  The parabolic-elliptic Keller-Segel equation with sensitivity saturation,
because of its pattern formation ability, is a challenge for numerical
simulations. We provide two finite-volume schemes whose goals are to preserve,
at the discrete level, the fundamental properties of the solutions, namely
energy dissipation, steady states, positivity and conservation of total mass.
These requirements happen to be critical when it comes to distinguishing
between discrete steady states, Turing unstable transient states, numerical
artifacts or approximate steady states as obtained by a simple upwind approach.
These schemes are obtained either by following closely the gradient flow
structure or by a proper exponential rewriting inspired by the
Scharfetter-Gummel discretization. An interesting feature is that upwind is
also necessary for all the expected properties to be preserved at the
semi-discrete level. These schemes are extended to the fully discrete level and
this leads us to tune precisely the terms according to explicit or implicit
discretizations. Using some appropriate monotony properties (reminiscent of the
maximum principle), we prove well-posedness for the scheme as well as all the
other requirements. Numerical implementations and simulations illustrate the
respective advantages of the three methods we compare.
