Improving the Stability of Predictor-Corrector Methods by Residue Smoothing

Abstract

Residue smoothing is usually applied in order to accelerate the convergence of iteration processes. Here, we show that residue smoothing can also be used in order to increase the stability region of predictor-corrector methods. We shall concentrate on increasing the real stability boundary. The iteration parameters and the smoothing operators are chosen such that the stability boundary becomes as large as c(m, q)m24g where m is the number of right-hand side evaluations per step, q the number of smoothing operations applied to each right-hand side evaluation, and c(m, q) a slowly varying function of m and q, of magnitude 1.3 in a typical case. Numerical results show that, for a variety of linear and nonlinear parabolic equations in one and two spatial dimensions, these smoothed predictor-corrector methods are at least competitive with conventional implicit methods.