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Penalty fluxes are dissipative numerical fluxes for high order discontinuousGalerkin (DG) methods which depend on a penalization parameter. We investigatethe dependence of the spectra of high order DG discretizations on thisparameter, and show that as its value increases, the spectra of the DGdiscretization splits into two disjoint sets of eigenvalues. One set convergesto the eigenvalues of a conforming discretization, while the other setcorresponds to spurious eigenvalues which are damped proportionally to theparameter. Numerical experiments also demonstrate that undamped spurious modespresent in both in the limit of zero and large penalization parameters aredamped for moderate values of the upwind parameter.

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