Mathematical Sciences: Numerical Solution of the Hele-Shaw Equations

所提出的研究涉及模拟Hele-Shaw单元中液体流动的各种自由边值问题的数值解。这种所谓的Hele-Shaw流本身就很重要，它们是涉及自由边界的更复杂现象的非常有用的模型。这项拟议的研究将产生稳健的数值格式，能够处理目前无法解析的许多不同类型的非线性现象。一种流体流入另一种流体时的“指进”现象是一种常见的、很容易观察到的现象，很难定量描述。这项拟议的研究将建议并评估不同类型的方法，用于数值研究支配不稳定手指随时间演变的数学方程组。对于这一特殊问题，一种成功的数值格式将对流体力学中的许多问题具有更广泛的适用性。

The proposed research involves the numerical solution of various free boundary value problems modelling flows of liquids in Hele-Shaw cells. Such Hele-Shaw flows, as they are called, are important in their own right, and they serve as very useful models of more complicated phenomena involving free boundaries. The proposed research will produce robust numerical schemes capable of handling many different types of nonlinear phenomena whose analytic resolution is currently impossible. The phenomenon of "fingering" of one fluid as it flows into another is a common, easily observed occurrence that is very difficult to describe quantitatively. The proposed research will suggest and evaluate different kinds of methods for studying numerically the set of mathematical equations that govern the evolution in time of an unstable finger. A successful numerical scheme for this particular problem will have much broader applicability to a host of problems in fluid dynamics.