Well-balanced numerical resolution of the two-layer shallow water equations under rigid-lid with wet–dry fronts

Abstract

In this paper, we present a well-balanced numerical scheme for the frictional two-layer shallow water equations (2LSWE) over variable bottom topography and under a rigid-lid (RL) to simulate internal waves propagating over wet and dry areas. Following the idea of Liang and Borthwick (2009) for the derivation of a pre-balanced formulation for one-layer flows, we derive a new formulation of the 2LSWE-RL using the interface elevation above datum and horizontal momentum as conservative variables. This new formulation mathematically balances the flux gradient and source terms so that the lake-at-rest steady state is automatically preserved in wet-bed applications. A proper discretization of the slope source term is adopted to produce well-balanced solutions in dry-bed areas where the lower layer depth vanishes. Using the Harten, Lax, and van Leer (HLL) approximate Riemann solver, we adopt a finite volume MUSCL-RK2 method to construct a second-order well-balanced numerical scheme ensuring the non-negativity of the layers depths with a special treatment of source terms. Finally, various numerical experiments are tested in both one and two-layer cases validating the properties of the numerical scheme.