SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page B82-B107, January 2021.
In this work, we develop novel structure-preserving numerical schemes for a class of nonlinear Fokker--Planck equations with nonlocal interactions. Such equations can cover many cases of importance, such as porous medium equations with external potentials, optimal transport problems, and aggregation-diffusion models. Based on the energetic variational approach, a trajectory equation is first derived by using the balance between the maximal dissipation principle and the least action principle. By a convex splitting technique, we propose energy dissipating numerical schemes for the trajectory equation. Rigorous numerical analysis reveals that the nonlinear numerical schemes are uniquely solvable, naturally respect mass conservation and positivity at the fully discrete level, and preserve steady states in an admissible convex set, where the discrete Jacobian of flow maps is positive. Under certain assumptions on smoothness and a positive Jacobian, the numerical schemes are shown to be second order accurate in space and first order accurate in time. Extensive numerical simulations are performed to demonstrate several valuable features of the proposed schemes. In addition to the preservation of physical structures, such as positivity, mass conservation, discrete energy dissipation, and steady states, numerical simulations further reveal that our numerical schemes are capable of solving degenerate cases of the Fokker--Planck equations effectively and robustly. It is shown that the developed numerical schemes have convergence order even in degenerate cases with the presence of solutions having compact support and can accurately and robustly compute the waiting time of free boundaries without any oscillation. The limitation of numerical schemes due to a singular Jacobian of the flow map is also discussed.

中文翻译：

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非线性Fokker-Planck方程非局部相互作用的保结构数值方法

SIAM科学计算杂志，第43卷，第1期，第B82-B107页，2021年1月。
在这项工作中，我们为一类具有非局部相互作用的非线性Fokker-Planck方程开发了新颖的保结构数值方案。这样的方程式可以涵盖许多重要的情况，例如具有外部电势的多孔介质方程式，最佳输运问题和聚集扩散模型。基于能量变分法，首先利用最大耗散原理与最小作用原理之间的平衡推导了轨迹方程。通过凸分裂技术，我们为轨迹方程提出了耗能数值方案。严格的数值分析表明，非线性数值方案是唯一可解的，在完全离散的水平上自然尊重质量守恒和正性，并在允许的凸集中保持稳态，流图的离散雅可比行列式为正。在对光滑度和正雅可比定律的某些假设下，数值方案显示为在空间上是二阶精确的，在时间上是一阶的。进行了广泛的数值模拟，以证明所提出方案的一些有价值的特征。除了保留物理结构（例如正性，质量守恒，离散能量耗散和稳态）之外，数值模拟还进一步表明，我们的数值方案能够有效且鲁棒地求解Fokker-Planck方程的退化情况。结果表明，所开发的数值方案即使在简并的情况下，在具有紧凑支持的解的存在下，也具有收敛阶，并且可以准确，可靠地计算自由边界的等待时间，而没有任何振荡。还讨论了由于流图的奇异雅可比定律而导致的数值方案的局限性。