Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
Philos Trans A Math Phys Eng Sci. 2013 Jan 13;371(1982):20120182. doi: 10.1098/rsta.2012.0182.
We discuss a new class of approaches for simulating multiscale kinetic problems, with particular emphasis on applications related to small-scale transport. These approaches are based on a decomposition of the kinetic description into an equilibrium part, which is described deterministically (analytically or numerically), and the remainder, which is described using a particle simulation method. We show that it is possible to derive evolution equations for the two parts from the governing kinetic equation, leading to a decomposition that is dynamically and automatically adaptive, and a multiscale method that seamlessly bridges the two descriptions without introducing any approximation. Our discussion pays particular attention to stochastic particle simulation methods that are typically used to simulate kinetic phenomena; in this context, these decomposition approaches can be thought of as control-variate variance-reduction formulations, with the nearby equilibrium serving as the control. Such formulations can provide substantial computational benefits in a broad spectrum of applications because a number of transport processes and phenomena of practical interest correspond to perturbations from nearby equilibrium distributions. In many cases, the computational cost reduction is sufficiently large to enable otherwise intractable simulations.
我们讨论了一类用于模拟多尺度动力学问题的新方法，特别强调了与小尺度输运相关的应用。这些方法基于对动力学描述的分解，即将平衡部分确定论地（解析或数值地）描述，其余部分则使用粒子模拟方法描述。我们表明，可以从控制动力学方程推导出这两部分的演化方程，从而得到一种动态和自动自适应的分解方法，以及一种无缝连接两种描述而不引入任何近似的多尺度方法。我们的讨论特别关注通常用于模拟动力学现象的随机粒子模拟方法；在这种情况下，这些分解方法可以被看作是控制变量方差减少公式，其中附近的平衡状态作为控制。在广泛的应用中，这种形式可以提供实质性的计算效益，因为许多实际感兴趣的输运过程和现象对应于来自附近平衡分布的扰动。在许多情况下，计算成本的降低足以实现原本难以处理的模拟。
