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A structure-preserving finite element framework for the Vlasov-Maxwell system

Katharina Kormann, Murtazo Nazarov, Junjie Wen

TL;DR

This work addresses the challenge of robust, high-order numerical simulation of the Vlasov-Maxwell system by introducing a structure-preserving, tensor-product finite element framework. It combines a Vlasov discretization on tensor-product spaces with curl- and div-conforming Maxwell spaces (Nédélec and Raviart-Thomas) and a novel residual-based, anisotropic viscosity to stabilize advection in high dimensions, while enforcing Gauss' laws through a regularized Maxwell system and a modified current. The key contributions are a computationally efficient residual-based viscosity that scales as $\mathcal{O}(N_x+N_v)$, a Kronecker-product assembly strategy for high-dimensional problems, and a comprehensive set of 1D2V/2D2V benchmarks demonstrating mass conservation, Gauss' law preservation, and optimal convergence across polynomial orders. The method offers a robust, conservative, high-order approach for plasma simulations in reduced dimensions, with potential impact on fusion-relevant modeling and high-dimensional kinetic calculations.

Abstract

We present a stabilized, structure-preserving finite element framework for solving the Vlasov-Maxwell equations. The method uses a tensor product of continuous polynomial spaces for the spatial and velocity domains, respectively, to discretize the Vlasov equation, combined with curl- and divergence-conforming Nédélec and Raviart-Thomas elements for Maxwell's equations on Cartesian grids. A novel, robust, consistent, and high-order accurate residual-based artificial viscosity method is introduced for stabilizing the Vlasov equations. The proposed method is tested on the 1D2V and 2D2V reduced Vlasov-Maxwell system, achieving optimal convergence orders for all polynomial spaces considered in this study. Several challenging benchmarks are solved to validate the effectiveness of the proposed method.

A structure-preserving finite element framework for the Vlasov-Maxwell system

TL;DR

This work addresses the challenge of robust, high-order numerical simulation of the Vlasov-Maxwell system by introducing a structure-preserving, tensor-product finite element framework. It combines a Vlasov discretization on tensor-product spaces with curl- and div-conforming Maxwell spaces (Nédélec and Raviart-Thomas) and a novel residual-based, anisotropic viscosity to stabilize advection in high dimensions, while enforcing Gauss' laws through a regularized Maxwell system and a modified current. The key contributions are a computationally efficient residual-based viscosity that scales as , a Kronecker-product assembly strategy for high-dimensional problems, and a comprehensive set of 1D2V/2D2V benchmarks demonstrating mass conservation, Gauss' law preservation, and optimal convergence across polynomial orders. The method offers a robust, conservative, high-order approach for plasma simulations in reduced dimensions, with potential impact on fusion-relevant modeling and high-dimensional kinetic calculations.

Abstract

We present a stabilized, structure-preserving finite element framework for solving the Vlasov-Maxwell equations. The method uses a tensor product of continuous polynomial spaces for the spatial and velocity domains, respectively, to discretize the Vlasov equation, combined with curl- and divergence-conforming Nédélec and Raviart-Thomas elements for Maxwell's equations on Cartesian grids. A novel, robust, consistent, and high-order accurate residual-based artificial viscosity method is introduced for stabilizing the Vlasov equations. The proposed method is tested on the 1D2V and 2D2V reduced Vlasov-Maxwell system, achieving optimal convergence orders for all polynomial spaces considered in this study. Several challenging benchmarks are solved to validate the effectiveness of the proposed method.

Paper Structure

This paper contains 20 sections, 5 theorems, 80 equations, 10 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Governing equations
  4. Viscous regularization of the Vlasov--Maxwell system
  5. Finite element approximations
  6. Finite element framework for Vlasov--Maxwell equations
  7. First-order artificial viscosity
  8. Nonlinear residual-based viscosity
  9. A novel residual-based viscosity
  10. Tensor product FE approximations for Vlasov--Maxwell equations
  11. Finite element approximation of the Maxwell’s equations
  12. Computation of the densities
  13. Numerical examples
  14. Vlasov--Maxwell in 1D2V and 2D2V phase-space
  15. Landau damping
  16. ...and 5 more sections

Key Result

Proposition 1

The following quantities are conservative for the solution of Vlasov--Maxwell equations:

Figures (10)

  • Figure 1: Illustration of the tensor product meshes.
  • Figure 2: Landau damping: electric energy with fitted damping and growth rates. The number of degrees of freedoms is $31\times61^2$ for all the results presented here.
  • Figure 3: Landau damping: time evolution of ${\mathcal{E}}$ (a) and Gauss' error (b). The solution is obtained using ${\mathbb Q}_1$ elements, and the number of degrees of freedoms is $33^2\times65^2$.
  • Figure 4: Weibel instability: the two electric and the magnetic energy together with the analytic growth rate. The number of degrees of freedoms is $31\times61^2$.
  • Figure 5: Streaming Weibel instability: the two electric and the magnetic energy together with the analytic growth rate. The number of degrees of freedoms is $31\times61^2$.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Proposition 1: Conservation
  • proof
  • remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • remark 2
  • remark 3
  • remark 4
  • ...and 7 more