We present a novel charge-conserving gather-scatter algorithm for particle-in-cell simulations on unstructured grids. Charge conservation is obtained from first principles, i.e., without the need for any post-processing or correction steps. The key ingredients that enable this property are (1) the use of (discrete) differential forms to represent the electromagnetic fields, currents, and charged particles and to provide consistent localization rules for the degrees of freedom thereof on the different types of geometrical elements (nodes, edges, facets) comprising the unstructured grid, and (2) use of Whitney forms as basic interpolants of such discrete differential forms to continuum space. The present algorithm discretizes the first-order Maxwell's system directly to avoid any spurious modes with secular growth (otherwise present on approaches based on a single field formulation) and on the use of a Galerkin-Whitney representation for the discrete Hodge star operators (i.e., the metric datum of the grid) that is applicable to irregular, unstructured grids. Examples are provided to verify exact preservation of discrete Gauss' law for all times.
Moon et al. also present a higher order, charge conserving algorithm for unstructured grids REF .
Exact Charge-Conserving Scatter-Gather Algorithm for Particle-in-Cell Simulations on Unstructured Grids: A Geometric Perspective
Computer Physics Communications, vol. 194, pp. 43-53, 2015
Mathematics Computer Science