Exponential Fitting of Matricial Multistep Methods for Ordinary Differential Equations

We study a class of explicit or implicit multistep integration formulas for solving N X 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 + 4>(x, y) provided Q — hD, h is the integration step, and 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 + °°). For arbitrary Q they are fitted exponentially at Q in a matricial sense. The implicit formulas are unconditionally fixed-ft 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.