Open access funding provided by Austrian Science Fund (FWF). This work was funded in part by NSF Grant DMS-1912779 (J.G.) and by Austrian Science Fund (FWF) Grant F65: Taming Complexity in Partial Differential Equations (C.W.).
SN Partial Differential Equations and Applications
Galerkin methods, Runge-Kutta formulas, Space and time, Causality (Physics), Hyperbolic differential equations
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.
Gopalakrishnan, J., Schöberl, J. & Wintersteiger, C. Structure aware Runge–Kutta time stepping for spacetime tents. SN Partial Differ. Equ. Appl. 1, 19 (2020). https://doi.org/10.1007/s42985-020-00020-4