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

TL;DR

This paper introduces an efficient optimization-based positivity-preserving limiter for semi-implicit discontinuous Galerkin schemes solving Fokker-Planck equations, ensuring positivity without sacrificing accuracy or conservation.

Contribution

It develops a novel optimization-based limiter that enforces positivity of cell averages in DG methods with minimal computational cost and high parallelizability.

Findings

The method maintains positivity effectively in numerical tests.

It achieves nearly optimal computational complexity of O(N).

The approach is flexible and suitable for parallel computing environments.

Abstract

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.