This paper is concerned with structure-preserving numerical approximations for a class of nonlinear nonlocal Fokker-Planck equations, which admit a gradient flow structure and find application in diverse contexts. The solutions, representing density distributions, must be non-negative and satisfy a specific energy dissipation law. We design an arbitrary high-order discontinuous Galerkin (DG) method tailored for these model problems. Both semi-discrete and fully discrete schemes are shown to admit the energy dissipation law for non-negative numerical solutions. To ensure the preservation of positivity in cell averages at all time steps, we introduce a local flux correction applied to the DDG diffusive flux. Subsequently, a hybrid algorithm is presented, utilizing a positivity-preserving limiter, to generate positive and energy-dissipating solutions. Numerical examples are provided to showcase the high resolution of the numerical solutions and the verified properties of the DG schemes.

翻译：本文研究一类非线性非局部Fokker-Planck方程的结构保持数值逼近方法，该类方程具有梯度流结构并在多个领域有应用。其解代表密度分布，必须非负且满足特定能量耗散律。我们针对这些模型问题设计了一种任意高阶的不连续Galerkin（DG）方法。半离散和全离散格式均被证明对非负数值解具有能量耗散律。为确保所有时间步的单元平均值保持正性，我们引入了应用于DDG扩散通量的局部通量修正。随后，提出了一种混合算法，利用保正限制器生成正且能量耗散的数值解。数值算例展示了DG格式的高分辨率及已验证的性质。