  In this paper we present a numerical scheme for nonlinear continuity
equations, which is based on the gradient flow formulation of an energy
functional with respect to the quadratic transportation distance. It can be
applied to a large class of nonlinear continuity equations, whose dynamics are
driven by internal energies, given external potentials and/or interaction
energies. The solver is based on its variational formulation as a gradient flow
with respect to the Wasserstein distance. Positivity of solutions as well as
energy decrease of the semi-discrete scheme are guaranteed by its construction.
We illustrate this properties with various examples in spatial dimension one
and two.
