[Paper Review] Systematic Improvement of Splitting Methods for the Hamilton Equations

Asif Mushtaq, Anne Kværnø+1 more|arXiv (Cornell University)|Apr 18, 2012

Numerical methods for differential equations10 references6 citations

TL;DR

This paper presents a systematic method to enhance the accuracy of splitting methods for Hamiltonian systems beyond the standard Störmer-Verlet scheme, achieving order $\tau^8$ without composition techniques. By modifying the kinetic and potential energy terms in the splitting via time-dependent coefficients derived from Taylor expansions of exact flows, the method preserves symplecticity while significantly improving long-term energy conservation and phase space accuracy.

ABSTRACT

We show how the standard (St{ö}rmer-Verlet) splitting method for differential equations of Hamiltonian mechanics (with accuracy of order $τ^2$ for a timestep of length $τ$) can be improved in a systematic manner without using the composition method. We give the explicit expressions which increase the accuracy to order $τ^8$, and demonstrate that the method work on a simple anharmonic oscillator.

Motivation & Objective

• To develop a systematic, non-composition-based method to improve the accuracy of symplectic splitting schemes for Hamiltonian systems.
• To address the limitation of standard splitting methods, which are only second-order accurate and suffer from long-term energy drift.
• To derive explicit expressions for higher-order corrections that maintain symplectic structure while increasing accuracy to $\tau^8$.
• To demonstrate the method's effectiveness on nonlinear systems like the anharmonic oscillator, showing superior energy and period conservation.

Proposed method

• The method modifies the kinetic and potential energy terms in the splitting scheme using time-dependent coefficients derived from Taylor expansions of the exact flow.
• It introduces correction terms $G_2, G_4, \dots, G_8$ to the generators of the flow, derived from nested commutators of the Hamiltonian components.
• The coefficients $m(\tau)$ and $k(\tau)$ are chosen such that the discrete evolution matches the exact solution of the harmonic oscillator up to $\tau^8$.
• The approach avoids composition by directly adjusting the splitting parameters rather than combining multiple steps.
• For nonlinear systems, the method requires solving nonlinear algebraic equations at each time step to determine the corrected parameters.
• The method preserves symplecticity by construction, ensuring long-term stability in energy and phase space trajectories.

Experimental results

Research questions

• RQ1 Can the accuracy of standard splitting methods for Hamiltonian systems be systematically improved beyond order $\tau^2$ without using composition techniques?
• RQ2 What explicit correction terms are required to achieve order $\tau^8$ accuracy in the splitting scheme while preserving symplecticity?
• RQ3 How do time-dependent modifications to the kinetic and potential energy terms affect the long-term conservation of energy and phase space structure?
• RQ4 To what extent does the method improve the period accuracy and energy conservation in nonlinear systems like the anharmonic oscillator?
• RQ5 What is the computational cost of applying higher-order corrections compared to standard schemes?

Key findings

• The method achieves order $\tau^8$ accuracy in the solution of Hamiltonian systems by systematically correcting the splitting parameters using exact harmonic oscillator flow expansions.
• Energy conservation is significantly improved: the quantity $(H - \frac{1}{2})/\tau^8$ remains small over long integration times, indicating high accuracy.
• For the anharmonic oscillator $H = \frac{1}{2}p^2 + \frac{1}{4}q^4$, the $\tau^6$-corrected scheme maintains energy conservation over more than 260,000 periods.
• The method preserves symplecticity, ensuring long-term stability and accurate phase space structure, even with large timesteps.
• The $\tau^6$-corrected scheme shows minimal deviation in oscillation period compared to the exact solution, outperforming lower-order schemes.
• The computational cost per step increases due to solving nonlinear equations, but the higher accuracy per step may justify the cost for high-precision simulations.