Abstract In this paper, we propose a two-level difference scheme for solving the two-dimensional Fokker–Planck equation. This equation is a parabolic type equation which governs the time evolution of probability density function of the stochastic processes. In addition, these equations preserve positivity and conservation. The Chang–Cooper discretization scheme is used, which ensures second-order accuracy, positiveness, and satisfies the conservation of the total probability. In particular, we investigate a two-level scheme with factor-three coarsening strategy. With coarsening by a factor-of-three we obtained simplified inter-grid transfer operators and thus have a significant reduction in CPU time. Numerical experiments are performed to validate efficiency of the proposed Chang–Cooper two-level algorithms to stationary and time-dependent Fokker–Planck equations, respectively.

中文翻译：

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二维 Fokker-Planck 方程的两级差分格式

摘要 在本文中，我们提出了一种求解二维福克-普朗克方程的两级差分格式。这个方程是一个抛物线型方程，它控制随机过程的概率密度函数的时间演化。此外，这些方程保留了正性和守恒性。采用 Chang-Cooper 离散化方案，保证二阶精度、正性，满足总概率守恒。特别是，我们研究了具有因子三粗化策略的两级方案。通过三倍粗化，我们获得了简化的网格间传输算子，从而显着减少了 CPU 时间。