Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which does not satisfy detailed balance. For the corresponding Fokker-Planck equation on a closed manifold, using Voronoi tessellation, we propose two upwind finite volume schemes with or without the information of the invariant measure. Both schemes possess stochastic $Q$-matrix structures and can be decomposed as a gradient flow part and a Hamiltonian flow part, enabling us to prove unconditional stability, ergodicity and error estimates. Based on the two upwind schemes, several numerical examples - including sampling accelerated by a mixture flow, image transformations and simulations for stochastic model of chaotic system - are conducted. These two structure-preserving schemes also give a natural random walk approximation for a generic irreversible drift-diffusion process on a manifold. This makes them suitable for adapting to manifold-related computations that arise from high-dimensional molecular dynamics simulations.

翻译：不可逆漂移-扩散过程在生物化学反应中极为常见。这类过程存在一个不满足细致平衡条件的非平衡稳态（不变测度）。针对封闭流形上相应的福克-普朗克方程，我们借助沃罗诺伊剖分提出了两种迎风有限体积格式，分别对应是否利用不变测度信息。这两种格式均具有随机$Q$-矩阵结构，并可分解为梯度流部分与哈密顿流部分，从而使得我们能够证明其无条件稳定性、遍历性及误差估计。基于这两种迎风格式，我们开展了若干数值实验——包括混合流加速采样、图像变换以及混沌系统随机模型的模拟。这两种保结构格式同时也为流形上一般不可逆漂移-扩散过程提供了自然的随机游走逼近方法，使其能够适应高维分子动力学模拟中出现的流形相关计算需求。