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

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.