In this paper, we introduce second order and fourth order space discretization via finite difference implementation of the finite element method for solving Fokker-Planck equations associated with irreversible processes. The proposed schemes are first order in time and second order and fourth order in space. Under mild mesh conditions and time step constraints for smooth solutions, the schemes are proved to be monotone, thus are positivity-preserving and energy dissipative. In particular, our scheme is suitable for capturing steady state solutions in large final time simulations.

翻译：本文通过有限元方法的有限差分离散实现，提出了求解不可逆过程相关福克-普朗克方程的二阶和四阶空间离散格式。所提格式在时间上为一阶精度，空间上分别为二阶和四阶精度。在温和网格条件及光滑解的时间步长约束下，我们证明这些格式是单调的，从而能够保持正性并满足能量耗散性质。特别地，该格式适用于长时间模拟中稳态解的计算。