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

An optimization-based positivity-preserving limiter in semi-implicit discontinuous Galerkin schemes solving Fokker-Planck equations