Structural schemes for hamiltonian systems

TL;DR

The paper presents a novel adaptation of the structural method to Hamiltonian systems, introducing a PE/SE split and two formulations, ZD and ZDS, to achieve high-order, unconditionally stable discretizations for scalar and multi-body problems. By coupling physical equations with compact structural constraints, the method delivers excellent long-time invariant preservation (energy and other invariants) and competes favorably with traditional symplectic integrators, including in non-separable cases. Extensive benchmarks across mass-spring, pendulum, Kepler, figure-eight, outer solar system, and charged particle in EM fields demonstrate superior accuracy, tunable order via the block size R, and favorable complexity and parallelization properties. The work highlights the method’s versatility, robustness, and potential for very high-precision long-time simulations with relatively modest stencil sizes. Its practical impact lies in providing a scalable, structure-preserving alternative for accurate, long-term simulations of complex Hamiltonian dynamics.

Abstract

We present an adaptation of the so-called structural method \cite{CMM23} for Hamiltonian systems, and redesign the method for this specific context, which involves two coupled differential systems. Structural schemes decompose the problem into two sets of equations: the physical equations, which describe the local dynamics of the system, and the structural equations, which only involve the discretization on a very compact stencil. They have desirable properties, such as unconditional stability or high-order accuracy. We first give a general description of the scheme for the scalar case (which corresponds to e.g. spring-mass interactions or pendulum motion), before extending the technique to the vector case (treating e.g. the $n$-body system). The scheme is also written in the case of a non-separable system (e.g. a charged particle in an electromagnetic field). We give numerical evidence of the method's efficiency, its capacity to preserve invariant quantities such as the total energy, and draw comparisons with the traditional symplectic methods.

Figures (8)

• Figure 1: Schematic representation of the two masses-spring system.
• Figure 2: Pendulum problem from \ref{['sec:pendulum']}: error on the Hamiltonian over time. Top panels: ZD (left) and ZDS (right) methods; bottom panels: classical symplectic (left) and non-symplectic (right) schemes.
• Figure 3: Kepler problem from \ref{['sec:2D_Kepler']}: errors on the invariants (Hamiltonian, angular momentum and Laplace-Runge-Lenz vector) over time. From left to right: ZD , ZDS and classical methods; top panels: errors on the Hamiltonian; middle panels: errors on the angular momentum; bottom panels: errors on the Laplace-Runge-Lenz vector. In this figure, the results of the uncorrected ZD and ZDS schemes from \ref{['sec:uncorrected_kepler']} are presented.
• Figure 4: Kepler problem from \ref{['sec:2D_Kepler']}: errors on the invariants (Hamiltonian, angular momentum and Laplace-Runge-Lenz vector) over time. From left to right: ZD and ZDS methods; top panels: errors on the Hamiltonian; middle panels: errors on the angular momentum; bottom panels: errors on the Laplace-Runge-Lenz vector. In this figure, the results of the projected ${\texttt{ZD}\xspace }$ and ${\texttt{ZDS}\xspace }$ schemes from \ref{['sec:Kepler_proj']} are presented.
• Figure 5: Two-dimensional three-body problem from \ref{['sec:2D_3body']}: errors on the invariants (Hamiltonian and angular momentum) over time. From left to right: ZD , ZDS and classical methods; top panels: errors on the Hamiltonian; bottom panels: errors on the angular momentum.

Theorems & Definitions (12)

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