Symmetric and symplectic exponential integrators for nonlinear Hamiltonian systems

TL;DR

This paper develops symmetric and symplectic exponential integrators tailored for nonlinear Hamiltonian systems, extending classical Runge-Kutta conditions, and demonstrates their superior numerical performance through experiments.

Contribution

It introduces new symmetric and symplectic exponential integrators up to order four, extending existing methods and analyzing their properties for nonlinear Hamiltonian systems.

Findings

New integrators exhibit remarkable numerical behavior.

Integrators outperform some existing Runge-Kutta methods.

Conditions for symmetry and symplecticity are extended from classical methods.

Abstract

This letter studies symmetric and symplectic exponential integrators when applied to numerically computing nonlinear Hamiltonian systems. We first establish the symmetry and symplecticity conditions of exponential integrators and then show that these conditions are extensions of the symmetry and symplecticity conditions of Runge-Kutta methods. Based on these conditions, some symmetric and symplectic exponential integrators up to order four are derived. Two numerical experiments are carried out and the results demonstrate the remarkable numerical behavior of the new exponential integrators in comparison with some symmetric and symplectic Runge-Kutta methods in the literature.