ENERGY-PRESERVING CONTINUOUS-STAGE EXPONENTIAL RUNGE--KUTTA INTEGRATORS FOR EFFICIENTLY SOLVING HAMILTONIAN SYSTEMS

As one of the most important properties, energy preservation is a natural requirement for numerical integrators of Hamiltonian systems. Considering the limited second-order accuracy of the existing exponential average vector filed (or discrete gradient) method, this paper is devoted to developing high-order energy-preserving exponential integrators for general semilinear Hamiltonian systems. To this end, we first formulate and analyze the continuous-stage exponential Runge-Kutta (ERK) method. After deriving the order conditions, energy-preserving conditions, and symmetric conditions for the continuous-stage ERK method, we construct a novel class of fourth-order symmetric energy-preserving continuous-stage ERK methods with a free parameter based on the established conditions and present a convergence theorem for the fixed-point iteration associated with the continuous-stage ERK method. Furthermore, we extend the proposed continuous-stage ERK method to noncanonical Hamiltonian systems and discuss implementation issues in detail. Finally, numerical results from the FPU problem, the charged-particle dynamics, and the Klein-Gordon equation demonstrate the high efficiency, good energy-preserving behavior, and applicability of large time stepsizes of the proposed fourth-order energy-preserving continuous-stage ERK method.

Automated summary

This paper focuses on developing high-order energy-preserving exponential integrators for general semilinear Hamiltonian systems. By constructing a novel class of fourth-order symmetric energy-preserving continuous-stage ERK methods, it achieves high efficiency, good energy-preserving behavior, and applicability of large time stepsizes.