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Canonical variables based numerical schemes for hybrid plasma models with kinetic ions and massless electrons

Yingzhe Li, Florian Holderied, Stefan Possanner, Eric Sonnendrücker

TL;DR

This paper develops and compares two canonical-momentum xpA formulations for a hybrid kinetic-ion/massless-electron plasma model, implementing PIC in phase space and FEEC for the vector potential. By employing operator splitting with midpoint time integration, it derives both a density-filtered Formulation I and an energy-conserving Formulation II based on an anti-symmetric bracket, showing Formulation II generally offers superior conservation and lower noise. The methods are validated across finite grid instability, R-mode, Bernstein waves, and ion cyclotron instability, demonstrating accurate dispersion and robust long-term behavior, with Formulation I enhanced by density filtering. The work advances structure-preserving discretizations for hybrid plasmas and highlights gauge choices as a practical lever to improve numerical properties, with potential for large-scale simulations and broader comparisons to related approaches.

Abstract

We study the canonical variables based numerical schemes of a hybrid model with kinetic ions and mass-less electrons. Two equivalent formulations of the hybrid model are presented with the vector potentials in different gauges and the distribution functions depending on canonical momentum (not velocity), which constitutes a pair of canonical variables with the position variable. Particle-in-cell methods are used for the distribution functions, and the vector potentials are discretized by the finite element methods in the framework of finite element exterior calculus. Splitting methods are used for the time discretizations. It is illustrated that the second formulation is numerically superior and the schemes constructed based on the anti-symmetric bracket proposed have better conservation properties and lower noise, although the filters can be used to improve the schemes of the first formulation.

Canonical variables based numerical schemes for hybrid plasma models with kinetic ions and massless electrons

TL;DR

This paper develops and compares two canonical-momentum xpA formulations for a hybrid kinetic-ion/massless-electron plasma model, implementing PIC in phase space and FEEC for the vector potential. By employing operator splitting with midpoint time integration, it derives both a density-filtered Formulation I and an energy-conserving Formulation II based on an anti-symmetric bracket, showing Formulation II generally offers superior conservation and lower noise. The methods are validated across finite grid instability, R-mode, Bernstein waves, and ion cyclotron instability, demonstrating accurate dispersion and robust long-term behavior, with Formulation I enhanced by density filtering. The work advances structure-preserving discretizations for hybrid plasmas and highlights gauge choices as a practical lever to improve numerical properties, with potential for large-scale simulations and broader comparisons to related approaches.

Abstract

We study the canonical variables based numerical schemes of a hybrid model with kinetic ions and mass-less electrons. Two equivalent formulations of the hybrid model are presented with the vector potentials in different gauges and the distribution functions depending on canonical momentum (not velocity), which constitutes a pair of canonical variables with the position variable. Particle-in-cell methods are used for the distribution functions, and the vector potentials are discretized by the finite element methods in the framework of finite element exterior calculus. Splitting methods are used for the time discretizations. It is illustrated that the second formulation is numerically superior and the schemes constructed based on the anti-symmetric bracket proposed have better conservation properties and lower noise, although the filters can be used to improve the schemes of the first formulation.
Paper Structure (15 sections, 68 equations, 8 figures)

This paper contains 15 sections, 68 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Canonical momentum based formulations
  3. Numerical discretization
  4. Phase-space discretization: Formulation I
  5. Phase-space discretization: Formulation II
  6. The comparisons of noise
  7. Numerical experiments
  8. Finite grid instability
  9. Parallel electromagnetic wave: R mode
  10. Perpendicular wave: ion Bernstein waves
  11. Ion cyclotron instability
  12. Conclusion
  13. Appendix
  14. The hybrid model with a background magnetic field
  15. $xvA$ formulation

Figures (8)

  • Figure 1: Finite grid instability. Time evolutions of the maximum values of $|A_z(0,0,x_3)|$, and density $n(0,0,x_3)$.
  • Figure 2: Finite grid instability. The contour plots of the scheme \ref{['eq:for1scheme']} with or without filtering and the scheme \ref{['eq:for2scheme']}.
  • Figure 3: Finite grid instability. Time evolutions of the ion temperature, relative energy error, and momentum error (the third component).
  • Figure 4: R-waves. The numerical dispersion relations of the R waves given by 1) the scheme \ref{['eq:for1scheme']} without filter, 2) the scheme \ref{['eq:for1scheme']} with filters, 3) the scheme \ref{['eq:for2scheme']}.
  • Figure 5: R-waves. Time evolutions of the momentum errors (the third component) and relative energy errors given by 1) the scheme \ref{['eq:for1scheme']} without filter, 2) the scheme \ref{['eq:for1scheme']} with filters, 3) the scheme \ref{['eq:for2scheme']}.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8