Bound-preserving and entropy-stable algebraic flux correction schemes for the shallow water equations with topography

TL;DR

This paper introduces a novel algebraic flux correction scheme for the shallow water equations with topography, ensuring well-balancedness, nonnegativity, and entropy stability, suitable for realistic wetting and drying simulations.

Contribution

It develops a well-balanced AFC scheme for SWE with topography, including a flux limiter and entropy stability, extending finite volume methods to finite element discretization.

Findings

Preserves lake at rest equilibrium up to machine precision

Guarantees nonnegativity of water heights under CFL condition

Satisfies semi-discrete entropy inequality in numerical schemes

Abstract

A well-designed numerical method for the shallow water equations (SWE) should ensure well-balancedness, nonnegativity of water heights, and entropy stability. For a continuous finite element discretization of a nonlinear hyperbolic system without source terms, positivity preservation and entropy stability can be enforced using the framework of algebraic flux correction (AFC). In this work, we develop a well-balanced AFC scheme for the SWE system including a topography source term. Our method preserves the lake at rest equilibrium up to machine precision. The low-order version represents a generalization of existing finite volume approaches to the finite element setting. The high-order extension is equipped with a property-preserving flux limiter. Nonnegativity of water heights is guaranteed under a standard CFL condition. Moreover, the flux-corrected space discretization satisfies a semi-discrete entropy inequality. New algorithms are proposed for realistic simulation of wetting and drying processes. Numerical examples for well-known benchmarks are presented to evaluate the performance of the scheme.