Nonclassical Shocks and Kinetic Relations: Finite Difference Schemes

Abstract

We consider hyperbolic systems of conservation laws that are not genuinely nonlinear. The solutions generated by diffusive-dispersive regularizations may include nonclassical (n.c.) shock waves that do not satisfy the classical Liu entropy criterion. We investigate the numerical approximation of n.c. shocks via conservative difference schemes constrained only by a single entropy inequality.The schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms. Limits of these schemes are characterized via the kinetic relation introduced earlier by the authors. We determine the kinetic function numerically for several examples of systems and schemes. This study demonstrates that the kinetic relation is a suitable tool for the selection of unique n.c. solutions and for the study of their sensitive dependence on the critical parameters: the ratios of diffusion/dispersion and diffusion/mesh size, the shock strength, and the order of discretization of the flux.

Locations

• SIAM Journal on Numerical Analysis - View
• CiteSeer X (The Pennsylvania State University) - View - PDF