Exponential fitting of matricial multistep methods for ordinary differential equations
Authors:
E. F. Sarkany and W. Liniger
Journal:
Math. Comp. 28 (1974), 10351052
MSC:
Primary 65L05
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 fixedh stable. We give two different algorithmic implementations of the methods in question. The first is based on implicit formulas alone and utilizes the NewtonRaphson method; it is well suited for stiff problems. The second implementation is a predictorcorrector 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:
http://dx.doi.org/10.1090/S00255718197403684411
PII:
S 00255718(1974)03684411
Keywords:
Ordinary differential equations,
matricial multistep methods,
exponential fitting,
unconditional fixedh stability
Article copyright:
© Copyright 1974
American Mathematical Society
