Primal Dual Methods for Wasserstein Gradient Flows

Abstract 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 porous media, materials science, and biological swarming. Our proceeds as follows: first, discretize time, either via JKO scheme or novel Crank–Nicolson-type introduce. Next, use Benamou–Brenier dynamical characterization Wasserstein distance to reduce computing solution discrete time equations solving fully minimization problems, strictly convex objective functions linear constraints. Third, compute minimizers by applying recently introduced, provably convergent primal dual three operators (Yan J Sci Comput 1–20, 2018). By leveraging PDEs’ underlying variational structure, our overcomes stability issues present previous work built on explicit discretizations, which suffer due equations’ strong nonlinearities degeneracies. is also naturally positivity mass preserving and, case scheme, energy decreasing. We prove that problem converge spatially continuous, spatial discretization refined. conclude simulations nonlinear PDEs geodesics one two dimensions illustrate key properties approach, including higher-order convergence method, when compared method.