Adaptive stochastic trajectory modelling in the chaotic advection regime
Adaptive stochastic trajectory modelling in the chaotic advection regime
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DOI：
10.1017/jfm.2015.75
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发表时间：
2015-03
影响因子：
3.7
通讯作者：
J. G. Esler
中科院分区：
文献类型：
--
作者：
J. G. Esler
出于改进和增强湍流中粒子分散的随机拉格朗日模型的目标，随机过程理论的技术被应用于模型输运问题。目的是找到一种有效且准确的方法来计算源和受体之间的总示踪剂传输，当两个位置之间的流动很弱时，导致直接随机拉格朗日模拟过于昂贵。提出了结合随机前向和后向轨迹计算信息的重要性采样方法。新方法的统一特征是，它们是根据以下观察结果开发的：完美的策略应该按照传输问题的前向解和伴随解的乘积按比例分配轨迹，这里的量称为“轨迹密度”$D(\boldsymbol{x},t)$。两种这样的方法应用于“硬”模型问题，其中规定的运动流处于大佩克莱特数混沌平流状态，并且输运问题需要模拟分离良好的轨迹的复杂分布。第一种是米尔斯坦的测量变换方法，涉及在轨迹方程中添加人工速度，同时校正新流下赋予每个粒子的权重。研究发现，虽然存在一个“完美”的人工速度 $\boldsymbol{v}^{\ast }$，并且它可以根据 $D$ 分布轨迹，但 $\boldsymbol{v}^{\ast }$ 数值估计中的小误差累积起来会导致该方法的困难。第二种方法是 Grassberger 的“与胜利者同行”分支过程，其中发现不太可能对净运输做出贡献的轨迹（失败者）被定期删除，而那些预计会做出重大贡献的轨迹（获胜者）则被分割。实现的主要挑战是找到一种算法来选择赢家和输家，这是通过一个选择来解决的，该选择明确地迫使分布朝向从先前的反向轨迹计算生成的 $D$ 的数值估计。结果是一种稳健且易于实现的算法，其典型方差比直接方法低三个数量级。
Motivated by the goal of improving and augmenting stochastic Lagrangian models of particle dispersion in turbulent flows, techniques from the theory of stochastic processes are applied to a model transport problem. The aim is to find an efficient and accurate method to calculate the total tracer transport between a source and a receptor when the flow between the two locations is weak, rendering direct stochastic Lagrangian simulation prohibitively expensive. Importance sampling methods that combine information from stochastic forward and back trajectory calculations are proposed. The unifying feature of the new methods is that they are developed using the observation that a perfect strategy should distribute trajectories in proportion to the product of the forward and adjoint solutions of the transport problem, a quantity here termed the ‘density of trajectories’ $D(\boldsymbol{x},t)$ . Two such methods are applied to a ‘hard’ model problem, in which the prescribed kinematic flow is in the large-Péclet-number chaotic advection regime, and the transport problem requires simulation of a complex distribution of well-separated trajectories. The first, Milstein’s measure transformation method, involves adding an artificial velocity to the trajectory equation and simultaneously correcting for the weighting given to each particle under the new flow. It is found that, although a ‘perfect’ artificial velocity $\boldsymbol{v}^{\ast }$ exists, which is shown to distribute the trajectories according to $D$ , small errors in numerical estimates of $\boldsymbol{v}^{\ast }$ cumulatively lead to difficulties with the method. A second method is Grassberger’s ‘go-with-the-winners’ branching process, where trajectories found unlikely to contribute to the net transport (losers) are periodically removed, while those expected to contribute significantly (winners) are split. The main challenge of implementation, which is finding an algorithm to select the winners and losers, is solved by a choice that explicitly forces the distribution towards a numerical estimate of $D$ generated from a previous back trajectory calculation. The result is a robust and easily implemented algorithm with typical variance up to three orders of magnitude lower than the direct approach.