A novel efficient structure-preserving exponential integrator for Hamiltonian systems

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.

💡 Research Summary

The paper introduces a new linearly implicit, time‑reversible exponential integrator tailored for semilinear Hamiltonian systems with polynomial nonlinearities. By embedding the Kahan discretization of the nonlinear term into the variation‑of‑constants exponential framework, the authors obtain a scheme that requires solving only a single linear system per time step while preserving key geometric properties: symmetry (time‑reversibility) and a near‑conservation of the Hamiltonian energy.

Methodology
For a system (\dot x = Ax + f(x)) with (A) representing a stiff linear operator and (f) a polynomial vector field, the classical variation‑of‑constants formula yields
(x(t_{n+1}) = e^{hA}x(t_n) + h\int_0^1 e^{(1-\tau)hA} f(x(t_n+\tau h)),d\tau).
The authors replace the integral by a Kahan‑type approximation of the nonlinear term, leading to the one‑step EKahan scheme:
(x_{n+1}=e^{hA}x_n + h\phi(hA)\big