On the penalty stabilization mechanism for upwind discontinuous Galerkin formulations of first order hyperbolic systems
This work provides theoretical insight into the stability and accuracy of high-order DG methods for hyperbolic systems, which is relevant for computational scientists using these methods.
The paper investigates how the penalization parameter in upwind discontinuous Galerkin methods affects the spectral properties of the discretization, showing that increasing the parameter splits the eigenvalues into a convergent set and a damped spurious set, with moderate values damping undamped spurious modes.
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.