Numerical Treatment of the Loss of Hyperbolicity of the Two-Layer Shallow-Water System

Abstract

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.

Dynamic coupling between horizontal vessel motion and two-layer shallow-water sloshing

“…So this condition is linked to the KH instability of the stratified flows [28,27]. See [14] for the numerical treatment of the loss of hyperbolicity of the two layer shallow water system, and see [10] for the nonlinear stability analysis of twolayer shallow water equations with a rigid-lid. This approach for the approximation of the eigenvalues is useful only for two-layer shallow fluid flows with densities very close to each other (Boussinesq approximation).…”

Numerical modelling of two-layer shallow water flow in microtidal salt-wedge estuaries: Finite volume solver and field validation

“…The practical problem is that too much friction can results in excessively diffused results and even produce spurious oscillations in the flow. In Castro et al (2011) a different strategy for maintaining the hyperbolic character was presented. This numerical workaround is based on the predictor/corrector algorithm, which consists in adding an extra friction term only to those individual cells in which complex eigenvalues are detected at a given time step (Castro et al, 2011).…”

“…In Castro et al (2011) a different strategy for maintaining the hyperbolic character was presented. This numerical workaround is based on the predictor/corrector algorithm, which consists in adding an extra friction term only to those individual cells in which complex eigenvalues are detected at a given time step (Castro et al, 2011). The additional friction should simulate the loss of momentum due to mixing, which is expected in real flows as a result of interfacial instabilities.…”

On the Hyperbolicity of Two- and Three-Layer Shallow Water Equations

“…In [5] it is shown that (2.5) is a good approximative criterion for the loss of hyperbolicity when r 12 is close to 1. The ratio κ can be interpreted as the 2/3-layer Shallow Water balance between the stabilizing influence of the difference in density and the destabilizing one of the velocity shear.…”

“…Several authors have suggested such models, e.g. LeVeque and Kim [6], Castro and Pares [5], Bouchut and Morales [2] and Audusse [1].…”