High-Order Schemes, Entropy Inequalities, and Nonclassical Shocks

We are concerned with the approximation of undercompressive, regularizationsensitive, nonclassical solutions of hyperbolic systems of conservation laws by high-order accurate, conservative, and semidiscrete finite difference schemes. Nonclassical shock waves can be generated by diffusive and dispersive terms kept in balance. Particular attention is given here to a class of systems of conservation laws including the scalar equations and the system 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 conservative 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 accurate, 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-order entropy conservative scheme by adding to it a mesh-independent, numerical viscosity, which preserves the order of accuracy of the base scheme. These schemes can only converge to solutions of the system of conservation laws 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 dispersion coefficient. Then, based on a third-order, entropy conservative scheme, we propose a general scheme for the numerical computation of nonclassical shocks. Importantly, our scheme satisfies a cell entropy inequality. Following Hayes and LeFloch [SIAM J. Numer. Anal., 35 (1998), pp. 2169–2194], we determine numerically the kinetic function which uniquely characterizes the dynamics of nonclassical shocks for each regularization of the conservation laws. Our results compare favorably with previous analytical and numerical results. Finally, we prove that there exists no fully discrete and entropy conservative scheme and we investigate the entropy stability of a class of fully discrete, Lax–Wendroff type schemes.