For high-order accurate schemes such as discontinuous Galerkin (DG) methods solving Fokker-Planck equations, it is desired to efficiently enforce positivity without losing conservation and high-order accuracy, especially for implicit time discretizations. We consider an optimization-based positivity-preserving limiter for enforcing positivity of cell averages of DG solutions in a semi-implicit time discretization scheme, so that the point values can be easily enforced to be positive by a simple scaling limiter on the DG polynomial in each cell. The optimization can be efficiently solved by a first-order splitting method with nearly optimal parameters, which has an $\mathcal{O}(N)$ computational complexity and is flexible for parallel computation. Numerical tests are shown on some representative examples to demonstrate the performance of the proposed method.

翻译：对于求解Fokker-Planck方程的高阶精确格式（如间断Galerkin (DG) 方法），尤其是在隐式时间离散下，需要在不破坏守恒性和高阶精度的前提下高效地强制保持解的正性。本文研究了一种基于优化的保正限制器，用于在半隐式时间离散格式中强制DG解在单元平均值上的正性，从而可以通过对每个单元内的DG多项式应用简单的缩放限制器，轻松地强制其点值为正。该优化问题可通过具有近似最优参数的一阶分裂方法高效求解，其计算复杂度为$\mathcal{O}(N)$，且便于并行计算。数值实验通过若干代表性算例展示了所提方法的性能。