Exponential Fitting of Matricial Multistep Methods for Ordinary Differential Equations

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.