  One route to numerically propagating quantum systems is time-dependent
density functional theory (TDDFT). The application of TDDFT to a particular
system's time evolution is predicated on $V$-representability which we have
analyzed in a previous publication. Here we describe a newly developed solver
for the scalar time-dependent Kohn-Sham potential. We present and interpret the
force-balance equation central to our numerical method, describe details of its
implementation, and present illustrative numerical results for one- and
two-electron systems. A new characterization of $V$-representability for
one-electron systems is also included along with possible improvements and
future directions.
