On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations

TL;DR

This paper demonstrates that moving-water well-balanced schemes outperform still-water schemes in accurately capturing small perturbations near moving-water equilibria in shallow water equations, reducing oscillations and improving numerical stability.

Contribution

It shows the advantages of moving-water well-balanced schemes over still-water schemes for shallow water equations near moving-water equilibria.

Findings

Moving-water schemes better capture small perturbations.

Still-water schemes generate spurious oscillations without fine meshes.

Moving-water schemes perform well in numerical tests.

Abstract

This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations. We concentrate on numerical examples with solutions near a moving-water equilibrium. For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh is used. On the other hand, moving- water well-balanced methods perform well in these tests. The numerical examples in this note clearly demonstrate the importance of utilizing moving-water well-balanced methods for solutions near a moving-water equilibrium.