Application of Distributed Adaptive Mesh Refinement Techniques to Problems Involving Diffusion

许多物理问题涉及到随时间移动的局部现象，如流体锋、界面和激波，这些现象通常不能用均匀网格上的有限差分方法很好地捕捉。 研究人员对此类问题的三类进行研究：多孔介质中的流动，特别是化学强化含水层修复;由于扩散通量消失而导致的流体前沿薄膜流动;以及可激发介质，特别是心脏中电脉冲传播的建模。 这三个问题都涉及扩散，扩散可以是非线性的，亚扩散的或超扩散的。 研究人员开发了有效的多级迭代方法和自适应网格细化工具来解决这些问题的并行处理机。 自从世纪前计算机问世以来，计算机的能力得到了显著的提高，而且性能的提高还在以惊人的速度继续着。 尽管如此，尽管公众普遍没有意识到这一点，但模拟许多现象（如天气）所需的计算资源远远超过了目前可用的资源。 获得大量计算机能力的一种方法是安排多台计算机同时处理一个问题。 然而，这是一个艰巨的任务，以编程这些计算机，使他们将正确地产生所需的模拟及时的方式。 在这项工作中，研究人员开发了有效的方法，使用多台计算机一起工作，以模拟地下污染物的流动，薄膜的行为和心脏的电生理学。 地下水污染是一个重大的环境问题。 薄膜问题出现在润滑和制造过程中。 对心脏电行为和肌肉行为的基本了解非常重要，可以更好地控制心律失常的机制。

A number of physical problems involve localized phenomenathat move in time, such as fluid fronts, interfaces and shocks.Often, these phenomena are not captured well by finite differencemethods on uniform grids. The investigators work on threeclasses of such problems: flow in porous media, particularlychemically enhanced aquifer remediation; thin film flow withfluid fronts due to vanishing diffusive fluxes; and excitablemedia, particularly the modeling of the propagation of electricalimpulses in the heart. All three of these problems involvediffusion; the diffusion can be nonlinear, subdiffusive orsuperdiffusive. The investigators develop efficient multileveliterative methods and adaptive mesh refinement tools to solvethese problems on parallel processor machines. The power of computers has dramatically increased sincethey were introduced a half century ago, and this improvement inperformance continues at a breathtaking pace. Nonetheless,although the public is generally unaware of this, thecomputational resources required to simulate many phenomena, likethe weather, greatly exceed those currently available. One wayto obtain a lot of computer power is to arrange for manycomputers to work simultaneously on a problem. However, it is adaunting task to program these computers so that they willcorrectly produce the desired simulation in a timely fashion. Inthis work, the investigators develop effective methods to usemany computers working together to simulate the flow ofcontaminants in the ground, the behavior of thin films, and theelectrophysiology of the heart. Groundwater contamination is asignificant environmental problem. Thin film problems arise inlubrication and in manufacturing processes. A fundamentalunderstanding of the electrical and muscular behavior of theheart is important and could lead to better control of themechanisms of arrhythmia.