Volume-preserving exponential integrators and their applications

It is well known that various dynamical systems preserve volume in phase space such as all Hamiltonian systems. This qualitative geometrical property of the analytical solution should be respected in the sense of Geometric Integration. This paper studies the volume-preserving property of exponential integrators for different vector fields. For exponential integrators, we first derive a necessary and sufficient condition of volume preservation. Then based on this condition, volume-preserving exponential integrators are discussed in detail for four kinds of vector fields. It is shown 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, some applications of volume-preserving exponential integrators are discussed. For solving highly oscillatory second-order systems, novel volume-preserving exponential integrators are derived, and for separable partitioned systems, extended Runge–Kutta–Nyström (ERKN) integrators of volume preservation are presented. Moreover, the volume preservation of Runge–Kutta–Nyström (RKN) methods is also discussed. Five illustrative numerical experiments are carried out to demonstrate the notable superiority of volume-preserving exponential integrators in comparison with volume-preserving Runge-Kutta methods.