Improving the Stability of Predictor-Corrector Methods by Residue Smoothing

Residue smoothing is usually applied in order to acceleratethe convergence of iteration processes. Here, we show that residuesmoothing can also be used in order to increase the stabilityregion of predictor-corrector methods. We shall concentrateon increasing the real stability boundary. The iteration parametersand the smoothing operators are chosen such that the stabilityboundary becomes as large as c(m, q)m²4^g where m is the numberof right-hand side evaluations per step, q the number of smoothingoperations applied to each right-hand side evaluation, and c(m,q) a slowly varying function of m and q, of magnitude 1.3 ina typical case. Numerical results show that, for a variety oflinear and nonlinear parabolic equations in one and two spatialdimensions, these smoothed predictor-corrector methods are atleast competitive with conventional implicit methods.