We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost.

翻译：本文针对标准Fokker-Planck方程提出了一种全离散有限体积格式。空间离散采用著名的平方根近似法，该方法属于两点通量近似框架。我们的时间离散方案具有创新性，基于一种特制的非线性中点规则，旨在精确捕捉模型的耗散结构。我们证明了该格式的适定性、解的正性，以及完全离散的能量耗散不等式（该不等式与连续情形保持一致）。随后在非结构网格的温和限制条件下（这些条件在实践中易于满足），我们严格证明了该格式的收敛性。数值模拟表明，我们的格式在时间和空间上均具有二阶精度，并且可以使用牛顿法以较低计算成本求解每个时间步产生的离散非线性系统。