A Central-Upwind Scheme for Two-layer Shallow-water Flows with Friction and Entrainment along Channels

TL;DR

This paper introduces a high-resolution central-upwind numerical scheme for simulating two-layer shallow-water flows with friction and entrainment, ensuring positivity and well-balanced properties for accurate channel flow modeling.

Contribution

It develops a novel semi-discrete central-upwind scheme that handles complex two-layer flows with non-conservative products, extending existing methods with improved stability and accuracy.

Findings

The scheme preserves water depth positivity.

It maintains well-balanced discretization of source terms.

Numerical experiments confirm robustness and accuracy.

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 {\it preserves the positivity} of the water depth for each layer, and it is {\it 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.