A novel efficient structure-preserving exponential integrator for Hamiltonian systems

TL;DR

This paper introduces a new linearly implicit exponential integrator for Hamiltonian systems that preserves geometric properties and offers computational efficiency, demonstrated through numerical experiments on various complex systems.

Contribution

It combines Kahan's method with exponential integrators to create a structure-preserving scheme that is computationally efficient and effective for semilinear Hamiltonian systems.

Findings

Balances computational cost and accuracy effectively

Preserves symmetry and nearly preserves energy

Demonstrates stability and efficiency in numerical experiments

Abstract

We propose a linearly implicit structure-preserving numerical method for semilinear Hamiltonian systems with polynomial nonlinearities, combining Kahan's method and exponential integrator. This approach efficiently balances computational cost, accuracy and the preservation of key geometric properties, including symmetry and near-preservation of energy. By requiring only the solution of a single linear system per time step, the proposed method offers significant computational advantages while comparing with the state-of-the-art symmetric energy-preserving exponential integrators. The stability, efficiency and long-term accuracy of the method are demonstrated through numerical experiments on systems such as the Henon-Heiles system, the Fermi-Pasta-Ulam system and the two-dimensional Zakharov-Kuznestov equation.