Volume-preserving exponential integrators
Provides a theoretical framework and practical methods for volume preservation in exponential integrators, benefiting researchers in geometric integration and numerical simulation of dynamical systems.
The paper derives necessary and sufficient conditions for exponential integrators to preserve volume in phase space, and develops novel volume-preserving exponential integrators for highly oscillatory second-order systems and separable partitioned systems. Numerical experiments show these integrators significantly outperform volume-preserving Runge-Kutta methods.
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--Nyström (ERKN) integrators of volume preservation are presented for separable partitioned systems. Moreover, the volume preservation of Runge--Kutta--Nyström (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.