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Exact local conservation of energy in fully implicit PIC algorithms

Luis Chacon, Guangye Chen

TL;DR

A temporally, spatially, and particle-discrete local energy conservation theorem is shown to exist, proving that these formulations (as originally proposed in the literature), in addition to being locally charge conserving, are strictly locally energy conserving as well.

Abstract

We consider the issue of strict, fully discrete \emph{local} energy conservation for a whole class of fully implicit local-charge- and global-energy-conserving particle-in-cell (PIC) algorithms. Earlier studies demonstrated these algorithms feature strict global energy conservation. However, whether a local energy conservation theorem exists (in which the local energy update is governed by a flux balance equation at every mesh cell) for these schemes is unclear. In this study, we show that a local energy conservation theorem indeed exists. We begin our analysis with the 1D electrostatic PIC model without orbit-averaging, and then generalize our conclusions to account for orbit averaging, multiple dimensions, and electromagnetic models (Darwin). In all cases, a temporally, spatially, and particle-discrete local energy conservation theorem is shown to exist, proving that these formulations (as originally proposed in the literature), in addition to being locally charge conserving, are strictly locally energy conserving as well. In contrast to earlier proofs of local conservation in the literature \citep{xiao2017local}, which only considered continuum time, our result is valid for the fully implicit time-discrete version of all models, including important features such as orbit averaging. We demonstrate the local-energy-conservation property numerically with a paradigmatic numerical example.

Exact local conservation of energy in fully implicit PIC algorithms

TL;DR

A temporally, spatially, and particle-discrete local energy conservation theorem is shown to exist, proving that these formulations (as originally proposed in the literature), in addition to being locally charge conserving, are strictly locally energy conserving as well.

Abstract

We consider the issue of strict, fully discrete \emph{local} energy conservation for a whole class of fully implicit local-charge- and global-energy-conserving particle-in-cell (PIC) algorithms. Earlier studies demonstrated these algorithms feature strict global energy conservation. However, whether a local energy conservation theorem exists (in which the local energy update is governed by a flux balance equation at every mesh cell) for these schemes is unclear. In this study, we show that a local energy conservation theorem indeed exists. We begin our analysis with the 1D electrostatic PIC model without orbit-averaging, and then generalize our conclusions to account for orbit averaging, multiple dimensions, and electromagnetic models (Darwin). In all cases, a temporally, spatially, and particle-discrete local energy conservation theorem is shown to exist, proving that these formulations (as originally proposed in the literature), in addition to being locally charge conserving, are strictly locally energy conserving as well. In contrast to earlier proofs of local conservation in the literature \citep{xiao2017local}, which only considered continuum time, our result is valid for the fully implicit time-discrete version of all models, including important features such as orbit averaging. We demonstrate the local-energy-conservation property numerically with a paradigmatic numerical example.

Paper Structure

This paper contains 20 sections, 124 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Continuum local energy conservation equation in the Vlasov-Darwin model
  3. Local energy conservation theorem in a fully discrete 1D electrostatic PIC model
  4. Kinetic energy local conservation law
  5. Electric field energy temporal update equation
  6. Total-energy local conservation law
  7. Generalization of local-energy conservation equation to include orbit-averaging
  8. Generalization to multiple dimensions
  9. Kinetic-energy conservation law
  10. Electrostatic energy local conservation law
  11. Total-energy local conservation law
  12. Generalization to EM (Darwin)
  13. Kinetic energy local conservation law
  14. Electrostatic energy local conservation law
  15. Magnetic energy local conservation law
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Modified two-stream instability problem: Electrostatic energy growth and saturation with timestep $\Delta t=0.2\omega_{ce}^{-1}$.
  • Figure 2: Modified two-stream instability problem: Simulation results of various terms in the energy balance equation at two different times: $\omega_{ce}t=10000,20000$. Panels a-e depict $(e_{k,g}^{n+1}-e_{k,g}^{n})/\Delta t$, $(\mathcal{E}_{g}^{n+1}-\mathcal{E}_{g}^{n})/\Delta t$, $\nabla_{h}\cdot\boldsymbol{\Gamma}_{g+1/2}$, $\nabla_{h}\cdot\boldsymbol{\Gamma}_{E,g+1/2}$ and $\nabla_{h}\cdot\boldsymbol{\Gamma}_{K,g+1/2}$ in Eq. \ref{['eq:energy-balance-es1d']} respectively. Panel f denotes the residue of the balance equation, which vanishes to numerical round-off.