A class ofA-stable methods

AbstractA 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 $$y_{i,r} = y_{n,r - 1} + h\sum\limits_{k = 1}^n {a_{ik} f(y_{k,r} ), n = 1,2, \ldots ,n, r = 1,2, \ldots ,} y_{n,0} given$$ whereaik are quadrature coefficients over the zeros ofPn −Pn−1 (v=1) orPn −Pn−2 (v=2), wherePn is the Legendre polynomial orthogonal on [0,1] and normalized such thatPm(1)=1. It is shown that $$\left| {y_{n,r} - y(rh) = 0(h^{2n - _v } )} \right|$$ wherey is the solution of $$\frac{{dy}}{{dt}} = f(y), t \mathbin{\lower.3ex\hbox{$\buildrel>\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 0, y(0) given.$$