Comparison of Very Smooth Cell-Model Trajectories Using Five Symplectic and Two Runge-Kutta Integrators

Abstract

Symplectic methods, which are precisely compatible with Liouville's phase-volume-conservation theorem, are often recommended for computational simulations of Hamiltonian mechanics. Lack of energy drift is an advantage of symplectic methods. But all numerical methods are susceptible to chaos, Lyapunov instability, which severely limits the maximum time for which solutions can be "accurate". The "advantages" of higher-order methods are lost for typical chaotic Hamiltonians. We illustrate these difficulties for a useful reproducible test case, the two-dimensional one-particle cell model. The motion is chaotic and occurs on a three-dimensional constant energy shell. We benchmark the problem with quadruple-precision trajectories using a fifth-order Runge-Kutta and the fourth-order Candy-Rozmus integrator. We compare these benchmark results for accurate particle trajectories to those from six double-precision algorithms, four symplectic and two Runge-Kutta.