  This paper contains a study of ADI methods in the presence of charge and
current sources. It is shown that there are four significantly distinct cases,
with four more related by duality. Of those, only one preserves divergence and,
thus, is guaranteed to be stable in the presence of moving charged particles.
Computational verification of this property is accomplished by implementation
in existing 3D-EMPIC simulation software. Of the other three cases, two are
verified unstable, as expected, and one remains stable, despite the lack of
divergence preservation. This other stable algorithm is shown to be related to
the divergence preserving case by a similarity transformation, effectively
providing the complement of the divergence preserving field in the
finite-difference energy quantity.
