Relaxation Schemes for Entropy Dissipative Systems of Viscous Conservation Laws (Communications in Computational Physics, 2025)
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 boththe 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 forlinear convection-diffusion equations. Lax-Wendroff type second-order finite-volumeschemes are developed to discretize the relaxation model. A number of numerical experiments, including viscous compressible flow problems from subsonic to supersonicspeeds, are used to validate the relaxation model and evaluate the performance of thecurrent schemes.
Keywords: Relaxation method; viscous conservation laws; entropy dissipation; Lax-Wendroff type solver; Navier-Stokes equations; generalized Riemann problem