Geometric integration of the Vlasov-Maxwell system with a variational particle-in-cell scheme

Abstract

A fully variational, unstructured, electromagnetic particle-in-cell integrator is developed for integration of the Vlasov-Maxwell equations. Using the formalism of Discrete Exterior Calculus [1], the field solver, interpolation scheme and particle advance algorithm are derived through minimization of a single discrete field theory action. As a consequence of ensuring that the action is invariant under discrete electromagnetic gauge transformations, the integrator exactly conserves Gauss's law.

Variational formulations of guiding-center Vlasov-Maxwell theory

“…In particular, the Hamiltonian structure of single-particle guiding-center dynamics has proved tremendously useful [1] in the theoretical and numerical analysis of magnetically-confined plasmas. Because of the recent development of variational integration numerical techniques [2][3][4][5][6], it is the primary purpose of the present paper to investigate the variational structures of the guiding-center Vlasov-Maxwell equations.…”

“…In particular, the Hamiltonian structure of single-particle guiding-center dynamics has proved tremendously useful [1] in the theoretical and numerical analysis of magnetically-confined plasmas. Because of the recent development of variational integration numerical techniques [2][3][4][5][6], it is the primary purpose of the present paper to investigate the variational structures of the guiding-center Vlasov-Maxwell equations.The guiding-center Vlasov-Maxwell equations describe the coupled time evolution of the guiding-center Vlasov distribution function f µ (X, p , t), and the electromagnetic fields E(x, t) and B(x, t). Here, X denotes the guiding-center position, while x denotes the field position, p denotes the parallel guiding-center (kinetic) momentum, µ denotes the guiding-center magnetic moment (which is a guiding-center invariant), and the guidingcenter gyroangle θ (which is canonically conjugate to the guiding-center gyroaction µ B/Ω) is an ignorable coordinate [1] (i.e., ∂f µ /∂θ ≡ 0).…”

The Hamiltonian Structure and Euler-Poincare Formulation of the Valsov-Maxwell and Gyrokinetic System

“…One approach, which has yielded fruitful results, is to discretize a variational principle and perform variations on the discrete action to derive an integration scheme. Some examples of field theoretic integrators constructed in this way are those for elastomechanics 29,30 electromagnetism 31 , fluids and magnetohydrodynamics 32,33 and a particle-in-cell (PIC) scheme for the Vlasov-Maxwell system 34 .…”

“…Analogously, a variational principle in Lagrangian form is used to construct a Lagrangian (particle-in-cell) integrator 34 . We note that in discretizing a variational principle it is obviously not desirable to be in an extended phase space, unless these extra dimensions can somehow be removed after a discretization.…”

Finite-dimensional collisionless kinetic theory

“…8 for a compatible fully-discrete Vlasov-Maxwell system.) Here compatibility refers to the fact that the truncated models are finite-dimensional Hamiltonian systems with Hamiltonians and Poisson brackets inherited from the corresponding continuum theory.…”