  We construct Two-Point Flux Approximation (TPFA) finite volume schemes to
solve the quadratic optimal transport problem in its dynamic form, namely the
problem originally introduced by Benamou and Brenier. We show numerically that
these type of discretizations are prone to form instabilities in their more
natural implementation, and we propose a variation based on nested meshes in
order to overcome these issues. Despite the lack of strict convexity of the
problem, we also derive quantitative estimates on the convergence of the
method, at least for the discrete potential and the discrete cost. Finally, we
introduce a strategy based on the barrier method to solve the discrete
optimization problem.
