A Robust Numerical Method for Integration of Point-Vortex Trajectories in Two Dimensions

TL;DR

This paper introduces a coordinate transformation-based numerical method for efficiently and accurately simulating 2D point-vortex trajectories, maintaining energy conservation even during close vortex interactions.

Contribution

It develops a novel approach using action-angle coordinates and Lie transform perturbation theory to improve simulation efficiency and energy conservation in vortex dynamics.

Findings

Significantly reduces computational time for close vortex pairs.

Maintains high energy conservation accuracy during simulations.

Demonstrates improved numerical stability over traditional methods.

Abstract

The venerable 2D point-vortex model plays an important role as a simplified version of many disparate physical systems, including superfluids, Bose-Einstein condensates, certain plasma configurations, and inviscid turbulence. This system is also a veritable mathematical playground, touching upon many different disciplines from topology to dynamic systems theory. Point-vortex dynamics are described by a relatively simple system of nonlinear ODEs which can easily be integrated numerically using an appropriate adaptive time stepping method. As the separation between a pair of vortices relative to all other inter-vortex length scales decreases, however, the computational time required diverges. Accuracy is usually the most discouraging casualty when trying to account for such vortex motion, though the varying energy of this ostensibly Hamiltonian system is a potentially more serious problem. We solve these problems by a series of coordinate transformations: We first transform to action-angle coordinates, which, to lowest order, treat the close pair as a single vortex amongst all others with an internal degree of freedom. We next, and most importantly, apply Lie transform perturbation theory to remove the higher-order correction terms in succession. The overall transformation drastically increases the numerical efficiency and ensures that the total energy remains constant to high accuracy.