Towards Fast and Stable Schemes for Brownian Dynamics with Hydrodynamic Interactions

聚合物流动的模拟是一个艰巨的计算问题。它需要一个数学模型，准确地捕捉聚合物和周围溶剂的影响。由此产生的动力学涉及流体动力、分子和热力之间的不平凡的相互作用，这些力产生了支配随机微分方程(SDE)。具有流体动力相互作用的布朗动力学指的是这些方程的时间积分。这些SDE具有高维、乘性噪声、非线性漂移、多时间尺度和与周围溶剂的复杂耦合等数值困难。虽然已经提出了一些方案来模拟这类系统，但没有一个方案满足数值稳定性的基本要求，这是本项目的主要动机。该项目将提供新的技术，以数值稳定和实用的方式解决这些SDE。有了荧光显微镜，科学家们现在可以短暂地观察单个DNA分子在各种流场中的运动。无论如何解释这些DNA运动的一瞥，它们都不够精确，不足以回答一些基本的问题，比如某些剪切流是如何导致DNA断裂或连接的？要回答这些问题，数值模拟是必不可少的，这个例子强调了定量模拟在新兴的DNA纳米技术领域发挥的重要作用。模拟与荧光显微镜相结合，使科学家能够全面研究各种流场中DNA动力学背后的机制。该项目将导致开发强大的新的数值工具来模拟流动中的DNA。它还将通过罗格斯-卡姆登的两个研究助学金提供学生培训。

Simulation of polymers in flow is a formidable computational problem. It requires a mathematical model that accurately captures both the polymer and the effect of the surrounding solvent. The resulting dynamics involve a nontrivial interplay between hydrodynamic, molecular and thermal forces that give rise to governing stochastic differential equations (SDE). Brownian dynamics with hydrodynamic interactions refers to the time-integration of these equations. These SDEs are fraught with numerical difficulties such as high dimensionality, multiplicative noise, nonlinear drifts, multiple time-scales and complex coupling with the surrounding solvent. While some schemes have been proposed to simulate such systems, none satisfy the fundamental requirement of numerical stability, which is a main motivation for this project. This project will deliver novel techniques to solve these SDEs in a numerically stable and practical way. With fluorescence microscopy, scientists can now briefly watch individual DNA molecules move in all sorts of flow fields. However illuminating these glimpses of DNA motion might be, they are not precise enough to answer fundamental questions such as how can certain shearing flows induce DNA to fragment or concatenate? To answer such questions, numerical simulation is essential, and this example emphasizes the significant role that quantitative simulations play in the burgeoning area of DNA nanotechnology. Simulations combined with fluorescence microscopy enable scientists to comprehensively study the mechanisms behind DNA dynamics in a variety of flow fields. This project will lead to the development of powerful new numerical tools to simulate DNA in flow. It will also provide student training through two research assistantships at Rutgers-Camden.