An analysis of discontinuous Galerkin methods for the compressible Euler equations: entropy and $$L_2$$ L 2 stability

Abstract

The objective of this article is to characterize the entropy and $$L_2$$ stability of several representative discontinuous Galerkin (DG) methods for solving the compressible Euler equations. Towards this end, three DG methods are constructed: one DG method with entropy variables as its unknowns, and two DG methods with conservative variables as their unknowns. These methods are employed in order to discretize the compressible Euler equations in space. Thereafter, the resulting semi-discrete formulations are analyzed, and the entropy and $$L_2$$ stability characteristics are evaluated. It is shown that the semi-discrete formulation of the DG method with entropy variables is entropy and $$L_2$$ stable. Furthermore, it is shown that the semi-discrete formulations of the DG methods with conservative variables are only guaranteed to be entropy and $$L_2$$ stable under the following assumptions: the entropy projection errors vanish, or the terms containing the entropy projection errors are non-positive. Thereafter, the semi-discrete formulation with entropy variables, and one of the semi-discrete formulations with conservative variables, are discretized in time with an ‘algebraically stable’ Runge–Kutta (RK) scheme. The resulting formulations are fully-discrete and can be immediately applied to practical problems. In this article, they are employed to simulate a vortex propagating for long distances. It is shown that temporal stability is maintained by the DG method with entropy variables, but the DG method with conservative variables exhibits instability.