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Article Dans Une Revue SIAM Journal on Numerical Analysis Année : 2014

Stability Theory for Difference Approximations of Euler--Korteweg Equations and Application to Thin Film Flows

Résumé

We study the stability of various difference approximations of the Euler Korteweg equations. This system of evolutionary 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-Lewy condition. 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. We validate our approach numerically on a shallow water system with surface tension which models thin films.

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Dates et versions

hal-01592282 , version 1 (23-09-2017)

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Pascal Noble, Jean-Paul Vila. Stability Theory for Difference Approximations of Euler--Korteweg Equations and Application to Thin Film Flows. SIAM Journal on Numerical Analysis, 2014, 52 (6), pp.2770-2791. ⟨10.1137/130918009⟩. ⟨hal-01592282⟩
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