A Second-Order Well-Balanced Finite Volume Scheme for the Multilayer Shallow Water Model with Variable Density

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.

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

“…As discussed before, according to [54], a direct discretization of transference terms may yield non-physical results, like relative densities below the unity, which directly conflicts with (2.2). Indeed, the transference terms can be interpreted as a vertical flux between layers, but by writing them as non-conservaive products, the information relative to the flux direction is lost.…”

“…In this work, we employ a model described in [5,54]. The considered model includes the effect of relative changes in density in the pressure terms of the system.…”

“…It is not the aim of this paper to make an exhaustive derivation of the model. The interested reader may refer to [5] or [54] for further details. The model is derived from the incompressible Euler equations with free surface, where two continuity equations are considered: the one corresponding to an incompressible flow with constant density ρ 0 and another one corresponding to an advected flow with variable density ρ a .…”

“…As in [54], here we use a path-conservative solver based on the hydrostatic reconstruction technique (see [2,20,65]). Let us briefly recall its definition,…”

“…The term T i+ 1 2 corresponds to the discretization of the transfer terms. As pointed in [54], those terms should be discretized in a vertical upwind manner as follows:…”