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

TL;DR

This paper explores the necessity of multiple shock wave theories in nonlinear hyperbolic systems, focusing on kinetic functions, equivalent equations, and high-order numerical schemes to better understand phase transitions and regularization effects.

Contribution

It introduces new insights into the numerical approximation of shock waves using high-order schemes and analyzes how kinetic functions relate to regularized models and complex fluid equations.

Findings

Numerical kinetic functions approach analytical ones with higher-order schemes.

Equivalent equations help understand the behavior of numerical schemes.

Kinetic functions are applicable to models like thin liquid films and van der Waals fluids.

Abstract

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.