SOME NEW METHODS FOR STIFF DIFFERENTIAL-EQUATIONS

A numerical method applied to the differential equation y’ = Xy can lead to a solution of the form y(x0 + nh) = Az”, where z and w = Ah satisfy an equation P(w,z) = 0. In general P(w,z) is a polynomial in both w and z, of degree M in w and N in z. Existing multistep and Runge-Kutta methods correspond to the cases M — 1 and N = 1 respectively. New methods are found by taking M2, N2z2. The approach here is first to find a suitable polynomial P(w,z) with the desired stability properties, and then to find a process which leads to this polynomial. Third- and fourth-order A- and L-stable processes are given of the semi-explicit and linearly implicit types. © 1979, Taylor & Francis Group, LLC. All rights reserved.