2017/11/09 by Pierson T. Guthrey, James A. Rossmanith, Guthrey, Pierson T. +1 · 1 citation
Mathematics · Engineering · #Numerical methods for differential equations #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods
paper · pdf · doi:10.48550/arxiv.1711.03447
Discontinuous Galerkin (DG) methods for hyperbolic partial differential\nequations (PDEs) with explicit time-stepping schemes, such as strong\nstability-preserving Runge-Kutta (SSP-RK), suffer from time-step restrictions\nthat are significantly worse than what a simple Courant-Friedrichs-Lewy (CFL)\nargument requires. In particular, the maximum stable time-step scales inversely\nwith the highest degree in the DG polynomial approximation space and becomes\nprogressively smaller with each added spatial dimension. In this work we\nintroduce a novel approach that we have dubbed the regionally implicit\ndiscontinuous Galerkin (RIDG) method to overcome these small time-step\nrestrictions. The RIDG method is based on an extension of the Lax-Wendroff DG\n(LxW-DG) method, which previously had been shown to be equivalent to a\npredictor-corrector approach, where the predictor is a locally implicit\nspacetime method (i.e., the predictor is something like a block-Jacobi update\nfor a fully implicit spacetime DG method). The corrector is an explicit method\nthat uses the spacetime reconstructed solution from the predictor step. In this\nwork we modify the predictor to include not just local information, but also\nneighboring information. With this modification we show that the stability is\ngreatly enhanced; in particular, we show that we are able to remove the\npolynomial degree dependence of the maximum time-step and show how this extends\nto multiple spatial dimensions. A semi-analytic von Neumann analysis is\npresented to theoretically justify the stability claims. Convergence and\nefficiency studies for linear and nonlinear problems in multiple dimensions are\naccomplished using a MATLAB code that can be freely downloaded.\n