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A dual grid geometric electromagnetic particle in cell method

Katharina Kormann, Eric Sonnendrücker

TL;DR

The paper addresses long-time, structure-preserving simulation of the $Vlasov-Maxwell$ system by developing a dual-grid mimetic finite-difference PIC method. It constructs a commuting discrete de Rham diagram on primal and dual Cartesian grids, derives semi-discrete equations from a discrete action principle, and employs a Hamiltonian-splitting time integrator to propagate the system while preserving key constraints. The main contributions include tensor-product mimetic operators, discrete Hodge mappings, a noncanonical Hamiltonian formulation, and detailed numerical validation (Hodge solver accuracy, wave dispersion, and Weibel instability) demonstrating robust conservation and high-order accuracy. The work enables high-order, geometry-preserving PIC on Cartesian grids with scalable AMReX integration, offering potential improvements in long-time plasma simulations and future extensions to adaptive mesh refinement and complex geometries.

Abstract

Geometric particle-in-cell discretizations have been derived based on a discretization of the fields that is conforming with the de Rham structure of the Maxwell's equation and a standard particle-in-cell ansatz for the fields by deriving the equations of motion from a discrete action principle. While earlier work has focused on finite element discretization of the fields based on the theory of Finite Element Exterior Calculus, we propose in this article an alternative formulation of the field equations that is based on the ideas conveyed by mimetic finite differences. The needed duality being expressed by the use of staggered grids. We construct a finite difference formulation based on degrees of freedom defined as point values, edge, face and volume integrals on a primal and its dual grid. Compared to the finite element formulation no mass matrix inversion is involved in the formulation of the Maxwell solver. In numerical experiments, we verify the conservation properties of the novel method and study the influence of the various parameters in the discretization.

A dual grid geometric electromagnetic particle in cell method

TL;DR

The paper addresses long-time, structure-preserving simulation of the system by developing a dual-grid mimetic finite-difference PIC method. It constructs a commuting discrete de Rham diagram on primal and dual Cartesian grids, derives semi-discrete equations from a discrete action principle, and employs a Hamiltonian-splitting time integrator to propagate the system while preserving key constraints. The main contributions include tensor-product mimetic operators, discrete Hodge mappings, a noncanonical Hamiltonian formulation, and detailed numerical validation (Hodge solver accuracy, wave dispersion, and Weibel instability) demonstrating robust conservation and high-order accuracy. The work enables high-order, geometry-preserving PIC on Cartesian grids with scalable AMReX integration, offering potential improvements in long-time plasma simulations and future extensions to adaptive mesh refinement and complex geometries.

Abstract

Geometric particle-in-cell discretizations have been derived based on a discretization of the fields that is conforming with the de Rham structure of the Maxwell's equation and a standard particle-in-cell ansatz for the fields by deriving the equations of motion from a discrete action principle. While earlier work has focused on finite element discretization of the fields based on the theory of Finite Element Exterior Calculus, we propose in this article an alternative formulation of the field equations that is based on the ideas conveyed by mimetic finite differences. The needed duality being expressed by the use of staggered grids. We construct a finite difference formulation based on degrees of freedom defined as point values, edge, face and volume integrals on a primal and its dual grid. Compared to the finite element formulation no mass matrix inversion is involved in the formulation of the Maxwell solver. In numerical experiments, we verify the conservation properties of the novel method and study the influence of the various parameters in the discretization.

Paper Structure

This paper contains 20 sections, 5 theorems, 79 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Vlasov--Maxwell system
  3. Mimetic finite differences
  4. Tensor-product construction of mimetic finite differences
  5. Reduction and derivative operators
  6. Interpolation-histopolation operators
  7. Discrete Hodge operators
  8. Semi-discrete Action principle
  9. Derivation of the equations of motion
  10. Discrete noncanonical Hamiltonian system
  11. Semi-discrete conservation properties
  12. Propagation in time
  13. Summary of the full scheme
  14. Numerical experiments
  15. Hodge solver
  16. ...and 5 more sections

Key Result

Theorem 1

Let us consider a tensor product grid on a Cartesian domain and let the restriction operators be defined as in Definition def:restriction and the derivative operators as in hD0-eq:relation_dual_derivative_ops. Then, relations eq:R1GR0-eq:tR3DR2 are satisfied.

Figures (3)

  • Figure 1: Comparison of the numerical dispersion relation for the L- and R-mode for varying order of the Maxwell solver (interpolation-histopolation Hodges).
  • Figure 2: Visualization of the conservation properties of the code for the Weibel instability test case.
  • Figure 3: Magnetic energy curves (second component) for various degree of the shape function (see legend) for two different resolutions (see subcaptions).

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Proof 1
  • Theorem 2
  • Proof 2
  • Remark 1
  • Theorem 3
  • Proof 3
  • Lemma 1
  • Proof 4
  • ...and 2 more