  This paper proposes a second-order accurate numerical scheme for the
Patlak-Keller-Segel system with various mobilities for the description of
chemotaxis. Formulated in a variational structure, the entropy part is novelly
discretized by a modified Crank-Nicolson approach so that the solution to the
proposed nonlinear scheme corresponds to a minimizer of a convex functional. A
careful theoretical analysis reveals that the unique solvability and
positivity-preserving property could be theoretically justified. More
importantly, such a second order numerical scheme is able to preserve the
dissipative property of the original energy functional, instead of a modified
one. To the best of our knowledge, the proposed scheme is the first
second-order accurate one in literature that could achieve both the numerical
positivity and original energy dissipation. In addition, an optimal rate
convergence estimate is provided for the proposed scheme, in which rough and
refined error estimate techniques have to be included to accomplish such an
analysis. Ample numerical results are presented to demonstrate robust
performance of the proposed scheme in preserving positivity and original energy
dissipation in blowup simulations.
