Stability Analysis of Discontinuous Galerkin Discrete Shock Profiles for Steady Scalar Conservation Laws

Abstract

We present a stability analysis of stationary discrete shock profiles for a discontinuous Galerkin method approximating scalar nonlinear hyperbolic conservation laws with a convex flux. We consider the Godunov method to evaluate the numerical flux and use either an explicit third-order Runge--Kutta scheme or an implicit backward Euler scheme for the time integration. Applying a linear stability analysis, we show that the steady solutions may become unstable when the numerical flux is not differentiable. In particular, the situation of a shock at an interface of the mesh corresponds to an unstable solution for a space discretization accuracy higher than second-order, whatever the time integration method and the physical flux. Spectra of the linearized operator indicate that the fourth-order numerical scheme with the explicit time integration is also unstable for a continuous range of shock positions around an interface.