Abstract
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.
Highlights
Shallow-water type systems are among the most widely used models to study geophysical flows
Similar assumptions as in the classical shallow water framework are made: the velocity is averaged in the vertical direction, vertical effects are neglected, and the pressure is assumed hydrostatic
We propose a path-conservative numerical scheme based on a modified hydrostatic reconstruction ([40]) combined with a second order reconstruction procedure
Summary
Shallow-water (or Saint–Venant) type systems are among the most widely used models to study geophysical flows (see [1,2]). In order to overcome this limitation, multilayer shallow water models have been developed in recent years This multilayer approach allows for capturing complex vertical effects and profiles that single layer shallow water models cannot (see [3,4,5]). Recent developments in multilayer shallow water models concern the local change of the number of layers according to the presence of relevant vertical effects (see [22]) Another approach to capture complex vertical profiles, but avoiding fully 3D models while calculating the free surface directly, are the so-called σ-coordinate models ([23]). Capturing the vertical profile in this situation is impossible for standard shallow water equations, and it is necessary to apply a model that takes into account these buoyancy effects In this framework, several authors have proposed multilayer shallow water models with variable density. In Appendix A, a study of performance and computational cost for different parallelization strategies is addressed and in Appendix B, a thorough description of the model derivation is provided
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