Well-balanced, anti-diffusive non-oscillatory central difference (adNOC) scheme for the shallow water equations with wet-dry fronts

TL;DR

This paper introduces a novel central differencing scheme for shallow water equations that effectively handles wet-dry fronts and non-flat bottom topography, ensuring stability and accuracy without complex Riemann solvers.

Contribution

The proposed adNOC scheme is a simple, fast, and robust method that maintains well-balanced properties and avoids negative water depths at wet-dry fronts.

Findings

Successfully handles wet-dry fronts without negative water depths

Exhibits good balance between fluxes and source terms

Performs well against analytical solutions in tests

Abstract

We present a central differencing scheme for the solution of the shallow water equations with non-flat bottom topography and moving wet-dry fronts. The problem is numerically challenging due to two reasons. First, the non-flat bottom topography requires accurate balancing of the source term of the momentum conservation equation accounting for the gravitational force and the flux gradient term accounting for the force due to pressure imbalance. Second, the modelling of moving wet-dry fronts involves handling of diminishing water height, which is numerically challenging to handle. The Riemann-solver free scheme is fast, simple and robust. It successfully avoids negative water depths at moving wet-dry boundaries and it exhibits good balancing between flux gradients and source terms. The performance of the scheme is verified with a number of test cases and the results compare favorably with published analytical solutions.