Bojan Popov, Texas A&M University

KAUST-IAMCS Workshop on Multiscale Modeling, Advanced Discretization Techniques, and Simulation of Wave Propagation

May 7-8, 2011

King Abdullah University of Science and Technology (KAUST)

Thuwal, Kingdom of Saudi Arabia

Entropy Viscosity

This is a joint work with Jean-Luc Guermond, Texas A&M, and Richard Pasquetti, University of Nice.

We introduce a new shock-capturing technique for solving nonlinear conservation laws. The method consists of adding a nonlinear viscosity to the Galerkin formulation of the nonlinear hyperbolic equation or system of equations. The key idea is that the added viscosity is proportional to the residual of the entropy equation and it is always limited from above by first order dissipation. The treatment of the nonlinear viscous term is explicit in time which makes the method very simple to implement for various discretizations: finite elements, spectral elements, and Fourier approximations. We prove that the method is convergent in some simple scalar cases and verify its performance numerically on various one and two dimensional benchmarks including scalar equations and nonlinear systems of conservation laws.