Relaxation Schemes for Entropy Dissipative Systems of Viscous Conservation Laws

Abstract

In this paper, a hyperbolic relaxation model is designed for a class of entropy dissipative systems of viscous conservation laws, such as the 1-D viscous Burgers and 2-D Navier-Stokes equations. An artificial variable is introduced to relax both the convective and viscous fluxes together. Based on the entropy dissipative property of the original system, a dissipation condition is proposed for the resulting relaxation model, and used to prove the entropy inequality of the relaxation model for linear convection-diffusion equations. Lax-Wendroff type second-order finite-volume schemes are developed to discretize the relaxation model. A number of numerical experiments, including viscous compressible flow problems from subsonic to supersonic speeds, are used to validate the relaxation model and evaluate the performance of the current schemes.