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The parabolic-elliptic Keller-Segel equation with sensitivity saturation,because of its pattern formation ability, is a challenge for numericalsimulations. We provide two finite-volume schemes whose goals are to preserve,at the discrete level, the fundamental properties of the solutions, namelyenergy dissipation, steady states, positivity and conservation of total mass.These requirements happen to be critical when it comes to distinguishingbetween discrete steady states, Turing unstable transient states, numericalartifacts or approximate steady states as obtained by a simple upwind approach.These schemes are obtained either by following closely the gradient flowstructure or by a proper exponential rewriting inspired by theScharfetter-Gummel discretization. An interesting feature is that upwind isalso necessary for all the expected properties to be preserved at thesemi-discrete level. These schemes are extended to the fully discrete level andthis leads us to tune precisely the terms according to explicit or implicitdiscretizations. Using some appropriate monotony properties (reminiscent of themaximum principle), we prove well-posedness for the scheme as well as all theother requirements. Numerical implementations and simulations illustrate therespective advantages of the three methods we compare.

Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations