Entropy Conservative and Entropy Stable Schemes for Nonconservative Hyperbolic Systems

The vanishing viscosity limit of nonconservative hyperbolic systems depends heavily on the specific form of the viscosity. Numerical approximations, such as the path consistent schemes of [C. Parés, SIAM J. Numer. Anal., 41 (2007), pp. 169–185], may not converge to the physically relevant solutions of the system. We construct entropy stable path consistent (ESPC) schemes to approximate nonconservative hyperbolic systems by combining entropy conservative discretizations with numerical diffusion operators that are based on the underlying viscous operator. Numerical experiments for the coupled Burgers system and the two-layer shallow water equations demonstrating the robustness of ESPC schemes are presented.