An Energy Stable and Well-balanced Scheme for the Ripa System

TL;DR

This paper introduces an energy stable, well-balanced finite volume scheme for the Ripa system of shallow water equations, ensuring physical property preservation and stability through carefully designed discretisation and stabilization techniques.

Contribution

The paper presents a novel explicit finite volume scheme that guarantees energy stability and well-balancing for the Ripa system, with positivity preservation and weak consistency.

Findings

Scheme preserves positivity of water height and temperature

Numerical tests confirm energy stability and well-balancing

Method effectively handles benchmark problems

Abstract

We design and analyse an energy stable, structure preserving and well-balanced scheme for the Ripa system of shallow water equations. The energy stability of the numerical solutions is achieved by introducing appropriate stabilisation terms in the discretisation of the convective fluxes of mass and momenta, the pressure gradient and the topography source term. A diligent choice of the interface values of the water height and the temperature ensures the well-balancing property of the scheme for three physically relevant hydrostatic steady states. The explicit in time and finite volume in space scheme preserves the positivity of the water height and the temperature, and it is weakly consistent with the continuous model equations in the sense of Lax-Wendroff. The results of extensive numerical case studies on benchmark test problems are presented to confirm the theoretical findings.