Exponential integrators with quadratic energy preservation for linear Poisson systems

Abstract

Motivated by the recent paper (Wang and Wu, 2018 [12]), where the authors proposed a class of functionally-fitted energy-preserving integrators for Poisson systems, in this paper we aim at developing exponential integrators with quadratic energy preservation for the important class of linear Poisson systems. By rewriting the linear Poisson system in a new formulation, we apply the symplectic exponential Runge-Kutta methods and thus obtain a novel class of exponential integrators for the linear Poisson system. It is proved that the obtained exponential integrator can exactly preserve the quadratic energy. In comparison with the functionally-fitted energy-preserving methods whose numerical performance depends on the fitted frequency, the new exponential integrators are independent of the fitted frequency and convenient to use. Furthermore, we investigate the convergence of the fixed-point iteration used in the implementation of the proposed exponential integrator, and show that the convergence region of the stepsize is independent of the norm of the skew-symmetric matrix. Finally, numerical results with the Euler equations of a free rigid body soundly support the remarkable performance of the exponential integrator in both the global errors and the energy preservation.