The Hamiltonian particle-mesh (HPM) method is generalized to the spherical shallow water equations, utilizing constrained particle dynamics on the sphere and smoothing with Merilees' double-periodic FFT formulation of O(J2 log J) in the latitudinal gridsize. The time step for the explicit, symplectic integrator depends only on the uniform smoothing length.

Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Frank, J.E, & Reich, S. (2003). The hamiltonian particle-mesh method for the spherical shallow water equations. Modelling, Analysis and Simulation [MAS]. CWI.