We describe a charge-conserving scatter-gather 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. This algorithm recovers, at a fundamental level, the scatter-gather algorithms presented recently by Campos-Pinto et al. (2014) (to first-order) and by Squire et al. (2012), but it is derived here in a streamlined fashion from a geometric viewpoint. Some ingredients reflecting this viewpoint are (1) the use of (discrete) differential forms of various degrees to represent fields, currents, and charged particles and provide localization rules for the degrees of freedom thereof on the various grid elements (nodes, edges, facets), (2) use of Whitney forms as basic interpolants from discrete differential forms to continuum space, and (3) use of a Galerkin formula for the discrete Hodge star operators (i.e., "mass matrices" incorporating the metric datum of the grid) applicable to generally irregular, unstructured grids. The expressions obtained for the scatter charges and scatter currents are very concise and do not involve numerical quadrature rules. Appropriate fractional areas within each grid element are identified that represent scatter charges and scatter currents within the element, and a simple geometric representation for the (exact) charge conservation mechanism is obtained by such identification. The field update is based on the coupled first-order Maxwell's curl equations to avoid spurious modes with secular growth (otherwise present in formulations that discretize the second-order wave equation). Examples are provided to verify preservation of discrete Gauss' law for all times.
Moon, H., Teixeira, F. L., & Omelchenko, Y. A. (2015). Exact charge-conserving scatter-gather algorithm for particle-in-cell simulations on unstructured grids: A geometric perspective. Computer Physics Communications, 194, 43–53. https://doi.org/10.1016/j.cpc.2015.04.014