[Paper Review] Rubrics for Charge Conserving Current Mapping in Finite Element Particle in Cell Methods

Zane D. Crawford, Scott O’Connor+2 more|arXiv (Cornell University)|Jan 28, 2021

Plasma Diagnostics and Applications4 citations

TL;DR

This paper establishes general mathematical rubrics for ensuring charge conservation in finite element particle-in-cell (FE-PIC) methods by leveraging de Rham complex mappings and discrete Hodge operators. It demonstrates that charge conservation is achieved when current mapping respects the duality between edge and face-based finite element bases, and presents a novel, inherently charge-conserving FE-PIC scheme validated through numerical results.

ABSTRACT

Modeling of kinetic plasmas using electromagnetic particle in cell methods (EM-PIC) is a problem that is well worn, in that methods developed have been used extensively both understanding physics and exploiting them for device design. EM-PIC tools have largely relied on finite difference methods coupled with particle representations of the distribution function. Refinements to ensure consistency and charge conservation have largely been an ad-hoc efforts specific to finite difference methods. Meanwhile, solution methods for field solver have grown by leaps and bounds with significant performance metrics compared to finite difference methods. Developing new EM-PIC computational schemes that leverage modern field solver technology means re-examining analysis framework necessary for self-consistent EM-PIC solution. In this paper, we prescribe general rubrics for charge conservation, demonstrate how these are satisfied in conventional finite difference PIC as well as finite element PIC, and prescribe a novel charge conserving finite element PIC. Our effort leverages proper mappings on to de-Rham sequences and lays a groundwork for understanding conditions that must be satisfied for consistency. Several numerical results demonstrate the applicability of these rubrics.

Motivation & Objective

• To develop general, solver-agnostic rubrics for charge conservation in electromagnetic particle-in-cell (EM-PIC) simulations.
• To identify the mathematical conditions under which current mapping preserves charge conservation in finite element methods.
• To extend charge-conserving principles from finite difference PIC to modern finite element field solvers.
• To provide a systematic framework for constructing consistent, divergence-free current deposition in FE-PIC schemes.
• To demonstrate the applicability and accuracy of the proposed rubrics through numerical validation.

Proposed method

• Formulates charge conservation as a requirement on the duality between discrete differential forms in the de Rham complex, ensuring that current mapping respects the Hodge star operator and curl relationships.
• Derives discrete Maxwell’s equations using edge-based (1-form) and face-based (2-form) finite element basis functions for electric and magnetic fields.
• Defines current degrees of freedom via line integrals of particle trajectories over edge-based basis functions, ensuring consistency with Ampere’s and Faraday’s laws.
• Applies the discrete Hodge star operator to map between primal and dual meshes, enabling accurate field and current interpolation.
• Uses the de Rham sequence to ensure that the discrete curl and divergence operators satisfy the same algebraic identities as their continuous counterparts.
• Validates the scheme using numerical experiments that confirm charge conservation and consistency with Maxwell’s equations over time.

Experimental results

Research questions

• RQ1 What mathematical conditions must be satisfied for current mapping to ensure charge conservation in finite element PIC methods?
• RQ2 How can charge-conserving current deposition be systematically derived for arbitrary finite element field solvers?
• RQ3 In what way do de Rham complex mappings and discrete Hodge operators enable consistent charge conservation in FE-PIC?
• RQ4 How does the proposed FE-PIC scheme compare to conventional FDTD-PIC in terms of charge conservation and consistency?
• RQ5 Can the proposed rubrics be generalized to other field solver types beyond finite element methods?

Key findings

• The paper establishes that charge conservation in FE-PIC is guaranteed when current mapping respects the duality between edge and face-based finite element basis functions via the de Rham complex.
• The proposed method ensures that the discrete continuity equation is satisfied at every time step, eliminating spurious charge generation.
• Numerical results confirm that the scheme maintains charge conservation to machine precision over long simulation times.
• The method recovers classical FDTD-PIC results when applied to structured grids, validating consistency with established schemes.
• The use of de Rham sequences and discrete Hodge operators ensures that the curl and divergence operators satisfy the same algebraic identities as in the continuous case.
• The framework enables the construction of new, inherently charge-conserving FE-PIC schemes without requiring post-processing or divergence cleaning.