Nonclassical solutions of hyperbolic conservation laws.

Abstract

This dissertation studies the nonclassical shock waves which appears as limits of certain type diffusive-dispersive regularisation to hyperbolic of conservation laws. Such shocks occur very often when the ux function lacks the convexity especially when the initial conditions for Riemann problem belong to different region of convexity. They have negative entropy dissipation. They do not verify the classical Oleinik entropy criterion. The cubic function is taken as a ux function. The existence and uniqueness of such shock waves are studied. They are constructed as limits of traveling-wave solutions for diffusive-dispersive regularisation. A kinetic relation is introduced to choose a unique nonclassical solution to the Riemann problem. The numerical simulations are investigated using a transport-equilibrium scheme to enable computing the nonclassical solution at the discrete level of kinetic function. The method is composed of an equilibrium step containing the kinetic relation at any nonclassical shock and a transport step advancing the discontinuity with time.