  Combining the classical theory of optimal transport with modern operator
splitting techniques, we develop a new numerical method for nonlinear, nonlocal
partial differential equations, arising in models of porous media, materials
science, and biological swarming. Our method proceeds as follows: First, we
discretize in time, either via the classical JKO scheme or via a novel
Crank-Nicolson type method we introduce. Next, we use the Benamou-Brenier
dynamical characterization of the Wasserstein distance to reduce computing the
solution of the discrete time equations to solving fully discrete minimization
problems, with strictly convex objective functions and linear constraints.
Third, we compute the minimizers by applying a recently introduced, provably
convergent primal dual splitting scheme for three operators [Yan 2018].
  By leveraging the PDEs' underlying variational structure, our method
overcomes stability issues present in previous numerical work built on explicit
time discretizations, which suffer due to the equations' strong nonlinearities
and degeneracies. Our method is also naturally positivity and mass preserving
and, in the case of the JKO scheme, energy decreasing. We prove that minimizers
of the fully discrete problem converge to minimizers of the spatially
continuous, discrete time problem as the spatial discretization is refined. We
conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one
and two dimensions that illustrate the key properties of our approach,
including higher order convergence our novel Crank-Nicolson type method, when
compared to the classical JKO method.
