Volume-preserving exponential integrators

TL;DR

This paper investigates the volume-preserving properties of exponential integrators, deriving conditions for volume preservation, and introduces new integrators that better preserve volume in various dynamical systems, especially oscillatory ones.

Contribution

It provides a necessary and sufficient condition for volume preservation of exponential integrators and develops novel volume-preserving methods for oscillatory and partitioned systems.

Findings

Symplectic exponential integrators can preserve volume for more vector fields than Hamiltonian systems.

New volume-preserving exponential integrators are derived for oscillatory second-order systems.

Numerical experiments show these integrators outperform volume-preserving Runge-Kutta methods.

Abstract

As is known that various dynamical systems including all Hamiltonian systems preserve volume in phase space. This qualitative geometrical property of the analytical solution should be respected in the sense of Geometric Integration. This paper analyses the volume-preserving property of exponential integrators in different vector fields. We derive a necessary and sufficient condition of volume preservation for exponential integrators, and with this condition, volume-preserving exponential integrators are analysed in detail for four kinds of vector fields. It turns out that symplectic exponential integrators can be volume preserving for a much larger class of vector fields than Hamiltonian systems. On the basis of the analysis, novel volume-preserving exponential integrators are derived for solving highly oscillatory second-order systems and extended Runge--Kutta--Nystr\"{o}m (ERKN) integrators of volume preservation are presented for separable partitioned systems. Moreover, the volume preservation of Runge--Kutta--Nystr\"{o}m (RKN) methods is also discussed. Four illustrative numerical experiments are carried out to demonstrate the notable superiority of volume-preserving exponential integrators in comparison with volume-preserving Runge-Kutta methods.