A class ofA-stable methods

Abstract

A class of methods for the numerical solution of systems of ordinary differential equations is given which—for linear systems—gives solutions which conserve the stability property of the differential equation. The methods are of a quadrature type

$$y_{i,r} = y_{n,r - 1} + h\sum\limits_{k = 1}^n {a_{ik} f(y_{k,r} ), n = 1,2, \ldots ,n, r = 1,2, \ldots ,} y_{n,0} given$$

wherea _ik are quadrature coefficients over the zeros ofP _n −P _n−1 (v=1) orP _n −P _n−2 (v=2), whereP _n is the Legendre polynomial orthogonal on [0,1] and normalized such thatP _m (1)=1. It is shown that

$$\left| {y_{n,r} - y(rh) = 0(h^{2n - _v } )} \right|$$

wherey is the solution of

$$\frac{{dy}}{{dt}} = f(y), t \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 0, y(0) given.$$