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

The application of discontinuous Galerkin (DG) schemes to hyperbolic systems of conservation laws requires a careful interplay between the space discretization, carried out with local polynomials and numerical fluxes at inter-cells, and a high-order time-integration to yield the final update. An important concern is how the scheme modifies the solution through the notions of numerical diffusion and dispersion. The analysis of these artifacts is well known for finite volume schemes, but it becomes more complex in the DG case. In particular, as far as we know, no analysis of this type has been considered for implicit integration with DG space discretization. The first part of this work intends to fill this gap, showing that the choice of the implicit Runge-Kutta scheme impacts deeply on the quality of the solution. We analyze dispersion-diffusion properties to select the best combination of the space-time discretization for high Courant numbers. In the second part of this work, we apply our findings to the integration of stiff hyperbolic systems with DG schemes. Implicit time-integration schemes leverage superior stability properties enabling the selection of time-steps based solely on accuracy requirements, thereby bypassing the need for minute time-steps. High-order schemes require the introduction of local space limiters which make the whole implicit scheme highly nonlinear. To mitigate the numerical complexity of these schemes, we propose to use appropriate space limiters that can be precomputed on a first-order prediction of the solution. This approach follows the methodology proposed by Puppo et al. (Commun. Comput. Phys., 2024) for high-order finite volume schemes. Numerical experiments explore the performance of this technique on scalar equations and systems.