Why many theories of shock waves are necessary. Kinetic functions, equivalent equations, and fourth-order models

We consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena. These models admit under-compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation. Building on earlier work by LeFloch and collaborators, we investigate the numerical approximation of these models by {\sl high-order} finite difference schemes, and uncover several new features of the kinetic function associated with with physically motivated second and third-order regularization terms, especially viscosity and capillarity terms. On one hand, the role of the equivalent equation associated with a finite difference scheme is discussed. We conjecture here and demonstrate numerically that the (numerical) kinetic function associated with a scheme approaches the (analytic) kinetic function associated with the given model --especially since its equivalent equation approaches the regularized model at a higher order. On the other hand, we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized Camassa-Holm model. Finally, we investigate to what extent a kinetic function can be associated with the equations of van der Waals fluids, whose flux-function admits two inflection points.