Long-Time Symplectic Integration: The Example of Four-Vortex Motion

Abstract

Detailed comparisons are made between long-time numerical integration of the motion of four identical point vortices obtained using both a fourth-order symplectic integration method of the implicit Runge-Kutta type and a standard fourth-order explicit Runge-Kutta scheme. We utilize the reduced hamiltonian formulation of the four-vortex problem due to Aref & Pomphrey. Initial conditions which give both fully chaotic and also quasi-periodic motions are considered over integration times of order 10^6-10^7 times the characteristic time scale of the evolution. The convergence, as the integration time step is decreased, of the Poincare section is investigated. When smoothness of the section compared to the converged image, and the fractional change in the hamiltonian H are used as diagnostic indicators, it is found that the symplectic scheme gives substantially superior performance over the explicit scheme, and exhibits only an apparent qualitative degrading in results up to integration time steps of order the minimum timescale of the evolution. It is concluded that this performance derives from the symplectic rather than the implicit character of the method.