Some Observations on the Interaction Between Linear and Nonlinear Stabilization for Continuous Finite Element Methods Applied to Hyperbolic Conservation Laws

Abstract

We discuss stability and accuracy of stabilized finite element methods for hyperbolic problems. In particular we focus on the interaction between linear and nonlinear stabilizations. First we show that the combination of linear and nonlinear stabilization can be designed to be invariant preserving. Then we show that such a combined method allows for the classical error estimates for smooth solutions of space semidiscretized formulations of the linear transport equation. Based on these ideas we propose a Runge–Kutta finite element method for the compressible Euler equations using entropy viscosity to ensure stability at shocks and gradient jump penalty to prevent propagation of high frequency content into the smooth part of the solution. In a numerical example we show that the method predicts the shock structure accurately, without high frequency pollution of the smooth parts.