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An explicit, energy-conserving particle-in-cell scheme

Lee F. Ricketson, Jingwei Hu

Abstract

We present an explicit temporal discretization of particle-in-cell schemes for the Vlasov equation that results in exact energy conservation when combined with an appropriate spatial discretization. The scheme is inspired by a simple, second-order explicit scheme that conserves energy exactly in the Eulerian context. We show that direct translation to particle-in-cell does not result in strict conservation, but derive a simple correction based on an analytically solvable optimization problem that recovers conservation. While this optimization problem is not guaranteed to have a real solution for every particle, we provide a correction that makes imaginary values extremely rare and still admits $\mathcal{O}(10^{-12})$ fractional errors in energy for practical simulation parameters. We present the scheme in both electrostatic -- where we use the Ampère formulation -- and electromagnetic contexts. With an electromagnetic field solve, the field update is most naturally linearly implicit, but the more computationally intensive particle update remains fully explicit. We also show how the scheme can be extended to use the fully explicit leapfrog and pseudospectral analytic time-domain (PSATD) field solvers. The scheme is tested on standard kinetic plasma problems, confirming its conservation properties.

An explicit, energy-conserving particle-in-cell scheme

Abstract

We present an explicit temporal discretization of particle-in-cell schemes for the Vlasov equation that results in exact energy conservation when combined with an appropriate spatial discretization. The scheme is inspired by a simple, second-order explicit scheme that conserves energy exactly in the Eulerian context. We show that direct translation to particle-in-cell does not result in strict conservation, but derive a simple correction based on an analytically solvable optimization problem that recovers conservation. While this optimization problem is not guaranteed to have a real solution for every particle, we provide a correction that makes imaginary values extremely rare and still admits fractional errors in energy for practical simulation parameters. We present the scheme in both electrostatic -- where we use the Ampère formulation -- and electromagnetic contexts. With an electromagnetic field solve, the field update is most naturally linearly implicit, but the more computationally intensive particle update remains fully explicit. We also show how the scheme can be extended to use the fully explicit leapfrog and pseudospectral analytic time-domain (PSATD) field solvers. The scheme is tested on standard kinetic plasma problems, confirming its conservation properties.

Paper Structure

This paper contains 22 sections, 83 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Vlasov Equation
  4. Particle-in-Cell Discretization
  5. Boris particle push
  6. Spatial discretizations and light-wave dispersion
  7. Implicit PIC and Energy Conservation
  8. Charge conservation
  9. The Method
  10. Motivating Eulerian Method
  11. PIC translation in electrostatic case
  12. $\Gamma_p^n$ correction
  13. Straightforward electromagnetic extension
  14. Charge Conservation
  15. Considerations of light-wave dispersion
  16. ...and 7 more sections

Figures (8)

  • Figure 1: Evolution of electrostatic potential energy for linear Landau damping test case. Excellent agreement is observed between all three schemes as well as the theoretical damping rate.
  • Figure 2: Fractional change in total energy over time for the three tested schemes.
  • Figure 3: Number of problematic particles with imaginary correction factors $\Gamma_p^n$ at each time-step for linear Landau damping with $\Delta t = 0.1$.
  • Figure 4: Maximum fractional energy errors for the linear Landau damping test as a function of time-step size.
  • Figure 5: Electrostatic potential energy for the two-stream instability test case, showing good agreement between all schemes and the theoretical linear growth rate.
  • ...and 3 more figures