High-order schemes, entropy inequalities, and nonclassical shocks

We are concerned with the approximation of undercompressive, regularization -sensitive, nonclassical solutions of hyperbolic systems of conservation la ws by high-order accurate, conservative, and semidiscrete finite difference schemes. Nonclassical shock waves can be generated by diffusive and disper sive terms kept in balance. Particular attention is given here to a class o f systems of conservation laws including the scalar equations and the syste m of nonlinear elasticity and to linear diffusion and dispersion in either the conservative or the entropy variables. First, we investigate the existence and the properties of entropy conservat ive schemes a notion due to Tadmor [Math. Comp., 49 (1987), pp. 91-103]. In particular we exhibit a new five-point scheme which is third-order accurat e, at least. Second, we study a class of entropy stable and high-order accurate schemes satisfying a single cell entropy inequality. They are built from any high-o rder entropy conservative scheme by adding to it a mesh-independent, numeri cal viscosity, which preserves the order of accuracy of the base scheme. Th ese schemes can only converge to solutions of the system of conservation la ws satisfying the entropy inequality. These entropy stable schemes exhibit mild oscillations near shocks and, interestingly, may converge to classical or nonclassical entropy solutions, depending on the sign of their dispersi on coefficient. Then, based on a third-order, entropy conservative scheme, we propose a gen eral scheme for the numerical computation of nonclassical shocks. Important ly, our scheme satis es a cell entropy inequality. Following Hayes and LeFl och [SIAM J. Numer. Anal., 35 (1998), pp. 2169-2194], we determine numerica lly the kinetic function which uniquely characterizes the dynamics of noncl assical shocks for each regularization of the conservation laws. Our result s compare favorably with previous analytical and numerical results. Finally, we prove that there exists no fully discrete and entropy conservat ive scheme and we investigate the entropy stability of a class of fully dis crete, Lax-Wendroff type schemes.