NONCLASSICAL SHOCKS AND KINETIC RELATIONS - FINITE-DIFFERENCE SCHEMES

We consider hyperbolic systems of conservation laws that are not genui nely nonlinear. The solutions generated by diffusive-dispersive regula rizations may include nonclassical (n.c.) shock waves that do not sati sfy the classical Liu entropy criterion. We investigate the numerical approximation of n.c. shocks via conservative difference schemes const rained only by a single entropy inequality. The schemes are designed b y comparing their equivalent equations with the continuous model and i nclude discretizations of the diffusive and dispersive terms. Limits o f 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 t hat the kinetic relation is a suitable tool for the selection of uniqu e n.c. solutions and for the study of their sensitive dependence on th e critical parameters: the ratios of diffusion/dispersion and diffusio n/mesh size, the shock strength, and the order of discretization of th e flux.