  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.
