Local, Explicit, and Charge-Conserving Electromagnetic Particle-In-Cell Algorithm on Unstructured Grids

Abstract

A novel electromagnetic particle-in-cell algorithm has been developed for fully kinetic plasma simulations on unstructured (irregular) meshes in complex body-of-revolution geometries. The algorithm, implemented in the BORPIC++ code, utilizes a set of field scalings and a coordinate mapping, reducing the Maxwell field problem in a cylindrical system to a Cartesian finite element Maxwell solver in the meridian plane. The latter obviates the cylindrical coordinate singularity in the symmetry axis. The choice of an unstructured finite element discretization enhances the geometrical flexibility of the BORPIC++ solver compared to the more traditional finite difference solvers. Symmetries in Maxwell's equations are explored to decompose the problem into two dual polarization states with isomorphic representations that enable code reuse. The particle-in-cell scatter and gather steps preserve charge-conservation at the discrete level. Our previous algorithm (BORPIC+) discretized the E and B field components of TE ϕ and TM ϕ polarizations on the finite element (primal) mesh [1,2]. Here, we employ a new field-update scheme. Using the same finite element (primal) mesh, this scheme advances two sets of field components independently: (1) E and B of TE ϕ polarized fields, (E z , E ρ , B ϕ ) and (2) D and H of TM ϕ polarized fields, (D ϕ , H z , H ρ ). Since these field updates are not explicitly coupled, the new field solver obviates the coordinate singularity, which otherwise arises at the cylindrical symmetric axis, ρ = 0 when defining the discrete Hodge matrices (generalized finite element mass matrices). A cylindrical perfectly matched layer is implemented as a boundary condition in the radial direction to simulate open space problems, with periodic boundary conditions in the axial direction. We investigate effects of charged particles moving next to the cylindrical perfectly matched layer. We model azimuthal currents arising from rotational motion of charged rings, which produce TM ϕ polarized fields. Several numerical examples are provided to illustrate the first application of the algorithm.

Finite element time-domain body-of-revolution Maxwell solver based on discrete exterior calculus

“…14. The proposed FETD-BOR solver is incorporated into a PIC algorithm [66,67,68] to simulate the wave-plasma interaction in the device [9]. The PIC algorithm is based on an unstructured grid and explained in detail in [9,24,25]. For this problem it suffices to consider the TE φ polarized field with m = 0.…”

“…It is highly desirable to develop BOR FE solvers in the time domain as well. Timedomain FE solvers are better suited for simulating broadband problems, for capturing transient processes such as those involved in beam-wave interactions [23,24,25], and for handling non-linear problems. However, the use of the second-order vector wave equation as a starting point for a time-domain FE formulation, as done in frequency-domain Maxwell FE solvers, is inadequate.…”

Electromagnetic Theory With Discrete Exterior Calculus

“…Therefore, all electromagnetic phenomena can be predicted and explained with this discrete electromagnetic theory with a appropriate discretization length scale. Besides, due to the charge conservation property, as will be shown, the numerical simulation of Maxwell's equations with DEC will not lead to spurious solutions [3,17].…”

Diagnosing numerical Cherenkov instabilities in relativistic plasma simulations based on general meshes

“…The fundamentals of present finite element time-domain (FETD) Maxwell field solver are briefly summarized in this appendix. A more comprehensive discussion of the Maxwell field solver, together with various details on the scatter, gather, and pusher steps of the EM-PIC algorithm can be found in [16,17,18].…”

“…Thus, the linear solve in (A.6) can be performed very quickly. Nevertheless, since the linear solve is needed at every step n of the time evolution, a sparse approximate inverse (SPAI) of [ ] may be computed priori to the start of the time-update to obviate the need for the linear solve [30,17].…”