A unified framework of energy-stable splitting exponential integrators for damped Hamiltonian systems

Abstract

In this work, we study long-time numerical integration of Hamiltonian systems subject to linear perturbations. By introducing an energy-induced metric, we establish a straightforward, coordinate-free criterion for dissipativity that ensures the decay of the physical energy for a wide class of linearly perturbed Hamiltonian systems. Since the conservative and dissipative effects cannot always be merged into a single gradient-structured dissipation and classical energy-stable methods developed for gradient flows can not directly extend to this setting, we propose a unified framework of two efficient and energy-stable splitting exponential integrators (SEI) to separately handle the dissipative and conservative parts: SEISAV (SEI based on the scalar auxiliary variable) and SEILM (SEI based on Lagrange multiplier). The SEISAV scheme composes the exact damping subflow with an exponential integrator based on the SAV update for the Hamiltonian subflow and requires solving only a one-dimensional linear algebraic equation at each time step. We prove the unconditional discrete decay for a modified energy that mirrors the continuous energy-dissipation mechanism. To enforce decay of the original energy rather than a modified one, we further develop SEILM by incorporating a Lagrange-multiplier formulation within the splitting exponential framework, leading to only a nonlinear algebraic equation at each time step. Numerical experiments on representative linearly damped Hamiltonian models confirm the predicted convergence rates, energy stability, and competitive efficiency relative to state-of-the-art schemes, indicating that the proposed framework provides a simple and robust approach for simulating linearly perturbed Hamiltonian dynamics.