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High-order Structure-preserving Methods for Damped Hamiltonian System

Lu Li

TL;DR

The paper addresses the challenge of numerically integrating damped Hamiltonian systems while preserving the intrinsic energy-dissipation structure. It blends exponential integrators with energy-preserving collocation to produce high-order, symmetric schemes that preserve the energy-dissipation rate under appropriate invariants, and extends the framework to semi-discrete Hamiltonian PDEs such as the damped Burger's and KdV equations. Key contributions include the construction of second, fourth, sixth, and eighth-order exponential energy-dissipation-preserving collocation (EEPC) methods, proofs of dissipation-preservation under specific conditions (e.g., $H(e^{Y_t}x_t)=e^{\phi_t}H(x_t)$), and extensive numerical validation showing accurate dissipation rates and higher-order accuracy. The methods demonstrate potential for efficient, stable simulation of large-scale damped Hamiltonian systems where preserving dissipative structure is crucial, with open questions around general damping forms and time-dependent skew-symmetric structures.

Abstract

We present a novel methodology for constructing arbitrarily high-order structure-preserving methods tailored for damped Hamiltonian systems. This method combines the idea of exponential integrator and energy-preserving collocation methods, effectively preserving the energy dissipation ratio introduced by the damping terms. We demonstrate the conservative properties of these methods and confirm their order of accuracy through numerical experiments involving the damped Burger's equation and Korteweg-de-Vries equation.

High-order Structure-preserving Methods for Damped Hamiltonian System

TL;DR

The paper addresses the challenge of numerically integrating damped Hamiltonian systems while preserving the intrinsic energy-dissipation structure. It blends exponential integrators with energy-preserving collocation to produce high-order, symmetric schemes that preserve the energy-dissipation rate under appropriate invariants, and extends the framework to semi-discrete Hamiltonian PDEs such as the damped Burger's and KdV equations. Key contributions include the construction of second, fourth, sixth, and eighth-order exponential energy-dissipation-preserving collocation (EEPC) methods, proofs of dissipation-preservation under specific conditions (e.g., ), and extensive numerical validation showing accurate dissipation rates and higher-order accuracy. The methods demonstrate potential for efficient, stable simulation of large-scale damped Hamiltonian systems where preserving dissipative structure is crucial, with open questions around general damping forms and time-dependent skew-symmetric structures.

Abstract

We present a novel methodology for constructing arbitrarily high-order structure-preserving methods tailored for damped Hamiltonian systems. This method combines the idea of exponential integrator and energy-preserving collocation methods, effectively preserving the energy dissipation ratio introduced by the damping terms. We demonstrate the conservative properties of these methods and confirm their order of accuracy through numerical experiments involving the damped Burger's equation and Korteweg-de-Vries equation.
Paper Structure (9 sections, 2 theorems, 43 equations, 7 figures)

This paper contains 9 sections, 2 theorems, 43 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Energy-preserving Collocation methods for Hamiltonian systems
  3. High-order exponential energy dissipation-preserving integrators
  4. Numerical experiments
  5. Damped Burger's equation
  6. Damped KdV equation
  7. Semi-discrete system based on the first Hamiltonian form
  8. Semi-discrete system based on the second Hamiltonian form
  9. Conclusion

Key Result

Theorem 1

For system H-damp-ODE, the exponential energy-preserving collocation methods EEP-C preserve the dissipative rate, i.e., $\frac{dH(x)}{dt}=-\eta(t)H(x)$, whenever $H$ satisfies with $\phi_t=\int_{t_0+h/2}^{t_0+(t-t_0)}\eta(s)ds$.

Figures (7)

  • Figure 1: Plots of the exponential energy dissipation-preserving collocation methods of various orders for the Burger's equation \ref{['damped-burger-semiH']} with $\gamma=0.25$ and $\Delta x=\pi/40$. Figures \ref{['solution-burgers']}, \ref{['energyDecay-burgers']} and \ref{['E_decay_rate_burgers']} utilize fixed time step $\Delta t=0.009$ is used.
  • Figure 2: Plots of the exponential energy dissipation-preserving collocation methods of various orders for the Burger's equation with $D(t)$ having constant unequal diagonal elements and $\Delta x=\pi/40$. In Fig \ref{['solution-burgers_gamma2']}, \ref{['energyDecay-burgers_gamma2']} and \ref{['E_decay_rate_burgers_gamma2']}, fixed time step $\Delta t=0.009$ is used.
  • Figure 3: Plots of the exponential energy dissipation-preserving collocation methods of various orders for the Burger's equation with time-dependent equal diagonal elements, and $\Delta x=\pi/40$. In Fig \ref{['solution-burgers_gamma3']}, \ref{['energyDecay-burgers_gamma3']} and \ref{['E_decay_rate_burgers_gamma3']}, fixed time step $\Delta t=0.009$ is used.
  • Figure 4: Plots of the exponential energy dissipation-preserving collocation methods of various orders for the KdV equation \ref{['KdV-H1']}, where $\gamma=0.01$ and $\Delta x=\pi/40$. In Fig \ref{['solution-kdv_Hamil1']}, \ref{['energyDecay-kdv_Hamil1']} and \ref{['E_decay_rate_kdv_Hamil1']}, fixed time step $\Delta t=0.009$ is used.
  • Figure 5: Plots of the exponential energy-preserving collocation methods of various orders for the KdV equation with the second Hamiltonian form \ref{['Eq-kdv-H2']}, where $\gamma=0.01$ and $\Delta x=\pi/40$. In Fig \ref{['solution-kdv_Hamil2']}, \ref{['energyDecay-kdv_Hamil2']} and \ref{['E_decay_rate_kdv_Hamil2']}, fixed time step $\Delta t=0.009$ is used.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof