We study the discretization, convergence, and numerical implementation of recent reformulations of the quadratic porous medium equation (multidimensional and anisotropic) and Burgers' equation (one-dimensional, with optional viscosity), as forward in time variants of the Benamou-Brenier formulation of optimal transport. This approach turns those evolution problems into global optimization problems in time and space, of which we introduce a discretization, one of whose originalities lies in the harmonic interpolation of the densities involved. We prove that the resulting schemes are unconditionally stable w.r.t. the space and time steps, and we establish a quadratic convergence rate for the dual PDE solution, under suitable assumptions. We also show that the schemes can be efficiently solved numerically using a proximal splitting method and a global space-time fast Fourier transform, and we illustrate our results with numerical experiments.

翻译：本文研究了二次多孔介质方程（多维各向异性情形）与Burgers方程（一维含可选粘性项）的近期重构形式——作为最优传输Benamou-Brenier公式的前向时间变体——的离散化、收敛性及数值实现方法。该途径将此类演化问题转化为时空全局优化问题，我们提出了其离散化方案，其创新点之一在于对涉及密度函数采用调和插值方法。我们证明了所得格式在空间与时间步长方面具有无条件稳定性，并在适当假设下建立了对偶偏微分方程解的二次收敛速率。同时，我们展示了该格式可通过近端分裂算法与全局时空快速傅里叶变换进行高效数值求解，并通过数值实验验证了理论结果。