Invariant domain preserving high-order spectral discontinuous approximations of hyperbolic systems

TL;DR

This paper introduces a limiting procedure for high-order spectral discontinuous Galerkin schemes that preserves invariant domains in hyperbolic conservation laws, ensuring stability and robustness.

Contribution

It develops a new limiting technique that guarantees invariant domain preservation for high-order spectral discontinuous Galerkin methods applied to hyperbolic systems.

Findings

The scheme maintains invariant domain properties under a specific time step condition.

Numerical experiments demonstrate improved robustness and stability.

The method is applicable to modal and nodal discontinuous Galerkin schemes.

Abstract

We propose a limiting procedure to preserve invariant domains with time explicit discrete high-order spectral discontinuous approximate solutions to hyperbolic systems of conservation laws. Provided the scheme is discretely conservative and satisfy geometric conservation laws at the discrete level, we derive a condition on the time step to guaranty that the cell-averaged approximate solution is a convex combination of states in the invariant domain. These states are then used to define local bounds which are then imposed to the full high-order approximate solution within the cell via an a posteriori scaling limiter. Numerical experiments are then presented with modal and nodal discontinuous Galerkin schemes confirm the robustness and stability enhancement of the present approach.