Abstract
We present a new high-resolution, non-oscillatory semi-discrete central-upwind scheme for one-dimensional two-layer shallow-water flows with friction and entrainment along channels with arbitrary cross sections and bottom topography. These flows are described by a conditionally hyperbolic balance law with non-conservative products. A detailed description of the properties of the model is provided, including entropy inequalities and asymptotic approximations of the eigenvalues of the corresponding coefficient matrix. The scheme extends existing central-upwind semi-discrete numerical methods for hyperbolic conservation and balance laws and it satisfies two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water depth for each layer, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. Along with the description of the scheme and proofs of these two properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.
Highlights
In this paper we present a new high-order numerical scheme for simulating two-layer shallow-water flows along channels with a bottom topography and varying width
Shallow-water flows are typically modeled by the Saint-Venant equations, a hyperbolic balance law that results from the depth averaging of the Euler equations
We propose a new high-order central-upwind scheme to compute two-layer shallow-water flows along channels with arbitrary geometry that incorporates the treatment of friction and entrainment terms
Summary
In this paper we present a new high-order numerical scheme for simulating two-layer shallow-water flows along channels with a bottom topography and varying width (see Fig. 1). For flows along channels with arbitrary geometry, the authors of [12] extended with great success (and high impact in the field) the Q-scheme for hyperbolic systems with source terms previously introduced in [11] Some of these schemes for shallow-water flows have been employed – and new ones created – for simulating, studying or recreating geophysical flows from the real world like gravitational currents or strait flows. The proposed numerical scheme evolves the cell averages of the flow variables with second order accuracy, and their implementation requires four main ingredients: a non-oscillatory reconstruction of point values from cell averages that preserves the positivity of the water depth, an evolution routine to advance the solution in time, estimates of the largest and smallest eigenvalues of the system, and the discretization of source terms and non-conservative products that balance the hyperbolic fluxes so as to recognize steady states at rest and add accuracy to the computation of flows near non stationary steady states. Numerical solutions for a variety of flow regimes are presented in Section 4, validating the scheme’s accuracy and robustness and demonstrating its ability to simulate a wide range of flows
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