Towards High-Order Unstructured Grid Particle-In-Cell Methods for

Jan Hesthaven
Brown University

Particle-in-cell (PIC) methods have, during the last few decades, been a highly successful computation tool for the modeling of a variety of plasma phenomena and devises, including fusion devises. However, the limitations of standard PIC methods, most often based on 2nd order accurate field solvers on Cartesian grids and simple particle shapes, are beginning to show up in large scale applications like high frequency, high power microwave generation and propagation, and high frequency particle accelerator modeling.
The reliance on a simple Cartesian grid, essential to the success of the method,
in the finite-difference time-domain (FDTD) solver severely limits the geometric
flexibility as well as the accuracy of the method. This effectively limits the
use of this tool to small and geometrically simple problems with dynamics containing only a limited range of dynamic scales. Other known patologies, caused by fundamental properties of the scheme, are finite-grid instabilities
and numerical Cherenkov radiation for relativistic particles. We shall discuss the ongoing development of a high-order accurate PIC
method on unstructured grids. The field solver is based on a Discontinuous Galerkin (DG) scheme for the time-domain Maxwell equations. The ensures geometric flexibility through a fully unstructured grid at arbitrary order of accuracy enabling efficient and accurate modeling of multiscale phenomena. Furthermore, some of
the known problems with classic methods are effectively eliminated by the very
properties of this new formulation.
The particle mover utilizes on high-order interpolation, efficient local search
algorithms to locate the particles, and a level set approach to implement elastic/inelastic interactions with boundaries. Charge and current redistribution computations use large smooth weight functions, dramatically decreasing the number of particles needed in a computational cell. Divergence control is done either through a fully hyperbolic Lagrange multiplier
technique or using a classical Boris projection scheme. We shall illustrate the performance of the two-dimensional and three-dimensional
algorithm through a few examples, e.g., plasma waves, Landau damping, magnetrons etc. Both non-relativistic and fully relativistic cases shall be considered as well as some simple magnetron cases with more complex geometries.
We shall also discuss the performance of the scheme for the GEM-challengde as an example of a fusion relevant application with magnetic reconnection This work is ongoing in collaboration with G. Jacobs (Brown) and G.
Lapenta (LANL).

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