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In this paper, we study the stability of various difference approximations ofthe Euler-Korteweg equations. This system of evolution PDEs is a classicalisentropic Euler system perturbed by a dispersive (third order) term. The Eulerequations are discretized with a classical scheme (e.g. Roe, Rusanov orLax-Friedrichs scheme) whereas the dispersive term is discretized with centeredfinite differences. We first prove that a certain amount of numerical viscosityis needed for a difference scheme to be stable in the Von Neumann sense. Thenwe consider the entropy stability of difference approximations. For thatpurpose, we introduce an additional unknown, the gradient of a function of thedensity. The Euler-Korteweg system is transformed into a hyperbolic systemperturbed by a second order skew symmetric term. We prove entropy stability ofLax-Friedrichs type schemes under a suitable Courant-Friedrichs-Levy condition.We validate our approach numerically on a simple case and then carry outnumerical simulations of a shallow water system with surface tension whichmodels thin films down an incline. In addition, we propose a spatialdiscretization of the Euler-Korteweg system seen as a Hamiltonian system ofevolution PDEs. This spatial discretization preserves the Hamiltonian structureand thus is naturally entropy conservative. This scheme makes possible thenumerical simulation of the dispersive shock waves of the Euler Kortewegsystem.

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