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Stability theory for difference approximations of some dispersive shallow water equations and application to capillary fluids
JeanPaul Vila and Pascal Noble (IMT)
In this talk ( Joint work with P. Noble), we study the stability of various difference approximations of the EulerKorteweg system which 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 LaxFriedrichs 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 a new unknown, the gradient of a function of the density: the EulerKorteweg is transformed into a hyperbolic system perturbed by a second order skew symmetric term. We prove entropy stability of LaxFriedrichs type schemes under suitable a CourantFriedrichsLevy condition. We validate our approach numerically on a simple test case and then carry out numerical simulations of a shallow water system with surface tension which models thin films down an incline. In addi tion, we propose a spatial discretization of the EulerKorteweg system seen as a Hamitonian system of PDEs which preserves the Hamiltonian structure: this makes possible the numerical simulation of so called dispersive shock waves of the Euler Korteweg system.

