ y' = - Dy + \phi (x,y)$ provided $ Q = hD,h$ is the integration step, and $ \phi $ belongs to a certain class of polynomials in the independent variable x. For arbitrary step number $ k \geqslant 1$, the coefficients of the formulas are given explicitly as functions of Q. The present formulas are generalizations of the Adams methods $ (Q = 0)$ and of the backward differentiation formulas $ (Q = + \infty )$. For arbitrary Q they are fitted exponentially at Q in a matricial sense. The implicit formulas are unconditionally fixed-h stable. We give two different algorithmic implementations of the methods in question. The first is based on implicit formulas alone and utilizes the Newton-Raphson method; it is well suited for stiff problems. The second implementation is a predictor-corrector approach. An error analysis is carried out for arbitrarily large Q. Finally, results of numerical test calculations are presented.

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Keywords: Ordinary differential equations, matricial multistep methods, exponential fitting, unconditional fixed-h stability
Article copyright: © Copyright 1974 American Mathematical Society