Shock capturing with entropy-based artificial viscosity for staggered grid discontinuous spectral element method

Abstract

This paper presents a shock capturing technique for a staggered grid discontinuous spectral element method (DSEM), which adds localized and smooth artificial viscosity to systems of nonlinear conservation laws. The artificial diffusivity model used in this work is a modified form of the entropy viscosity (EV) presented by Guermond et al. (2011). We extend the application of this method to high-order discontinuous schemes for the simulation of high speed flows with discontinuities on staggered grids. Direct implementation of the entropy viscosity method in DSEM leads to a non-smooth artificial viscosity, which in turn leads to oscillations and instability of the solution. To smoothen the artificial viscosity, the EV method is coupled with a spectral filter and an interface treatment technique. The resulting artificial viscosity is locally large near discontinuities and transitions smoothly to zero in smooth flow regions. The method enables using elements with orders higher than unity while avoiding adaptive mesh refinement and preserving the locality and compactness of the discontinuous Galerkin (DG) scheme. The method is implemented for the inviscid compressible Euler equations in two space dimensions and its effectiveness is demonstrated through its application to a series of benchmark problems.