High-Order, Implicit Time Integration of Discrete, Chaotic Dynamical Systems

A wide range of implicit time integration methods, including multi-step, implicit Runge-Kutta, and Galerkin finite-time element schemes, is evaluated in the context of chaotic dynamical systems. The schemes are applied to solve the Lorenz equations, the equation of motion of a Duffing oscillator, and the Kuramoto-Sivashinsky system, with the goal of finding the most computationally efficient method that results in the least expensive model for a chosen level of accuracy. It is found that the quasi-period of a chaotic system strongly limits the time-step size that can be used in the simulations, and all schemes fail once the time-step size reaches a significant fraction of that period. In these conditions, the computational cost per time-step becomes one of the most important factors determining the efficiency of the schemes. The cheaper, second-order schemes are shown to have an advantage over the higher-order schemes at large time-step sizes, with one possible exception being the fourth-order continuous Galerkin scheme. The higher-order schemes become more efficient than the lower-order schemes as accuracy requirements tighten. If going beyond the second-order is necessary for reasons other than computational efficiency, the fourth-order methods are shown to perform better than the third-order ones at all time-step sizes.