  In this work, we introduce semi-implicit or implicit finite difference
schemes for the continuity equation with a gradient flow structure. Examples of
such equations include the linear Fokker-Planck equation and the Keller-Segel
equations. The two proposed schemes are first order accurate in time,
explicitly solvable, and second order and fourth order accurate in space, which
are obtained via finite difference implementation of the classical continuous
finite element method. The fully discrete schemes are proven
positivity-preserving and energy-dissipative: the second order scheme can
achieve so unconditionally; the fourth order scheme only requires a mild time
step and mesh size constraint. Furthermore, the fourth order scheme is the
first high order spatial discretization that can achieve both positivity and
energy decay properties, which is suitable for long time simulation and to
obtain accurate steady state solutions.
