On the Convergence of Space-Time Discontinuous Galerkin Schemes for Scalar Conservation Laws
Provides theoretical justification for a widely used numerical method, removing a restrictive stabilization requirement.
Proved convergence of space-time discontinuous Galerkin schemes for scalar conservation laws to the unique entropy solution for all polynomial orders, without streamline-diffusion stabilization, matching practical usage.
We prove convergence of a class of space-time discontinuous Galerkin schemes for scalar hyperbolic conservation laws. Convergence to the unique entropy solution is shown for all orders of polynomial approximation, provided strictly monotone flux functions and a suitable shock-capturing operator are used. The main improvement, compared to previously published results of similar scope, is that no streamline-diffusion stabilization is used. This is the way discontinuous Galerkin schemes were originally proposed, and are most often used in practice.