Volume-preserving exponential integrators

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.