Many-Stage Optimal Stabilized Runge–Kutta Methods for Hyperbolic Partial Differential Equations

Abstract

A novel optimization procedure for the generation of stability polynomials of stabilized explicit Runge–Kutta methods is devised. Intended for semidiscretizations of hyperbolic partial differential equations, the herein developed approach allows the optimization of stability polynomials with more than hundred stages. A potential application of these high degree stability polynomials are problems with locally varying characteristic speeds as found for non-uniformly refined meshes and spatially varying wave speeds. To demonstrate the applicability of the stability polynomials we construct 2N-storage many-stage Runge–Kutta methods that match their designed second order of accuracy when applied to a range of linear and nonlinear hyperbolic PDEs with smooth solutions. These methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods.