Explicit methods in extended phase space for inseparable Hamiltonian problems

TL;DR

This paper introduces an explicit leapfrog integration method using an extended phase space for inseparable Hamiltonian systems, demonstrating improved stability and accuracy over traditional methods in various applications.

Contribution

It develops a novel explicit symplectic integration approach for inseparable Hamiltonian systems via an extended phase space, enhancing long-term stability and applicability.

Findings

Method performs well on geodesic Hamiltonian problems.

Method outperforms LSODE and implicit midpoint in certain cases.

Extended phase space approach improves stability and accuracy.

Abstract

We present a method for explicit leapfrog integration of inseparable Hamiltonian systems by means of an extended phase space. A suitably defined new Hamiltonian on the extended phase space leads to equations of motion that can be numerically integrated by standard symplectic leapfrog (splitting) methods. When the leapfrog is combined with coordinate mixing transformations, the resulting algorithm shows good long term stability and error behaviour. We extend the method to non-Hamiltonian problems as well, and investigate optimal methods of projecting the extended phase space back to original dimension. Finally, we apply the methods to a Hamiltonian problem of geodesics in a curved space, and a non-Hamiltonian problem of a forced non-linear oscillator. We compare the performance of the methods to a general purpose differential equation solver LSODE, and the implicit midpoint method, a symplectic one-step method. We find the extended phase space methods to compare favorably to both for the Hamiltonian problem, and to the implicit midpoint method in the case of the non-linear oscillator.