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Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank\u2013Nicolson-type method we introduce. Next, we use the Benamou\u2013Brenier 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 in J Sci Comput 1\u201320, 2018). By leveraging the PDEs\u2019 underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations\u2019 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\u2013Nicolson-type method, when compared to the classical JKO method.<\/jats:p>","DOI":"10.1007\/s10208-021-09503-1","type":"journal-article","created":{"date-parts":[[2021,3,31]],"date-time":"2021-03-31T20:02:44Z","timestamp":1617220964000},"page":"389-443","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":47,"title":["Primal Dual Methods for Wasserstein Gradient Flows"],"prefix":"10.1007","volume":"22","author":[{"given":"Jos\u00e9 A.","family":"Carrillo","sequence":"first","affiliation":[]},{"given":"Katy","family":"Craig","sequence":"additional","affiliation":[]},{"given":"Li","family":"Wang","sequence":"additional","affiliation":[]},{"given":"Chaozhen","family":"Wei","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,31]]},"reference":[{"key":"9503_CR1","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198502456.001.0001","volume-title":"Functions of bounded variation and free discontinuity problems","author":"L Ambrosio","year":"2000","unstructured":"L.\u00a0Ambrosio, N.\u00a0Fusco, and D.\u00a0Pallara, Functions of bounded variation and free discontinuity problems, vol.\u00a0254, Clarendon Press Oxford, 2000."},{"key":"9503_CR2","doi-asserted-by":"crossref","unstructured":"L.\u00a0Ambrosio, N.\u00a0Gigli, and G.\u00a0Savar\u00e9, Gradient flows: in metric spaces and in the space of probability measures, Springer Science & Business Media, 2008.","DOI":"10.1016\/S1874-5717(07)80004-1"},{"key":"9503_CR3","doi-asserted-by":"publisher","first-page":"1259","DOI":"10.4310\/CMS.2020.v18.n5.a5","volume":"18","author":"R Bailo","year":"2020","unstructured":"R.\u00a0Bailo, J.\u00a0A. 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