AI 中文总结
研究福克 - 普朗克方程线性二阶有限差分格式无法保持正性问题，通过扩展DF求解器，拆分算子，应用限幅器得到FCDF格式，无条件正定且守恒质量、保持精度，解决小步长限制，经实验验证方法有效性。
AI 中文摘要
根据戈东诺夫定理，福克 - 普朗克方程的线性二阶有限差分格式无法保持正性。对角蛙（DF）框架此前通过最终正性绕过了这一障碍，但需要严格的最小时间步长。本文通过DF求解器的非线性扩展解决了小步长限制问题。将二阶方向算子拆分为单调M矩阵核心和反扩散通量校正，在隐式带状求解中迭代应用扎莱萨克型限幅器，得到的通量校正DF（FCDF）格式（变体A和B）在所有时间步长上无条件正定，精确守恒离散质量并保持二阶精度，限幅器仅在未解析层激活，确保全局$L_1$收敛在单元佩克莱数上均匀保持二阶，避免了Chang - Cooper格式中的一阶退化。该方法的皮卡迭代在纯对流步长限制下是收缩的，为支持任意步长开发了活动集重新表述，使用半光滑牛顿迭代求解系统，计算成本仅与正性约束的节点数成比例，最后引入缺陷校正时间步长方法恢复二阶时间精度。对奥恩斯坦 - 乌伦贝克和平流主导基准的数值实验证实了这些结论。
英文摘要
By Godunov's theorem, linear second-order finite-difference schemes for the Fokker-Planck equation cannot preserve positivity. The Diagonal Frog (DF) framework previously bypassed this barrier using eventual positivity, but required a strict minimum time step. This paper resolves the small-step limitation using a nonlinear extension of the DF solvers. We split the second-order directional operator into a monotone M-matrix core and an antidiffusive flux correction. A Zalesak-type limiter is then applied iteratively within the implicit banded solve. The resulting Flux-Corrected DF (FCDF) schemes (variants A and B) are unconditionally positive across all time steps. Because the limiter acts on fluxes rather than point values, these schemes conserve discrete mass exactly and maintain second-order accuracy. Crucially, the limiter activates only within unresolved layers. This ensures the global $L_1$ convergence remains second-order uniformly in the cell Péclet number, avoiding the first-order degradation seen in the Chang-Cooper scheme. The method's Picard iteration is contractive under a purely convective step restriction. To support arbitrary step sizes, we develop an active-set reformulation. This solves the system using a semismooth Newton iteration, where computational cost scales only with the number of nodes where positivity binds. Finally, we introduce a defect-corrected time stepping approach that restores second-order time accuracy. Numerical experiments on Ornstein-Uhlenbeck and advection-dominated benchmarks confirm our claims.
Comments30 pages, 2 figures, 9 tables
