Stability theory for difference approximations of some dispersive shallow water equations and application to thin film flows

TL;DR

This paper analyzes the stability of difference schemes for the Euler-Korteweg equations, demonstrating the need for numerical viscosity, establishing entropy stability, and applying these methods to shallow water and thin film flow simulations.

Contribution

It introduces a stability analysis for difference approximations of dispersive shallow water equations, including entropy stability conditions and a Hamiltonian-preserving discretization scheme.

Findings

Numerical viscosity is essential for scheme stability.

Entropy stability is achieved under Courant-Friedrichs-Lewy conditions.

Hamiltonian-preserving discretization enables simulation of dispersive shock waves.

Abstract

In this paper, we study the stability of various difference approximations of the Euler-Korteweg equations. This system of evolution PDEs is a classical isentropic Euler system perturbed by a dispersive (third order) term. The Euler equations are discretized with a classical scheme (e.g. Roe, Rusanov or Lax-Friedrichs scheme) whereas the dispersive term is discretized with centered finite differences. We first prove that a certain amount of numerical viscosity is needed for a difference scheme to be stable in the Von Neumann sense. Then we consider the entropy stability of difference approximations. For that purpose, we introduce an additional unknown, the gradient of a function of the density. The Euler-Korteweg system is transformed into a hyperbolic system perturbed by a second order skew symmetric term. We prove entropy stability of Lax-Friedrichs type schemes under a suitable Courant-Friedrichs-Levy condition. We validate our approach numerically on a simple case and then carry out numerical simulations of a shallow water system with surface tension which models thin films down an incline. In addition, we propose a spatial discretization of the Euler-Korteweg system seen as a Hamiltonian system of evolution PDEs. This spatial discretization preserves the Hamiltonian structure and thus is naturally entropy conservative. This scheme makes possible the numerical simulation of the dispersive shock waves of the Euler Korteweg system.