On the convergence of a shock capturing discontinuous galerkin method for nonlinear hyperbolic systems of conservation laws

In this paper, we present a shock capturing discontinuous Galerkin method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [A. Hiltebrand and S. Mishra, Numer. Math., 126(2014), pp. 103-151], our approach is new in that we do not use streamline diffusion stabilization. It is proved that an artificial viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued solution. The result is valid for general systems and for the arbitrary order discontinuous Galerkin method.

Duke Scholars

Author Georg May DKU Faculty