在分子扩散消失极限下计算有效扩散系数
Calculating Effective Diffusivities in the Limit of Vanishing Molecular Diffusion
AI总结：
针对周期无散速度场中粒子在分子扩散消失极限下的有效扩散系数计算问题，提出一种考虑体积保持性质的随机分裂方法，并通过修正方程进行后向误差分析，数值实验表明新方法优于标准欧拉方法。
摘要：
本文研究在分子扩散消失极限下，通过蒙特卡罗方法数值计算周期无散速度场中粒子有效扩散系数的问题。在该极限下，传统数值方法通常失效，因为它们不能准确表示底层确定性动力学的几何结构。我们提出一种随机分裂方法，该方法在无噪声且可忽略惯性效应时考虑了运动方程的保体积性质。然后，将该方法扩展到噪声具有非平凡时间相关结构且不能忽略惯性效应的情况。使用修正方程进行后向误差分析。新的随机几何积分器在数值实验中优于基于欧拉的标准积分器。利用新积分器，通过数值实验研究了若干具有物理意义的渐近极限。
英文摘要：
In this paper we study the problem of the numerical calculation (by Monte Carlo Methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity filed, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume preserving property of the equations motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non trivial time-correlation structure and when inertial effects cannot be neglected. Modified equations are used to perform backward error analysis. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators.
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