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

Lu Li, Xiaoli Li, Zaijiu Shang, Quanquan Xu

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.

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.
Paper Structure (19 sections, 13 theorems, 119 equations, 7 figures, 4 algorithms)

This paper contains 19 sections, 13 theorems, 119 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The SAV/LM reformulation
  3. The standard SAV approach
  4. The Lagrange multiplier approach
  5. The exponential integrators based on the SAV/LM framework
  6. The standard EISAV method
  7. The EILM approach
  8. The two splitting schemes based on EISAV and EILM
  9. Splitting EISAV method
  10. Splitting EILM method
  11. Energy stability
  12. Numerical experiments
  13. Example 1: Damped nonlinear Klein-Gordon equation
  14. Example 2: Continuous $\alpha$-FPU model with damping terms
  15. Example 2.1 -- $\mathcal{S}$ is a skew-adjoint operator
  16. ...and 4 more sections

Key Result

Proposition 1

\newlabelSAV The extended system Extendsys-SAV inherits the conservation/dissipation structure of eq:model-gradient flow, i.e.,

Figures (7)

  • Figure 5.1: Comparison of convergence orders (left) and computational efficiency (right) for the damped nonlinear Klein-Gordon equation.
  • Figure 5.2: Discrete energy evolution (left) and energy error (right) for the damped nonlinear Klein-Gordon equation.
  • Figure 5.3: Comparison of convergence orders and computational efficiency for the damped $\alpha$-FPU equation in Example 2.1.
  • Figure 5.4: Left: discrete energy evolution for all methods applied to the $\alpha$-FPU formulation in Example 2.1; right: CPU time versus accuracy for all methods applied to the $\alpha$-FPU formulation in Example 2.1.
  • Figure 5.5: Left: discrete energy evolution for all methods applied to the $\alpha$-FPU formulation in Example 2.2; right: CPU time versus accuracy for all splitting schemes applied to the $\alpha$-FPU formulations in Examples 2.1 and 2.2.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Lemma 4.1
  • proof
  • Theorem 4.2
  • ...and 10 more