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Numerous infinite dimensional dynamical systems arising in different fieldshave been shown to exhibit a gradient flow structure in the Wasserstein space.We construct Two Point Flux Approximation Finite Volume schemes discretizingsuch problems which preserve the variational structure and have second orderaccuracy in space. We propose an interior point method to solve the discretevariational problem, providing an efficient and robust algorithm. We presenttwo applications to test the scheme and show its order of convergence.

TPFA Finite Volume Approximation of Wasserstein Gradient Flows