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We describe a charge-conserving scatter-gather algorithm for particle-in-cellsimulations on unstructured grids. Charge conservation is obtained from firstprinciples, i.e., without the need for any post-processing or correction steps.This algorithm recovers, at a fundamental level, the scatter-gather algorithmspresented recently by Campos-Pinto et al. [1] (to first-order) and by Squire etal. [2], but it is derived here in a streamlined fashion from a geometricviewpoint. 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 freedomthereof on the various grid elements (nodes, edges, facets), (2) use of Whitneyforms as basic interpolants from discrete differential forms to continuumspace, and (3) use of a Galerkin formula for the discrete Hodge star operators(i.e., "mass matrices" incorporating the metric datum of the grid) applicableto generally irregular, unstructured grids. The expressions obtained for thescatter charges and scatter currents are very concise and do not involvenumerical quadrature rules. Appropriate fractional areas within each gridelement are identified that represent scatter charges and scatter currentswithin the element, and a simple geometric representation for the (exact)charge conservation mechanism is obtained by such identification. The fieldupdate is based on the coupled first-order Maxwell's curl equations to avoidspurious modes with secular growth (otherwise present in formulations thatdiscretize the second-order wave equation). Examples are provided to verifypreservation of discrete Gauss' law for all times.

Exact Charge-Conserving Scatter-Gather Algorithm for Particle-in-Cell Simulations on Unstructured Grids: A Geometric Perspective