Variational integrator for the rotating shallow-water equations on the sphere

TL;DR

This paper introduces a variational integrator tailored for the rotating shallow-water equations on a sphere, emphasizing conservation and accuracy through a discretization based on the Euler-Poincaré framework.

Contribution

It presents a novel variational integrator for shallow-water equations on a sphere, utilizing a discretization of the Euler-Poincaré equations on arbitrary meshes.

Findings

The integrator accurately models shallow-water dynamics.

It exhibits excellent conservation properties.

Numerical tests confirm its effectiveness.

Abstract

We develop a variational integrator for the shallow-water equations on a rotating sphere. The variational integrator is built around a discretization of the continuous Euler-Poincar\'{e} reduction framework for Eulerian hydrodynamics. We describe the discretization of the continuous Euler-Poincar\'{e} equations on arbitrary simplicial meshes. Standard numerical tests are carried out to verify the accuracy and the excellent conservational properties of the discrete variational integrator.