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Structure-preserving finite element approximations of a hybrid relativistic cold fluid-particle model

Tileuzhan Mukhamet, Katharina Kormann

TL;DR

The paper develops structure-preserving finite element discretizations for a hybrid cold relativistic fluid–particle plasma model, exploiting a Poisson-bracket Hamiltonian structure to derive invariant-preserving weak formulations. It introduces two spatial discretizations (flux-free and flux-based) and two time-stepping schemes (an energy-conserving implicit AVF method and an explicit SSP-RK3 method), with Gauss-cleaning to enforce Maxwell constraints when particles are present. The methods conserve total mass and energy and maintain weak Gauss's law and divB at the discrete level, and numerical experiments demonstrate convergence, conservation, and applicability to plasma wake-field simulations. The results indicate a robust framework for long-time simulations of hybrid plasmas that balances efficiency and fidelity against fully kinetic approaches.

Abstract

We derive mixed finite element discretizations of a cold relativistics fluid model from approximations of the Poisson bracket that preserve mass, energy and the divergence constraints. For time-discretization we derive an implicit energy-conserving average-vector field method or apply an explicit strong-stability preserving Runge-Kutta scheme. We also consider a coupling of the fluid model to relativistic particles. We perform a numerical study of the scheme which shows convergence and conservation properties of the proposed methods and apply the new scheme to a plasma wake field simulation.

Structure-preserving finite element approximations of a hybrid relativistic cold fluid-particle model

TL;DR

The paper develops structure-preserving finite element discretizations for a hybrid cold relativistic fluid–particle plasma model, exploiting a Poisson-bracket Hamiltonian structure to derive invariant-preserving weak formulations. It introduces two spatial discretizations (flux-free and flux-based) and two time-stepping schemes (an energy-conserving implicit AVF method and an explicit SSP-RK3 method), with Gauss-cleaning to enforce Maxwell constraints when particles are present. The methods conserve total mass and energy and maintain weak Gauss's law and divB at the discrete level, and numerical experiments demonstrate convergence, conservation, and applicability to plasma wake-field simulations. The results indicate a robust framework for long-time simulations of hybrid plasmas that balances efficiency and fidelity against fully kinetic approaches.

Abstract

We derive mixed finite element discretizations of a cold relativistics fluid model from approximations of the Poisson bracket that preserve mass, energy and the divergence constraints. For time-discretization we derive an implicit energy-conserving average-vector field method or apply an explicit strong-stability preserving Runge-Kutta scheme. We also consider a coupling of the fluid model to relativistic particles. We perform a numerical study of the scheme which shows convergence and conservation properties of the proposed methods and apply the new scheme to a plasma wake field simulation.

Paper Structure

This paper contains 17 sections, 8 theorems, 95 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Hamiltonian structure and weak formulation
  3. Poisson bracket of the cold relativistic plasma model
  4. Weak equations of motion
  5. Spatial discretization
  6. A flux-free formulation
  7. A formulation with fluxes
  8. Temporal discretization
  9. An energy-conserving implicit scheme
  10. Explicit Runge-Kutta
  11. Numerical examples
  12. Convergence of the fluid system.
  13. Conservation properties
  14. Plasma wake simulation
  15. Acknowledgments
  16. ...and 2 more sections

Key Result

Lemma 3.1

The Particle-in-cell equation eq:qqnrt_X_k-eq:qqnrt_U_k satisfy the following charge balance equation

Figures (7)

  • Figure 1: $L^2$-errors in the electric field, magnetic field, fluid density, fluid momentum at time t=$0.5$ with the boundary conditions $\bm n \cdot \bm M|_{\partial \Omega}=0$, $\bm n \times \bm E|_{\partial \Omega} = 0$, $\bm n \cdot \bm B|_{\partial \Omega}=0$. The plots to the left are for the method (\ref{['eq:dt_qqnrt_rho']}-\ref{['eq:dt_qqnrt_w']}), the plots to the right are for the method (\ref{['eq:dt_dgdiv_rho']}-\ref{['eq:dt_dgdiv_w']}).
  • Figure 2: Errors in the total mass, total energy, Gauss's law, and divB constraint for t=$[0,0.3]$. The plots to the left are for the method (\ref{['eq:dt_qqnrt_rho']} - \ref{['eq:dt_qqnrt_U_k']}), the plots to the right are for the method (\ref{['eq:dt_dgdiv_rho']} - \ref{['eq:dt_dgdiv_U_k']}).
  • Figure 3: Errors in the total mass, total energy, Gauss's law, and divB constraint for t=$[0,0.3]$ for the method (\ref{['eq:qqnrt_rho']}-\ref{['eq:qqnrt_U_k']}) integrated in time with SSP RK method of order three. The speed of light $c=10$. The plots to the left are for $N=10^5$ particles, while the plots to the right are for $N=0$, i.e. without particles
  • Figure 4: Errors in the total mass, total energy, Gauss's law, and divB constraint for t=$[0,0.3]$ for the method (\ref{['eq:qqnrt_rho']}-\ref{['eq:qqnrt_U_k']}) integrated in time with SSP RK method of order three. The speed of light $c=10$. Gauss's cleaning is applied every 100 time-steps. The plots are for $N=10^5$ particles.
  • Figure 5: Errors in the total mass, total energy, Gauss's law, and divB constraint for t=$[0,0.3]$ for the method (\ref{['eq:qqnrt_rho']}-\ref{['eq:qqnrt_U_k']}) integrated in time with SSP RK method of order three. The speed of light $c=1$. The plots to the left are for $N=10^5$ particles, while the plots to the right are for $N=0$, i.e. without particles.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Lemma 3.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.1
  • proof
  • ...and 4 more