  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. [1] (to first-order) and by Squire et
al. [2], 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.
