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arXiv · 2402.12140

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

Daniel Doehring·Gregor J. Gassner·Manuel Torrilhon
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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 in non-uniformly refined meshes and different 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. The methods are constructed to reduce the amplification of round off errors which becomes a significant concern for these many-stage methods.

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Daniel Doehring, Gregor J. Gassner, Manuel Torrilhon. 2024-03-18. Many-Stage Optimal Stabilized Runge-Kutta Methods for Hyperbolic Partial Differential Equations. https://doi.org/10.1007/s10915-024-02478-5

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