Stabilizing Discontinuous Galerkin Methods Using Dafermos’ Entropy Rate Criterion: I—One-Dimensional Conservation Laws

Abstract

AbstractA novel approach for the stabilization of the discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. The approach is centered around the efficient solution of linear or nonlinear optimization problems in every timestep as a correction to the basic discontinuous Galerkin scheme. The thereby enforced Dafermos criterion results in improved stability compared to the basic method while retaining a high order of accuracy in numerical experiments for scalar conservation laws. Further modification of the optimization problem allows also to enforce classical entropy inequalities for the scheme. The proposed stabilization is therefore an alternative to flux-differencing to enforce entropy inequalities. As the shock-capturing abilities of the scheme are also enhanced is the method also an alternative to finite-volume subcells, artificial viscosity, modal filtering, and other shock capturing procedures in one space dimension. Tests are carried out for Burgers’ equation.