  We propose a fully discrete variational scheme for nonlinear evolution
equations with gradient flow structure on the space of finite Radon measures on
an interval with respect to a generalized version of the Wasserstein distance
with nonlinear mobility. Our scheme relies on a spatially discrete
approximation of the semi-discrete (in time) minimizing movement scheme for
gradient flows. Performing a finite-volume discretization of the continuity
equation appearing in the definition of the distance, we obtain a
finite-dimensional convex minimization problem usable as an iterative scheme.
We prove that solutions to the spatially discrete minimization problem converge
to solutions of the spatially continuous original minimizing movement scheme
using the theory of $\Gamma$-convergence, and hence obtain convergence to a
weak solution of the evolution equation in the continuous-time limit if the
minimizing movement scheme converges. We illustrate our result with numerical
simulations for several second- and fourth-order equations.
