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A space-time discontinuous Galerkin finite element method for two-fluid problems

A space-time discontinuous Galerkin finite element method for two fluid flow problems is presented. By using a combination of level set and cut-cell methods the interface between two fluids is tracked in space-time. The movement of the interface in space-time is calculated by solving the level set equation, where the interface geometry is identified with the 0-level set. To enhance the accuracy of the interface approximation the level set function is advected with the interface velocity, which for this purpose is extended into the domain. Close to the interface the mesh is locally refined in such a way that the 0-level set coincides with a set of faces in the mesh. The two fluid flow equations are solved on this refined mesh. The procedure is repeated until both the mesh and the flow solution have converged to a reasonable accuracy. The method is tested on linear advection and Euler shock tube problems involving ideal gas and compressible bubbly magma. Oscillations around the interface are eliminated by choosing a suitable interface flux.