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.

翻译：我们建立双点通量接近(TPFA)限量计划,以解决动态形式的四极最佳运输问题,即Benamou和Brenier最初引入的问题。我们从数字上表明,这些类型的离散性在更自然地实施时容易形成不稳定,我们提议以嵌套模类为基础的变异,以克服这些问题。尽管问题缺乏严格的混杂性,但我们也从数量上估计了该方法的趋同情况,至少是离散潜力和离散成本。最后,我们引入了基于屏障方法的战略,以解决离散优化问题。