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GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods

Michael Kraus, Katharina Kormann, Philip J. Morrison, Eric Sonnendrücker

arXiv:1609.03053v3math.NAphysics.comp-phphysics.plasm-ph

TL;DR

The paper addresses how to discretize the Vlasov–Maxwell system while retaining its noncanonical Hamiltonian structure and associated conservation properties. It develops a FEEC-compatible semi-discrete Poisson bracket and combines it with Hamiltonian splitting methods for time integration. The resulting framework preserves bracket structure, Casimirs, divergence constraints, Gauss’ law, and exact charge conservation under compatible finite-element choices.

  • Problem

    The challenge is to preserve the Vlasov–Maxwell system’s Hamiltonian structure, divergence constraints, and charge-conservation properties when using finite particles and finite-element field spaces.

  • Method

    The method discretizes the noncanonical Poisson bracket with finite particle characteristics and compatible FEEC field spaces, then uses Hamiltonian splitting and Poisson time integrators.

  • Results

    The semi-discrete system preserves antisymmetry, the Jacobi identity, and Casimir invariants, while the framework preserves div B and Gauss’ law and supports exact charge conservation.

  • Takeaways & Limitations

    The conservation properties hold independently of the particular finite-element basis when the associated spaces satisfy the required compatibility conditions.

  • Takeaways & Limitations

    Suitable Hamiltonian splittings are not readily available for some cases, and relativistic, collisional, and other non-Hamiltonian extensions remain under development.

Abstract

from arXiv · show

We present a novel framework for Finite Element Particle-in-Cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi identity, as well as conservation of its Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from Finite Element Exterior Calculus ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge invariant, feature exact charge conservation and show excellent long-time energy and momentum behaviour. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the Finite Element basis, as long as the corresponding Finite Element spaces satisfy certain compatibility conditions.

1 Introduction

The Vlasov–Maxwell system has Hamiltonian and variational structure, motivating discretizations that preserve its structural properties. Prior geometric approaches use variational, Poisson, finite-element, and exterior-calculus ideas, while this work develops a general FEEC-based framework.

  • The Vlasov–Maxwell system couples kinetic equations for charged particles with Maxwell equations for electrodynamic phenomena.
  • Its Hamiltonian and variational structures imply conserved quantities associated with symmetries and Casimir functionals.
  • Discrete Maxwell fields require compatible finite-element spaces to preserve the continuity equation, vector-calculus identities, Gauss’ law, and div B.
  • Earlier geometric PIC methods used variational discretizations, discrete exterior calculus, Whitney interpolants, or direct bracket discretization, with differing conservation properties.
  • This work unifies preceding ideas in a general and flexible framework based on Finite Element Exterior Calculus.
  • The paper preserves Hamiltonian structure through a semi-discrete Poisson bracket and Poisson time integrators, rather than treating energy and momentum conservation as the primary goal.

2 The Vlasov–Maxwell System

The Vlasov–Maxwell equations describe charged-particle dynamics coupled to electromagnetic fields and possess a noncanonical Hamiltonian structure with constraints and conserved quantities. The section introduces the bracket formulation, divergence constraints, Casimirs, and momentum maps underlying the discretization.

  • The non-relativistic Vlasov equation for each charged species couples nonlinearly to Maxwell’s equations for the electric and magnetic fields.
  • Gauss’ law and div B are divergence constraints that must remain satisfied because Maxwell’s equations are ill posed when the compatibility condition fails.
  • The Vlasov–Maxwell equations arise from a bilinear, antisymmetric Poisson bracket satisfying Leibniz’ rule and the Jacobi identity.
  • The Hamiltonian is the sum of particle kinetic energy and electric- and magnetic-field energies, and it generates the evolution of functionals.
  • The proposed discretization replaces the continuum distribution with finitely many particle positions and places fields in compatible finite-dimensional subspaces.
  • Casimirs Poisson-commute with every functional and include distribution-function integrals and quantities associated with Gauss’ law and div B.
  • Momentum maps arise from symmetries preserving the Hamiltonian and are conserved precisely when they Poisson-commute with it.

3 Finite Element Exterior Calculus

Finite Element Exterior Calculus organizes compatible finite-element spaces as a discrete deRham complex for stable Maxwell discretization. This structure supports discrete field relations and enforces magnetic-divergence preservation relevant to the Poisson bracket.

  • FEEC is a framework for mixed finite-element methods that uses geometric and topological ideas to analyze stability and convergence.
  • Electromagnetic quantities are represented as differential forms, with E and H as 1-forms, D and B as 2-forms, and charge density as a 3-form.
  • A discrete deRham complex reproduces Im(grad) ⊆ Ker(curl) and Im(curl) ⊆ Ker(div) through compatible finite-dimensional spaces V0, V1, V2, V3.
  • The framework accommodates multiple compatible element sequences, including Nédélec, Raviart–Thomas, mimetic spectral, and spline finite elements.
  • Projecting Maxwell equations onto compatible spaces yields discrete Faraday, Ampère, and Gauss laws, with dual relations handled through weak formulations.
  • Compatible spaces preserve the relation Bh = curl Ah and ensure div B = 0 in strong form.
  • Finite-element bases and dual degrees of freedom represent the discrete fields and their coefficients in finite-dimensional spaces.

4 Discretization of the Hamiltonian Structure

The paper discretizes the Vlasov–Maxwell Poisson bracket using particles and compatible finite-element field spaces, producing a finite-dimensional semi-discrete system. The resulting structure preserves antisymmetry, the Jacobi identity, divergence constraints, Gauss’ law, charge continuity, and Casimir invariants under stated compatibility conditions.

  • 4.1 Discretization: The continuous bracket is discretized by replacing the distribution function with finitely many particle characteristics and the fields with coefficients in finite-element subspaces.The dynamical variables are particle positions and velocities together with electric- and magnetic-field degrees of freedom.
  • 4.3 Discrete Poisson Bracket: Replacing the functional derivatives yields a semi-discrete Poisson bracket for the particle and field degrees of freedom.Its matrix representation uses a Poisson matrix J depending on particle positions and magnetic-field coefficients.
  • 4.4 Jacobi Identity: The discrete Poisson matrix is immediately antisymmetric, while the Jacobi identity follows from divergence-free magnetic fields and finite-element de Rham compatibility.The curl of the one-form basis must be represented in the two-form basis through the curl matrix, a condition satisfied by spaces forming a de Rham complex.
  • 4.4 Jacobi Identity: Choosing initially divergence-free magnetic fields and using a discrete de Rham complex preserves div Bh for all times.The magnetic field must be closed as a 2-form, but it need not be exact or represent the curl of a vector potential.
  • 4.6–4.7 Discrete Constraints: The discretization conserves discrete Gauss’ law and includes a discrete continuity equation for charge conservation.Gauss’ law appears both through conserved Casimirs and through the discrete evolution relation.
  • 4.7 Discrete Casimir Invariants: The identified discrete Casimir invariants remain valid when the finite-element spaces form a complex, even if the de Rham sequence is not exact.Exactness is useful for identifying the invariants but is not required for J(u)DC(u)=0.

5 Hamiltonian Splitting

The discrete Vlasov–Maxwell system is split into Hamiltonian subsystems whose exact flows compose into Poisson structure-preserving integrators. Lie–Trotter and Strang compositions provide first- and second-order methods, while modified-Hamiltonian analysis explains their energy behaviour.

  • Subsystem splitting: The discrete Vlasov–Maxwell equations are split into particle, electric-field, and magnetic-field subsystems.The state is written as u = (X, V, e, b)⊤, and the subsystems are generated by Hp, HE, and HB.
  • Subsystem splitting: Each subsystem’s exact solution is a Poisson map, so their composition yields Poisson structure-preserving integration methods.This construction preserves the Poisson structure because compositions of Poisson maps remain Poisson maps.
  • Subsystem splitting: The particle Hamiltonian is further split because general magnetic-field coefficients prevent exact integration of the full particle subsystem.Each velocity component can then be treated separately because its evolution does not depend on the corresponding velocity component.
  • Composition methods: Lie–Trotter composition gives a first-order integrator, whereas symmetric Strang composition gives a second-order integrator.Higher-order methods can be constructed by composition, and a free parameter α = 0.1932 is reported to produce particularly small error for one second-order composition.
  • Composition methods: The Lie splitting shares properties with Strang splitting because its exact subsystem flows are related by the group property of the flow.This relation does not generally hold when arbitrary symplectic or Poisson integrators replace the exact subsystem solutions.
  • Modified Hamiltonian: The Lie–Trotter integrator preserves a modified energy H̃ = H + Δt H̃1 to O(Δt^2), while the original energy is preserved only to O(Δt).The first-order correction H̃1 is expressed through Poisson brackets among the split Hamiltonians.

6 Example: Vlasov–Maxwell in 1D2V

A 1D2V reduction of the Vlasov–Maxwell system yields a compatible reduced Poisson formulation and a corresponding finite-element discretization. The reduced bracket retains the Jacobi identity, while the discrete equations and split flows correspond directly to the reduced continuous system.

  • Reduction assumptions: The 1D2V reduction assumes x = (x, 0, 0), v = (v1, v2, 0), E = (E1, E2, 0), B = (0, 0, B3), and f = f(x, v1, v2, t).Under these assumptions, the Vlasov and Maxwell equations reduce to a lower-dimensional system.
  • Reduced continuous system: The reduced geometry makes div B = 0 manifest, and the reduced bracket with its Hamiltonian generates the reduced Vlasov–Maxwell equations.The bracket and Hamiltonian are obtained by applying the stated reduction to the full formulation.
  • Reduced Jacobi identity: The reduced bracket satisfies the Jacobi identity because reduced functionals are closed under the full Vlasov–Maxwell bracket.This closure argument avoids relying only on direct calculation of the reduced expression.
  • Reduced Jacobi identity: Not every functional reduction has closure: functionals depending on E and f but not B produce a bracket that depends on B.For this restricted class, omitting the final term also destroys the Jacobi identity; adding a projector can remedy the issue.
  • Finite-element discretization: The finite-element field solver represents E_h with degree-p splines and D_h, B_h with degree-(p−1) splines.The discretization uses separate differential-form components for E1, E2, and B3 on an equidistant grid.
  • Finite-element discretization: The discrete reduced Poisson bracket satisfies the Jacobi identity, and its equations of motion correspond directly to the reduced continuous equations.The discrete Hamiltonian is formulated in the degrees of freedom u = (X, V1, V2, d, e, b).

7 Numerical Experiments

Numerical experiments test the Hamiltonian splitting method on Weibel, streaming Weibel, and strong Landau damping problems. The method verifies analytic growth rates, preserves Gauss’ law exactly, improves energy accuracy with splitting order, and conserves momentum only approximately.

  • Weibel Instability: The Weibel simulation verifies the analytic growth rate 0.02784 using 100,000 particles, 32 grid points, degree-3 and degree-2 splines, and ∆t = 0.05.The analytic growth rate is compared with the numerical electric and magnetic energies.
  • Weibel Instability: Hamiltonian splitting satisfies Gauss’ law at every time step, whereas the Boris–Yee scheme does not.Gauss’ law is measured as the maximum coefficient difference between electric fields computed by Ampere’s and Poisson’s equations.
  • Weibel Instability: Energy error improves with splitting order, but neither Hamiltonian splitting nor Boris–Yee conserves energy exactly.The total-energy error over time is shown for the various integrators.
  • Streaming Weibel Instability: The streaming Weibel simulation verifies the reported growth rate 0.03 for the second electric-field component.The simulation uses 20,000,000 particles, 128 grid points, degree-3 and degree-2 splines, and ∆t = 0.01.
  • Streaming Weibel Instability: For streaming Weibel, Hamiltonian splitting again conserves Gauss’ law, while integrators show approximately the same energy behavior as in the Weibel test.The comparison uses maximum errors through time 200.
  • Strong Landau Damping: Strong Landau damping produces damping and growth rates in good agreement with other codes, while fourth-order methods provide excellent energy conservation.The electric energy and fitted rates are reported alongside the time evolution of the energy error.
  • Momentum Conservation: The numerical scheme does not conserve momentum exactly, although Strang splitting reaches a maximum deviation of 1.6×10^-16 in the reported simulation.The error remains small during the linear phase, with 20,000,000 particles, 128 grid points, and ∆t = 0.01.

8 Summary

The framework combines a Hamiltonian finite-element semi-discretization with Hamiltonian splitting in time to produce a geometric Particle-in-Cell method. It preserves key constraints and Casimirs exactly while maintaining favourable long-time energy behaviour, with extensions to other spaces and systems left for future work.

  • Framework: The method first discretizes the noncanonical Poisson bracket, preserving the Jacobi identity and important Casimir invariants so the finite-dimensional system remains Hamiltonian.The time discretization then applies Hamiltonian splitting methods while retaining exact Casimir conservation.
  • Conservation: Exact conservation of Casimirs yields exact conservation of Gauss’ law and div B = 0 in the fully discrete method.These properties make the scheme a genuine Poisson integrator.
  • Long-time behaviour: Energy is not preserved exactly, but backward error analysis shows that its error is independent of the degrees of freedom, particle count, and number of time steps.
  • Numerical verification: The numerical properties were verified in 1d2v experiments using splines as basis functions for the electromagnetic fields.The implementation cost is comparable to existing charge-conserving methods, while the particle pusher requires exact computation of some line integrals.
  • Extensions: The framework can use other finite-element spaces forming a deRham complex, including mimetic spectral, N´ed´elec, and Raviart–Thomas elements.Extensions to gyrokinetic and relativistic systems, and to non-Hamiltonian effects such as collisions, are identified as future research directions.

A The Boris–Yee scheme

The Boris–Yee scheme advances the conforming finite-element Vlasov–Maxwell variables with time staggering. Initialization samples particle data, solves Poisson’s equation for the initial electric displacement, and uses a half-step update that preserves second-order accuracy overall.

  • Time staggering: The scheme uses staggered variables, with particle positions and field displacement, electric field, velocity, and magnetic field evaluated at alternating integer and half-integer times.
  • Time step: Starting from the previous staggered state, the Vlasov–Maxwell system is propagated through the specified time-step equations.
  • Time step: The magnetic field is computed first, followed by the electric displacement according to the scheme’s staged update.
  • Initialization: Initialization samples X0 and V0, sets the initial fields, and solves Poisson’s equation for d0 before performing a half-step update.The half-step introduces an error of order (∆t)2 only in the first step, so the overall scheme remains second order.
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