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The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems

Lijie Mei, Xinyuan Wu, Yaolin Jiang

TL;DR

The paper addresses the challenge of constructing explicit symplectic integrators for general nonseparable Hamiltonian systems. It develops a constructive existence proof by embedding the original problem in an extended phase space, then mapping explicit symplectic solutions back to the original space via nonlinear projections while preserving the symplectic structure. A backward-error analysis shows linear growth of global errors and near conservation of first integrals for (near-)integrable cases, with careful treatment of nonintegrable dynamics into regular and ergodic regimes. Numerical experiments on completely integrable and nonintegrable nonseparable Hamiltonians demonstrate good long-term behavior and superior efficiency of the proposed explicit integrators compared to implicit counterparts, validating the theoretical findings.

Abstract

The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems is an open and important problem in both numerical analysis and computing in science and engineering, as explicit integrators are usually more efficient than the implicit integrators of the same order of accuracy. Up to now, all responses to this problem are negative. That is, there exist explicit symplectic integrators only for some special nonseparable Hamiltonian systems, whereas the universal design involving explicit symplectic integrators for general nonseparable Hamiltonian systems has not yet been studied sufficiently. In this paper, we present a constructive proof for the existence of explicit symplectic integrators for general nonseparable Hamiltonian systems via finding explicit symplectic mappings under which the special submanifold of the extended phase space is invariant. It turns out that the proposed explicit integrators are symplectic in both the extended phase space and the original phase space. Moreover, on the basis of the global modified Hamiltonians of the proposed integrators, the backward error analysis is made via a parameter relaxation and restriction technique to show the linear growth of global errors and the near-preservation of first integrals. In particular, the effective estimated time interval is nearly the same as classical implicit symplectic integrators when applied to (near-) integrable Hamiltonian systems. Numerical experiments with a completely integrable nonseparable Hamiltonian and a nonintegrable nonseparable Hamiltonian illustrate the good long-term behavior and high efficiency of the explicit symplectic integrators proposed and analyzed in this paper.

The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems

TL;DR

The paper addresses the challenge of constructing explicit symplectic integrators for general nonseparable Hamiltonian systems. It develops a constructive existence proof by embedding the original problem in an extended phase space, then mapping explicit symplectic solutions back to the original space via nonlinear projections while preserving the symplectic structure. A backward-error analysis shows linear growth of global errors and near conservation of first integrals for (near-)integrable cases, with careful treatment of nonintegrable dynamics into regular and ergodic regimes. Numerical experiments on completely integrable and nonintegrable nonseparable Hamiltonians demonstrate good long-term behavior and superior efficiency of the proposed explicit integrators compared to implicit counterparts, validating the theoretical findings.

Abstract

The existence of explicit symplectic integrators for general nonseparable Hamiltonian systems is an open and important problem in both numerical analysis and computing in science and engineering, as explicit integrators are usually more efficient than the implicit integrators of the same order of accuracy. Up to now, all responses to this problem are negative. That is, there exist explicit symplectic integrators only for some special nonseparable Hamiltonian systems, whereas the universal design involving explicit symplectic integrators for general nonseparable Hamiltonian systems has not yet been studied sufficiently. In this paper, we present a constructive proof for the existence of explicit symplectic integrators for general nonseparable Hamiltonian systems via finding explicit symplectic mappings under which the special submanifold of the extended phase space is invariant. It turns out that the proposed explicit integrators are symplectic in both the extended phase space and the original phase space. Moreover, on the basis of the global modified Hamiltonians of the proposed integrators, the backward error analysis is made via a parameter relaxation and restriction technique to show the linear growth of global errors and the near-preservation of first integrals. In particular, the effective estimated time interval is nearly the same as classical implicit symplectic integrators when applied to (near-) integrable Hamiltonian systems. Numerical experiments with a completely integrable nonseparable Hamiltonian and a nonintegrable nonseparable Hamiltonian illustrate the good long-term behavior and high efficiency of the explicit symplectic integrators proposed and analyzed in this paper.

Paper Structure

This paper contains 23 sections, 9 theorems, 168 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Some symplectic integrators in the literature
  3. Symplectic integrators derived in the original phase space
  4. Splitting methods for separable Hamiltonian systems
  5. Symplectic methods based on generating functions
  6. Symplectic Runge--Kutta methods
  7. Symplectic extended phase space methods
  8. Extended phase space methods
  9. Semiexplicit symplectic methods
  10. Existence of explicit symplectic integrators
  11. Symplectic integrators with projection
  12. Symplectic integrator with standard projection
  13. Symplectic integrator with generalized projection
  14. Are Pihajoki's or Tao's explicit methods symplectic?
  15. Symmetry of symplectic integrators with projection
  16. ...and 8 more sections

Key Result

Lemma 3.1

Suppose that the mapping $\Phi:\ T^*\mathbb{R}^{2d}\rightarrow T^*\mathbb{R}^{2d}$ is symplectic in the extended phase space. If $\Phi(\mathcal{N})\subset \mathcal{N}$, i.e., for any $p,q\in\mathbb{R}^{d}$, there exist $\tilde{p},\tilde{q}\in \mathbb{R}^{d}$ such that $\Phi(p,p,q,q) =(\tilde{p},\til

Figures (8)

  • Figure 1: The exact solutions of the original Hamiltonian, the Poincaré sections at the surface $x=0$ of the extended Hamiltonian, and the exponential error growth of the 2nd-order Pihajoki's original extended phase space integrator for Problem \ref{['problem-1']}.
  • Figure 2: The global errors and energy errors with $h=0.01$.
  • Figure 3: The convergence orders and the efficiency curves.
  • Figure 4: Dependence of the fourth-order integrator ExpSymp4 on the factors $(\lambda_0,\mu_0)$.
  • Figure 5: The performance of different strategies for the symplectic integrator.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1
  • Definition 3.3
  • Theorem 3.4
  • proof
  • Remark 3.2
  • Definition 4.1: Symplectic integrator with standard projection
  • ...and 18 more