发表机构
University of California Riverside（加州大学河滨分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文从Fokker–Planck方程能量耗散定律出发，提出保结构详细平衡主方程离散化方法，构造的半/全离散格式具质量守恒等特性，数值实验验证其收敛性与性能。
AI 中文摘要
我们直接从Fokker–Planck方程的能量耗散定律出发，开发了一种变分马尔可夫构造方法，用于构建其详细平衡主方程离散化格式。该方法不直接离散微分算子，而是通过两个方向的跃迁率来表示相邻网格点之间的质量转移。在每条边上，离散能量耗散定律决定了净通量，而详细平衡条件则确定了速率比；随后通过从相邻网格点值进行局部重构来唯一确定这两个速率。对数均值重构可得到经典的Scharfetter–Gummel/Wang–Peskin–Elston速率，而其他替代重构则可产生其他可逆格式。由此得到的半离散系统可保持质量守恒、非负性、满足详细平衡条件，并耗散离散自由能。这种基于边的相同构造方法可扩展至依赖状态的迁移率、退化迁移率、非局部相互作用能以及高维问题。对应的隐式和半隐式全离散格式是线性的、质量守恒的、保正性的且能量稳定的，仅当非局部相互作用能的核不是半负定时，才需要时间步长限制。数值实验证实了所有详细平衡格式的预期空间收敛率，并验证了其在不连续势、饱和效应、非局部相互作用及二维问题下的平衡精度、保正性和自由能衰减特性。
英文摘要
We develop a variational--Markov construction of detailed-balance master-equation discretizations for Fokker--Planck equations directly from their energy--dissipation laws. Rather than discretizing the differential operator, we represent mass transfer between neighboring grid points by two directional jump rates. On each edge, the discrete energy--dissipation law determines the net flux, while detailed balance fixes the rate ratio; a local reconstruction of edge quantities from neighboring grid-point values then uniquely determines both rates. A logarithmic-mean reconstruction recovers the classical Scharfetter--Gummel/Wang--Peskin--Elston rates, while alternative reconstructions yield other reversible schemes. The resulting semi-discrete systems conserve mass, preserve nonnegativity, satisfy detailed balance, and dissipate a discrete free energy. The same edgewise construction extends to state-dependent and degenerate mobilities, nonlocal interaction energies and higher-dimensional problems. The corresponding implicit and semi-implicit fully discrete schemes are linear, mass-conservative, positivity-preserving, and energy-stable, with a time-step restriction required only for nonlocal interaction energies whose kernel is not negative semidefinite. Numerical experiments confirm the expected spatial convergence rates of all detailed-balance schemes and verify their equilibrium accuracy, positivity preservation, and free-energy decay with discontinuous potentials, saturation, nonlocal interactions, and two-dimensional problems.
Comments37 pages, 8 figures
