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A novel efficient structure-preserving exponential integrator for Hamiltonian systems

Pan Zhang, Fengyang Xiao, Lu Li

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.

A novel efficient structure-preserving exponential integrator for Hamiltonian systems

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.

Paper Structure

This paper contains 14 sections, 15 theorems, 107 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The new linearly implicit structure-preserving symmetric method
  3. The one-step EKahan scheme for cubic Hamiltonians
  4. Norms and growth constants
  5. Lipschitz assumptions
  6. The multistep EKahan scheme for higher-degree polynomial Hamiltonians
  7. Stacked state and history norm
  8. $k$-step assumptions
  9. Numerical test
  10. Hénon-Heiles system
  11. FPU system with a cubic Hamiltonian
  12. FPU system with a fourth-order Hamiltonian
  13. Zakharov--Kuznetsov equation
  14. Conclusion

Key Result

Lemma 1

Assuming that $f$ is a quadratic vector field and that the exact solution of semilinear ODE is sufficiently smooth on $[t_n,t_{n+1}]$, then the EKahan scheme EKahan scheme has local truncation error $O(h^3)$.

Figures (10)

  • Figure 1: Hénon-Heiles equation with $T=100$ and $h_{0}=0.02$. Left: energy error defined in \ref{['Eerror-plot']}; right: the residual of equation \ref{['step-wise-E']} in Theorem \ref{['theorem-energy']}.
  • Figure 2: Global error and computational time of different methods for the Hénon-Heiles equation with $T=100$ and different time step sizes.
  • Figure 3: FPU system with $p=1$, $T=100$, and $h_{2}=0.25$. Figure \ref{['fig:FPU energy_error_gamma=0_beta=0']}: the energy error defined in \ref{['Eerror-plot']} computed by all schemes for the conservative system; Figure \ref{['fig:FPU est']}: the residual of equation \ref{['step-wise-E']} in Theorem \ref{['theorem-energy']} for the conservative system; Figure \ref{['fig:FPU energy_error_gamma=0.1_beta=0']} and Figure \ref{['fig:FPU energy_error_gamma=0_beta=2']}: the energy defined in Theorem \ref{['theorem-energy']} of all schemes for the two dissipative systems.
  • Figure 4: Global error and computational time of different methods for the FPU system with $p=1$, $\gamma=0$, $\beta=0$, $T=100$ and different time step sizes $h_{i}$.
  • Figure 5: Numerical solutions of the FPU system computed over $t \in [0, 500]$ with time step $h_2 = 0.25$.
  • ...and 5 more figures

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Lemma 2
  • Theorem 2
  • proof
  • Corollary 1
  • Remark 1
  • Lemma 3
  • ...and 14 more