  We propose fully discrete, implicit-in-time finite-volume schemes for a
general family of non-linear and non-local Fokker-Planck equations with a
gradient-flow structure, usually known as aggregation-diffusion equations, in
any dimension. The schemes enjoy the positivity-preservation and
energy-dissipation properties, essential for their practical use. The
first-order scheme verifies these properties unconditionally for general
non-linear diffusions and interaction potentials, while the second-order scheme
does so provided a CFL condition holds. Sweeping dimensional splitting permits
the efficient construction of these schemes in higher dimensions while
preserving their structural properties. Numerical experiments validate the
schemes and show their ability to handle complicated phenomena typical in
aggregation-diffusion equations, such as free boundaries, metastability,
merging and phase transitions.
