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.
Published in Springer Journal of Scientific Computing
Authors: Daniel Doehring, Gregor Gassner, Manuel Torrilhon
arXiv:arXiv:2402.12140