This paper aims to investigate the diffusion behavior of particles moving in stochastic flows under a structure-preserving scheme. We compute the effective diffusivity for normal diffusive random flows and establish the power law between spatial and temporal variables for cases with anomalous diffusion phenomena. From a Lagrangian approach, we separate the corresponding stochastic differential equations (SDEs) into sub-problems and construct a one-step structure-preserving method to solve them. Then by modified equation systems, the convergence analysis in calculating the effective diffusivity is provided and compared between the structure-preserving scheme and the Euler-Maruyama scheme. Also, we provide the error estimate for the structure-preserving scheme in calculating the power law for a series of super-diffusive random flows. Finally, we calculate the effective diffusivity and anomalous diffusion phenomena for a series of 2D and 3D random fields.

翻译：本文旨在研究结构保持格式下粒子在随机流中运动的扩散行为。我们计算了正常扩散随机流的有效扩散率，并针对存在反常扩散现象的情形建立了空间与时间变量之间的幂律关系。基于拉格朗日方法，我们将对应的随机微分方程分解为若干子问题，并构建了一步结构保持格式进行求解。随后通过修正方程系统，给出了计算有效扩散率时的收敛性分析，并比较了结构保持格式与Euler-Maruyama格式的差异。此外，我们针对一系列超扩散随机流，给出了结构保持格式在计算幂律关系时的误差估计。最后，我们对一系列二维与三维随机场计算了其有效扩散率及反常扩散现象。