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.

Observation-based correction of dynamical models using thermostats

“…Point vortices have been studied extensively in [ 41 – 43 ]. We work on a spherical domain [ 44 – 48 ], viewing the vortex position as a vector in R 3 with unit norm. The equations of motion have a Lie–Poisson structure where the Hamiltonian is given by and each Γ i represents the circulation of a single point vortex.…”

“…A numerical integrator for the point vortex system is constructed as in Myerscough & Frank [ 48 ] based on splitting the differential equations into integrable subproblems (see related ideas in [ 53 , 54 ]). The backward error analysis of symplectic integrators has an analogous development for Poisson systems, implying approximate conservation of the Hamiltonian [ 8 ].…”

“…Point vortices have been studied extensively in [41][42][43]. We work on a spherical domain [44][45][46][47][48], viewing the vortex position as a vector in R 3 with unit norm. The equations of motion have a Lie-Poisson structure…”

“…This fact is exploited in our earlier work [48] to construct a splitting method for integration of the point vortex system with favourable (geometric) properties. The modified double bracket thermostat is also a combination of pairwise vortex interactions.…”

“…Time integration of the thermostatted system In this appendix, we detail the method of integration for the perturbation dynamics and thermostat variable. The unperturbed dynamics are formed by the combination of pairwise interactions: This fact is exploited in our earlier work [ 48 ] to construct a splitting method for integration of the point vortex system with favourable (geometric) properties. The modified double bracket thermostat is also a combination of pairwise vortex interactions.…”