Towards Fast and Stable Schemes for Brownian Dynamics with Hydrodynamic Interactions

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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.