Explicit symplectic approximation of nonseparable Hamiltonians: algorithm and long time performance

TL;DR

This paper introduces explicit symplectic integrators for arbitrary Hamiltonians that are of any even order, symplectic in an extended phase space, and demonstrate excellent long-term behavior, supported by theoretical analysis and numerical experiments.

Contribution

It proposes a novel class of explicit symplectic integrators applicable to nonseparable Hamiltonians, extending the scope of symplectic methods beyond traditional separable cases.

Findings

Error bound of order Tδ^lω for integrable systems

Satisfactory statistical behavior observed in nonlinear Schrödinger equation experiments

Integrators are explicit, symplectic, and of arbitrary even order

Abstract

Explicit symplectic integrators have been important tools for accurate and efficient approximations of mechanical systems with separable Hamiltonians. For the first time, the article proposes for arbitrary Hamiltonians similar integrators, which are explicit, of any even order, symplectic in an extended phase space, and with pleasant long time properties. They are based on a mechanical restraint that binds two copies of phase space together. Using backward error analysis, KAM theory, and additional multiscale analysis, an error bound of $\mathcal{O}(T\delta^l \omega)$ is established for integrable systems, where $T$, $\delta$, $l$ and $\omega$ are respectively the (long) simulation time, step size, integrator order, and some binding constant. For non-integrable systems with positive Lyapunov exponents, such an error bound is generally impossible, but satisfactory statistical behaviors were observed in a numerical experiment with a nonlinear Schr\"{o}dinger equation.