An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification
Résumé
We present an explicit scheme for a two-dimensional multilayer shallow water model with density stratification, for general meshes and collocated variables. The proposed strategy is based on a regularized model where the transport velocity in the advective fluxes is shifted proportionally to the pressure potential gradient. Using a similar strategy for the potential forces, we show the stability of the method, in the sense of a discrete dissipation of the mechanical energy, in the general multilayer and non-linear frame. Based on a linear analysis, and with the objective of minimizing the diffusive losses in realistic contexts, sufficient conditions are exhibited on the regularizing terms to ensure linear stability. These results are subsequently validated by numerical investigations. The other main result stands in the consistency with respect to the asymptotics reached at small and large time scales in low Froude regimes, which governs large scale oceanic circulation. Additionally, robustness and well balanced results for motionless steady states are also ensured. These stability properties tend to provide a very robust and efficient approach, easy to implement and particularly well suited for large scale simulations. Two numerical experiments are proposed to highlight the scheme efficiency: a first experiment of fast gravitational modes and a second of slow Rossby modes simulating the displacement of a baroclinic vortex subject to the Coriolis force.
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