The derivative discontinuity of the exchange-correlation (xc) energy at
integer particle number is a property of the exact, unknown xc functional of
density functional theory (DFT) which is absent in many popular local and
semilocal approximations. In lattice DFT, approximations exist which exhibit a
discontinuity in the xc potential at half filling. However, due to convergence
problems of the Kohn-Sham (KS) self-consistency cycle, the use of these
functionals is mostly restricted to situations where the local density is away
from half filling. Here a numerical scheme for the self-consistent solution of
the lattice KS Hamiltonian with a local xc potential with rapid (or
quasi-discontinuous) density dependence is suggested. The problem is formulated
in terms of finite-temperature DFT where the discontinuity in the xc potential
emerges naturally in the limit of zero temperature. A simple parametrization is
suggested for the xc potential of the uniform 1D Hubbard model at finite
temperature which is obtained from the solution of the thermodynamic Bethe
ansatz. The feasibility of the numerical scheme is demonstrated by application
to a model of fermionic atoms in a harmonic trap. The corresponding density
profile exhibits a plateau of integer occupation at low temperatures which
melts away for higher temperatures.Comment: 14 pages, 11 figure