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Sharp Error Estimates on a Stochastic Structure-Preserving Scheme in Computing Effective Diffusivity of 3D Chaotic Flows
Multiscale Modeling and Simulation ( IF 1.6 ) Pub Date : 2021-07-20 , DOI: 10.1137/19m1275516
Zhongjian Wang , Jack Xin , Zhiwen Zhang

Multiscale Modeling &Simulation, Volume 19, Issue 3, Page 1167-1189, January 2021.
In this paper, we study the problem of computing the effective diffusivity for particles moving in chaotic flows. Instead of solving a convection-diffusion type cell problem in the Eulerian formulation (arising from homogenization theory for parabolic equations), we compute the motion of particles in the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). A robust numerical integrator based on a splitting method was proposed to solve the SDEs and rigorous error analysis for the numerical integrator was provided using the backward error analysis technique in our previous work. However, the upper bound on the error estimate is not sharp. To improve our result, we propose a new and uniform in time error analysis for the numerical integrator that allows us to get rid of the exponential growth factor in our previous error estimate. Our new error analysis is based on a probabilistic approach, which interprets the solution process generated by our numerical integrator as a Markov process. By exploring the ergodicity of the solution process, we prove the convergence analysis of our method in computing effective diffusivity over infinite time. We present numerical results to verify the accuracy and efficiency of the proposed method in computing effective diffusivity for several chaotic flows, especially the Arnold--Beltrami--Childress flow and Kolmogorov flow in three-dimensional space.


中文翻译:

计算 3D 混沌流有效扩散率的随机结构保持方案的尖锐误差估计

多尺度建模与仿真,第 19 卷,第 3 期,第 1167-1189 页,2021 年 1 月。
在本文中,我们研究了计算在混沌流中运动的粒子的有效扩散率的问题。我们不是在欧拉公式中解决对流扩散类型的单元问题(源自抛物线方程的均质化理论),而是在拉格朗日公式中计算粒子的运动,该公式由随机微分方程 (SDE) 建模。提出了一种基于分裂方法的稳健数值积分器来解决 SDE,并在我们之前的工作中使用后向误差分析技术为数值积分器提供了严格的误差分析。然而,误差估计的上限并不明显。为了改善我们的结果,我们为数值积分器提出了一种新的统一的时间误差分析,它使我们能够摆脱先前误差估计中的指数增长因子。我们新的误差分析基于概率方法,该方法将数值积分器生成的求解过程解释为马尔可夫过程。通过探索求解过程的遍历性,我们证明了我们的方法在计算无限时间内有效扩散率的收敛性分析。我们提供数值结果来验证所提出的方法在计算几种混沌流的有效扩散率时的准确性和效率,特别是三维空间中的 Arnold--Beltrami--Childress 流和 Kolmogorov 流。它将我们的数值积分器生成的求解过程解释为马尔可夫过程。通过探索求解过程的遍历性,我们证明了我们的方法在计算无限时间内有效扩散率的收敛性分析。我们提供数值结果来验证所提出的方法在计算几种混沌流的有效扩散率时的准确性和效率,特别是三维空间中的 Arnold--Beltrami--Childress 流和 Kolmogorov 流。它将我们的数值积分器生成的求解过程解释为马尔可夫过程。通过探索求解过程的遍历性,我们证明了我们的方法在计算无限时间内有效扩散率的收敛性分析。我们提供数值结果来验证所提出的方法在计算几种混沌流的有效扩散率时的准确性和效率,特别是三维空间中的 Arnold--Beltrami--Childress 流和 Kolmogorov 流。
更新日期:2021-07-20
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