I³MS - Rossmanith Seminar
Prof. Dr. James Rossmanith - The Regionally-Implicit Discontinuous Galerkin Method: Improving the Stability of High-Order DG-FEM
Department of Mathematics,
Iowa State University, USA
Discontinuous Galerkin (DG) methods for hyperbolic partial differential equations (PDEs) with explicit time-stepping schemes such as strong stability-preserving Runge-Kutta (SSP-RK) suffer from time-step restrictions that are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL) argument requires. In particular, the maximum stable time-step scales inversely with the highest degree in the DG polynomial approximation space and becomes progressively smaller with each added spatial dimension. In this work we introduce a novel approach that we have dubbed the regionally implicit discontinuous Galerkin method (RIDG) to overcome these small time-step restrictions. The RIDG is method is based on an extension of the Lax-Wendroff DG (LxW-DG) method, which previously had been shown to be equivalent (for linear constant coefficient problems) to a predictor-corrector approach, where the predictor is a locally implicit space-time method (i.e., the predictor is something like a block-Jacobi update for a fully implicit space-time DG method). The corrector is an explicit method that uses the space-time reconstructed solution from the predictor step. In this work we modify the predictor to include not just local information, but also neighboring information. With this modification we show that the stability is greatly enhanced; in particular, we show that we are able to remove the polynomial degree dependence of the maximum time-step and show vastly improved time-steps in multiple spatial dimensions. A semi-analytic von Neumann analysis is presented and several tests are shown to verify the efficiency of the proposed scheme.
This work is joint with Pierson Guthrey.