Dissipation-dispersion analysis of fully-discrete implicit discontinuous Galerkin methods and application to stiff hyperbolic problems

TL;DR

This paper analyzes the dissipation and dispersion effects of fully-discrete implicit discontinuous Galerkin methods on hyperbolic systems, highlighting the impact of Runge-Kutta choices and proposing space limiters for improved stability and efficiency.

Contribution

It provides the first dissipation-dispersion analysis for implicit DG methods and introduces a precomputed space limiter approach for stiff hyperbolic problems.

Findings

Implicit Runge-Kutta choice affects solution quality

Optimal space-time discretization identified for high Courant numbers

Precomputed space limiters improve nonlinear scheme performance

Abstract

The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between space discretization, carried out with local polynomials and numerical fluxes at inter-cells, and time-integration to yield the final update. An important concern is how the scheme modifies the solution through the notions of numerical dissipation-dispersion. As far as we know, no analysis of these artifacts has been considered for implicit integration of DG methods. The first part of this work intends to fill this gap, showing that the choice of the implicit Runge-Kutta impacts deeply on the quality of the solution. We analyze one-dimensional dissipation-dispersion to select the best combination of the space-time discretization for high Courant numbers. Then, we apply our findings to the integration of one-dimensional stiff hyperbolic systems. Implicit schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements. High-order schemes require the introduction of local space limiters which make the whole implicit scheme highly nonlinear. To mitigate the numerical complexity, we propose to use appropriate space limiters that can be precomputed on a first-order prediction of the solution. Numerical experiments explore the performance of this technique on scalar equations and systems.