An Unstructured Body-of-Revolution Electromagnetic Particle-in-Cell Algorithm with Radial Perfectly Matched Layers and Dual Polarizations

TL;DR

The work presents BORPIC++, a fully kinetic particle-in-cell algorithm for axisymmetric plasmas on unstructured meshes in body-of-revolution geometries. It employs a cylindrical-to-Cartesian domain mapping and a dual-polarization decomposition ($TE$-$\phi$ and $TM$-$\phi$) to use a Cartesian finite-element Maxwell solver in the meridian plane while removing the axis singularity. The method features open radial boundaries via a cylindrical radial PML, charge-conserving gather/scatter operations, and a discretization strategy based on differential forms (Whitney elements) that yields stable, structure-preserving updates for both polarizations. The results validate accurate wave absorption, realistic ring and gyromotion dynamics, and efficient potential for parallelization via sparse approximate inverses, enabling large-scale, open-boundary axisymmetric PIC simulations with complex geometries and currents.$

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-phi and TM-phi polarizations on the finite element (primal) mesh. 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-phi polarized fields. Several numerical examples are provided to illustrate the first application of the algorithm.

Figures (19)

• Figure 1: Approximation of the motion of charged particles in the presence of a strong axial magnetic field.
• Figure 2: Localization of the field degrees of freedom on the primal and dual meshes.
• Figure 3: The regular-mesh counterpart of the field discretization on an irregular mesh.
• Figure 4: Causal field dependency after time discretization, for both polarizations.
• Figure 5: The overall update schematic of the proposed BORPIC algorithm.