G. Grün, M. Lenz, and M. Rumpf.
A finite volume scheme for surfactant driven thin film flow.
In R. Herbin and D. Kröner, editors, Finite Volumes for
Complex Applications III, pages 567-574. Hermes Penton Sciences, 2002.
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The flow of a thin viscous film under the influence of a surface active agent (surfactant) is described by a system of degenerate parabolic equations. A robust and effective numerical scheme based on a finite volume discretization in space and a suitable operator splitting in time is presented. The convective part, which models the effects of Marangoni forces, is treated by an higher order explicit up-wind scheme with a limited linear reconstruction. For the fourth order parabolic part, which corresponds to the classical thin film problem, we formulate a finite volume scheme that entails the same conservation properties continuous solutions have, i.e. energy and entropy estimates. The scheme and the fundamental estimates are derived in the relevant 2D case. Numerical simulations and the convergence result are currently restricted to 1D.