Three new ideas for anisotropic adaption of unstructured triangular grids are presented, with particular emphasis on fluid flow computations. © 1997 John Wiley & Sons, Ltd.

In this work, a characterization of the hyperbolicity region for the two layer shallow-water system is proposed and checked. Next, some path-conservative finite volume schemes (see [11]) that can be used even if the system is not hyperbolic are presented, but they are not in general L 2 linearly stable in that case. Then, we introduce a simple but efficient strategy to enforce the hyperbolicity of the two-layer shallow-water system consisting in adding to the system an extra amount of friction at every cell in which complex eigenvalues are detected at a given time step. The implementation is performed by a predictor/corrector strategy: first a numerical scheme is applied to the unmodified two-layer system, regardless of the hyperbolic character of the system. Next, we check if the predicted cell averages are in the hyperbolic region or not. If not, the mass-fluxes are corrected by adding a quadratic friction law between layers whose coefficient is computed so that the corrected cell average is as near as possible of the boundary of the hyperbolicity region. Finally, some numerical test have been performed to assess the efficiency of the proposed strategy.Short title : Numerical treatment of the loss of hyperbolicity of the two-layer SWS.

In this work, we consider a multilayer shallow water model with variable density. It consists of a system of hyperbolic equations with non-conservative products that takes into account the pressure variations due to density fluctuations in a stratified fluid. A second-order finite volume method that combines a hydrostatic reconstruction technique with a MUSCL second order reconstruction operator is developed. The scheme is well-balanced for the lake-at-rest steady state solutions. Additionally, hints on how to preserve a general class of stationary solutions corresponding to a stratified density profile are also provided. Some numerical results are presented, including validation with laboratory data that show the efficiency and accuracy of the approach introduced here. Finally, a comparison between two different parallelization strategies on GPU is presented.

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