Abstract
We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.
Highlights
For simulating wave phenomena, the state of the art relies heavily on efficient and accurate numerical solution techniques for hyperbolic systems
We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions
We report on the discrete stability properties of these methods applied to linear hyperbolic equations
Summary
The state of the art relies heavily on efficient and accurate numerical solution techniques for hyperbolic systems. This paper is concerned with those solution techniques that proceed by subdividing the spacetime into tentshaped subregions satisfying a causality condition. By constraining the height of the tent pole, erected vertically in an increasing time direction, one can ensure that the tent encloses the domain of dependence of all its points. This constraint on the tent pole height is a causality condition that a numerical scheme using such tents should satisfy. The spacetime subdivision into tents may be unstructured, allowing such schemes to advance in time by different amounts at. This article is part of the topical collection ‘‘Waves 2019 – invited papers’’ edited by Manfred Kaltenbacher and Markus Melenk
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