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Diagnosing symplecticity in simulations of high-dimensional Hamiltonian systems

William Barham, J. W. Burby

TL;DR

The paper introduces a computable diagnostic based on the first Poincaré integral invariant to quantify preservation of the symplectic form in Hamiltonian simulations. It demonstrates, through high- and low-regularity analyses and PIC-based experiments, that time-stepping with piecewise linear interpolation fails to be symplectic, while higher-order interpolation restores symplecticity in Strang-splitting schemes. The diagnostic converges spectrally for smooth data and reveals a critical link between spatial regularity and symplecticity preservation, with smoothing unable to fix linear-interpolation failures. Practically, the work underscores that truly structure-preserving PIC methods require globally $C^1$ interpolation (e.g., cubic B-splines) to ensure long-time symplectic behavior and energy stability.

Abstract

Integrals of the Liouville $1$-form, known as the first Poincaré integral invariant, provide a computable figure of merit for monitoring the conservation of symplecticity in the numerical integration of Hamiltonian systems. These integrals may be approximated with spectral convergence in the number of sample points, limited only by the regularity of the Hamiltonian. We devise a numerical integral invariant diagnostic for checking preservation of symplecticity in particle-in-cell (PIC) kinetic plasma simulation codes. As a first application of this diagnostic tool, we check the preservation of symplecticity in symplectic electrostatic particle-in-cell (PIC) methods. Surprisingly, such PIC methods fail to have symplectic time-advance maps if the charge is interpolated to the grid using linear shape functions, as is commonly done in practice. It is found that at least quadratic interpolation is needed for a structure-preserving PIC method to truly be symplectic.

Diagnosing symplecticity in simulations of high-dimensional Hamiltonian systems

TL;DR

The paper introduces a computable diagnostic based on the first Poincaré integral invariant to quantify preservation of the symplectic form in Hamiltonian simulations. It demonstrates, through high- and low-regularity analyses and PIC-based experiments, that time-stepping with piecewise linear interpolation fails to be symplectic, while higher-order interpolation restores symplecticity in Strang-splitting schemes. The diagnostic converges spectrally for smooth data and reveals a critical link between spatial regularity and symplecticity preservation, with smoothing unable to fix linear-interpolation failures. Practically, the work underscores that truly structure-preserving PIC methods require globally interpolation (e.g., cubic B-splines) to ensure long-time symplectic behavior and energy stability.

Abstract

Integrals of the Liouville -form, known as the first Poincaré integral invariant, provide a computable figure of merit for monitoring the conservation of symplecticity in the numerical integration of Hamiltonian systems. These integrals may be approximated with spectral convergence in the number of sample points, limited only by the regularity of the Hamiltonian. We devise a numerical integral invariant diagnostic for checking preservation of symplecticity in particle-in-cell (PIC) kinetic plasma simulation codes. As a first application of this diagnostic tool, we check the preservation of symplecticity in symplectic electrostatic particle-in-cell (PIC) methods. Surprisingly, such PIC methods fail to have symplectic time-advance maps if the charge is interpolated to the grid using linear shape functions, as is commonly done in practice. It is found that at least quadratic interpolation is needed for a structure-preserving PIC method to truly be symplectic.

Paper Structure

This paper contains 33 sections, 6 theorems, 160 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Hamiltonian systems and symplecticity
  4. A conserved loop integral
  5. A conserved loop integral
  6. Approximating the loop integral
  7. Pseudospectral approximation:
  8. Finite difference approximation:
  9. Convergence of the the loop integral approximations
  10. Approximating the loop integral as a function of time
  11. A computable diagnostic for symplecticity
  12. A diagnostic for symplecticity
  13. Illustrative example: the shearing of phase-space loops
  14. Conservative, nearly-symplectic systems
  15. On the symplecticity of low regularity Hamiltonian systems
  16. ...and 18 more sections

Key Result

Lemma 1

Let $f, g \in H^r_{\mathrm{per}}([0,1])$ for some $r > 1/2$. Define the truncated Fourier series where $\hat{f}_k$ and $\hat{g}_k$ are the Fourier coefficients of $f$ and $g$, respectively: Then one finds that

Figures (9)

  • Figure 1: Visualization of the behavior of the nonlinear pendulum.
  • Figure 2: Comparison of approximate loop integral conservation by the nonlinear pendulum for two different initial loops.
  • Figure 3: Interpolated potential conservation results. Sweep in the number of points approximating the loop. The approximation of the diagnostic for the linear case saturates at an error of about $10^{-6}$, indicating a failure to converge the symplectic diagnostic.
  • Figure 4: Interpolated potential conservation results. Sweep in the time-step size.
  • Figure 5: Results for nonlinear pendulum array test.
  • ...and 4 more figures

Theorems & Definitions (20)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Lemma 1: Fourier Galerkin error estimate
  • proof
  • Lemma 2: Aliasing formula for DFT approximation of Fourier coefficients
  • ...and 10 more