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

Penalty fluxes are dissipative numerical fluxes for high order discontinuous Galerkin (DG) methods which depend on a penalization parameter (Warburton, 2013; Ye et al., 2016). We investigate the dependence of the spectra of high order DG discretizations on this parameter, and show that as its value increases, the spectrum 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 τ. 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.