Development of discontinuous Galerkin methods for hyperbolic systems that preserve a curl or a divergence constraint: the case of linear systems

TL;DR

This paper demonstrates that discontinuous Galerkin methods can preserve differential constraints like divergence and curl in hyperbolic systems at the discrete level, maintaining high accuracy and constraint fidelity.

Contribution

It introduces a framework for using discontinuous Galerkin methods with appropriate approximation spaces to exactly preserve divergence and curl constraints in hyperbolic systems.

Findings

Constraints are preserved at machine precision in numerical tests.

The method maintains high order accuracy.

Discrete divergence and curl are exactly preserved under mild assumptions.

Abstract

Some hyperbolic systems are known to include implicit preservation of differential constraints: these are for example the time conservation of the curl or the divergence of a vector that appear as an implicit constraint. In this article, we show that this kind of constraint can be easily conserved at the discrete level with the classical discontinuous Galerkin method, provided the right approximation space is used for the vectorial space, and under some mild assumption on the numerical flux. For this, we recall a discrete de-Rham framework in which discontinuous approximation spaces for vectors fits. The discrete adjoint divergence and curl are proven to be exactly preserved by the discontinuous Galerkin method under a small assumption on the numerical flux. Numerical tests are performed on the wave system, the two dimensional Maxwell system and the induction equation, and confirm that the differential constraints are preserved at machine precision while keeping the high order of accuracy.