An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment

Abstract

Abstract.We consider the system of partial differential equations governing the one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values.Mathematics Subject Classification. 74S10, 35L60, 74G15.

Central-Upwind Schemes for Two-Layer Shallow Water Equations

“…This example is taken from [4] (see also [31]). The time-splitting scheme proposed there produces a stationary shock (see Figure 7 in [4]), which seems to be unphysical. The solutions computed by our central-upwind scheme on three different grids with ∆x = 1/50, 1/100, and 1/500, do not contain such a shock.…”

“…An interesting approach to overcome the above difficulties has been recently proposed in [1], where two artificial equations have been added to the system (1.1) so that the extended system becomes hyperbolic and thus could be solved by a second-order Roe-type scheme in a rather straightforward manner. A timesplitting approach, proposed in [4], is another way to implement upwinding without having the full eigenstructure of the system readily available.…”

“…Some of the above methods, [1,4,5,6,12], are well-balanced in the sense that they exactly preserve stationary steady-state solutions given by tationally friendly form and describe the central-upwind scheme for the modified system. The description includes a special discretization of the bottom topography function B ( §2.1), the nonconservative products ( §2.4), and a well-balanced discretization of the geometric source term ( §2.5).…”

On an Intermediate Field Capturing Riemann Solver Based on a Parabolic Viscosity Matrix for the Two-Layer Shallow Water System

“…Each layer is assumed to have a constant density, ρ i , i = 1, 2 (ρ 1 < ρ 2 ). The unknowns q i (x, t) and h i (x, t) represent respectively the mass-flow and the thickness of the ith layer at the section of coordinate x at time t. The numerical resolution of two-layer or multilayer shallow water systems has been object of an intense research during the last years: see for instance [1], [3], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [17], [22], [24] . .…”

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

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