  We propose a variational finite volume scheme to approximate the solutions to
Wasserstein gradient flows. The time discretization is based on an implicit
linearization of the Wasserstein distance expressed thanks to Benamou-Brenier
formula, whereas space discretization relies on upstream mobility two-point
flux approximation finite volumes. Our scheme is based on a first discretize
then optimize approach in order to preserve the variational structure of the
continuous model at the discrete level. Our scheme can be applied to a wide
range of energies, guarantees non-negativity of the discrete solutions as well
as decay of the energy. We show that our scheme admits a unique solution
whatever the convex energy involved in the continuous problem, and we prove its
convergence in the case of the linear Fokker-Planck equation with positive
initial density. Numerical illustrations show that it is first order accurate
in both time and space, and robust with respect to both the energy and the
initial profile.
