Integrable decomposition methods and ensemble averaging for non-integrable N-vortex problems

Abstract

We propose a new class of algorithms to integrate non-integrable N-body systems based on a time-splitting method in which each sub-step advances an integrable sub-cluster. Since each sub-step is integrable, numerical instabilities are reduced because there are no positive Lyapunov exponents causing exponential growth of perturbations. We describe the method using N-vortex simulations, where the largest integrable sub-problem is a 3-vortex interaction (N _crit = 3). Each time-step, Δt, is broken into N!/(N-3)!3! sub-steps, where we cycle through all possible integrable sub-clusters. With this method, we show that quantities known to be theoretically conserved, typically fluctuate randomly around a constant mean. By contrast, standard Runge-Kutta solvers typically exhibit secular drift of conserved quantities, creating errors that build in time and are difficult to systematically correct. One has a choice as to how to numerically advance the sub-clusters and this choice ultimately influences properties of the overall scheme, such as whether or not it is symplectic or energy-preserving. The fluctuations in the conserved quantities are shown to be (approximately) normally distributed Gaussian random processes having positive discrete entropy for any fixed Δt. The order in which we cycle through the sub-clusters influences the numerical simulation and we exploit this by ensemble averaging the chaotic trajectories obtained from different cycling sequences.

This article was chosen from Selected Proceedings of the 4th International Workshop on Vortex Flows and Related Numerical Methods (UC Santa-Barbara, 17-20 March 2002) ed E Meiburg, G H Cottet, A Ghoniem and P Koumoutsakos.