On the penalty stabilization mechanism for upwind discontinuous Galerkin formulations of first order hyperbolic systems

TL;DR

This paper analyzes how penalty parameters affect the spectral properties of high order discontinuous Galerkin methods for hyperbolic systems, showing that increasing penalization separates eigenvalues into conforming and spurious sets, with damping of unwanted modes.

Contribution

It provides a detailed spectral analysis of penalty fluxes in DG methods, revealing how eigenvalues split and how damping of spurious modes depends on the penalization parameter.

Findings

Eigenvalues split into conforming and spurious sets as penalty increases

Spurious eigenvalues are damped proportionally to the penalization parameter

Moderate upwind parameters effectively damp undesired modes

Abstract

Penalty fluxes are dissipative numerical fluxes for high order discontinuous Galerkin (DG) methods which depend on a penalization parameter. We investigate the dependence of the spectra of high order DG discretizations on this parameter, and show that as its value increases, the spectra of the DG discretization splits into two disjoint sets of eigenvalues. One set converges to the eigenvalues of a conforming discretization, while the other set corresponds to spurious eigenvalues which are damped proportionally to the parameter. Numerical experiments also demonstrate that undamped spurious modes present in both in the limit of zero and large penalization parameters are damped for moderate values of the upwind parameter.