Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems (Q2845595)

The authors consider numerical methods for solving problems for first-order quasi-linear hyperbolic systems with nonlinear or linearly degenerated characteristics fields. In the case of the non-conservative type of equations, oscillations and shocks may occur, which lead to a determination of weak solutions. In the introduction, existing finite difference schemes for solving the considered non-conservative systems are presented to clarify the difficulties associated with the choice of the underlying viscosity operator and other aspects, in order to avoid possible non-convergence of the difference schemes to the physically relevant solutions. To overcome these problems, the so,called entropy stable path consistent (ESPC) schemes are constructed taking into account the appropriate numerical viscosity term to be entropy stable. The ESPC schemes are applied for solving numerically the coupled Burgers non-conservative system and the two-layer shallow water problem. A comparison with other numerical methods is also provided.