Stationarity preservation and the low Mach number behaviour of the Discontinuous Galerkin method on Cartesian grids

TL;DR

This paper investigates how the Discontinuous Galerkin method preserves stationary states and behaves at low Mach numbers for linear acoustics, revealing conditions for stationarity preservation and accuracy reduction.

Contribution

It extends previous studies by analyzing the stationarity preservation of DG methods on Cartesian grids and their low Mach number behavior both theoretically and experimentally.

Findings

DG is stationarity preserving above a polynomial degree threshold

Choice of numerical flux affects accuracy at stationary states

Explains low Mach number behavior of DG for Euler equations

Abstract

Due to added numerical stabilization (diffusion), the stationary states of numerical methods for hyperbolic problems need not be consistent discretizations of those of the PDEs. A closely related phenomenon is the lack of consistency of common finite volume methods for the Euler equations in the limit of low Mach number. In this work, the stationary states of the Discontinuous Galerkin (DG) method for linear acoustics on Cartesian grids are explored theoretically and experimentally, thus extending previous studies in the context of first-order finite difference methods. It is found that for a polynomial degree above some threshold, DG is stationarity preserving, but depending on the choice of numerical flux can suffer from a reduction of the order of accuracy at stationary state. This allows to explain the behaviour of the method for the Euler equations at low Mach number.