On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws
Provides theoretical convergence guarantees for a class of numerical methods used in computational fluid dynamics, addressing a known gap in stability analysis.
This paper proves that a shock capturing discontinuous Galerkin method without streamline diffusion stabilization converges to an entropy measure-valued solution for nonlinear hyperbolic systems, establishing entropy stability and consistency for arbitrary order methods.
In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) 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 [14], our approach is new in that we do not use streamline diffusion (SD) 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 (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.