Explicit, parallel Poisson integration of point vortices on the sphere

Abstract

Point vortex models are frequently encountered in conceptual studies in geophysical fluid dynamics, but also in practical applications, for instance, in aeronautics. In spherical geometry, the motion of vortex centres is governed by a dynamical system with a known Poisson structure. We construct Poisson integration methods for these dynamics by splitting the Hamiltonian into its constituent vortex pair terms. From backward error analysis, the method is formally known to provide solutions to a modified Poisson system with the correct bracket, but with a modified Hamiltonian function. Different orderings of the pairwise interactions are considered and also used for the construction of higher order methods. The energy and momentum conservation of the splitting schemes is demonstrated for several test cases. For particular orderings of the pairwise interactions, the schemes allow scalable parallelization. This results in a linear–as opposed to quadratic–scaling of computation time with system size when scaling the number of processors accordingly.