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

DOI:
https://doi.org/10.1090/S0025-5718-1974-0368441-1

MathSciNet review:
0368441

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Abstract: We study a class of explicit or implicit multistep integration formulas for solving 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 provided is the integration step, and belongs to a certain class of polynomials in the independent variable *x*. For arbitrary step number , the coefficients of the formulas are given explicitly as functions of *Q*. The present formulas are generalizations of the Adams methods and of the backward differentiation formulas . 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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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0368441-1

Keywords:
Ordinary differential equations,
matricial multistep methods,
exponential fitting,
unconditional fixed-*h* stability

Article copyright:
© Copyright 1974
American Mathematical Society