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