Computing Effective Diffusivity of Chaotic and Stochastic Flows Using Structure-Preserving Schemes
Computing Effective Diffusivity of Chaotic and Stochastic Flows Using Structure-Preserving Schemes
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使用结构保持方案计算混沌和随机流的有效扩散率
DOI：
10.1137/18m1165219
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Zhiwen Zhang
中科院分区：
文献类型：
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作者：
Zhongjian Wang;J. Xin;Zhiwen Zhang
本文研究了在混沌和随机流动中运动的粒子的有效扩散系数的计算问题。此外，对混沌平流中的残余扩散现象进行了数值研究。残余扩散是指由于流线的混沌混合，在分子零扩散极限下的非零有效（均质）扩散。在这个极限下，由于平流扩散方程的解呈现出明显的梯度，传统的数值方法通常失效。我们不是在欧拉公式中求解Fokker-Planck方程，而是在拉格朗日公式中计算粒子的运动，拉格朗日公式由随机微分方程（SDEs）建模。我们提出了一种新的基于随机分裂方法的数值积分器来求解相应的SDEs，其中确定性子问题是辛保持的，而随机子问题可以看作是扰动。我们使用反向误差分析技术对新的数值积分器进行了严格的误差分析，并表明我们的方法优于标准的基于欧拉的积分器。数值结果表明，该方法对几种典型的物理兴趣的混沌和随机流动问题具有准确性和有效性。
In this paper we study the problem of computing the effective diffusivity for a particle moving in chaotic and stochastic flows. In addition we numerically investigate the residual diffusion phenomenon in chaotic advection. The residual diffusion refers to the non-zero effective (homogenized) diffusion in the limit of zero molecular diffusion as a result of chaotic mixing of the streamlines. In this limit traditional numerical methods typically fail since the solutions of the advection-diffusion equation develop sharp gradients. Instead of solving the Fokker-Planck equation in the Eulerian formulation, we compute the motion of particles in the Lagrangian formulation, which is modelled by stochastic differential equations (SDEs). We propose a new numerical integrator based on a stochastic splitting method to solve the corresponding SDEs in which the deterministic subproblem is symplectic preserving while the random subproblem can be viewed as a perturbation. We provide rigorous error analysis for the new numerical integrator using the backward error analysis technique and show that our method outperforms standard Euler-based integrators. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several typical chaotic and stochastic flow problems of physical interests.