Mathematical Sciences: Presidential Young Investigator Award

总统青年研究员将在计算流体力学领域进行研究；他尤其将研究不同激波相互作用以及与流体界面和涡旋片相互作用时的激波绕射和分叉。这项研究的目的有两个。第一个目标是发现由激波和流体结构的复杂相互作用产生的不同类型的波型，而第二个目标是开发数值方法，使人们能够跟踪相互作用产生的波的时间演变。流体动力学中的许多现象是不同类型的波相互作用以及与内部流体结构(如界面)相互作用的结果。但这也很棘手，因为计算机通常会产生无意义的信息，除非使用的数值方法是可靠的。PI将用“智能”的数值方法来研究这些问题。这些方法内置了足够的分析和物理理论，它们可以产生对正确物理解的精确近似。

The Presidential Young Investigator will conduct research in the area of computational fluid dynamics; in particular, he will study shock diffractions and bifurcations as different shocks interact with each other and with fluid interfaces and vortex sheets. This is very important and timely research, and it is of sufficient complexity that the PI will employ analytical methods from geometry and bifurcation theory, in addition to computational techniques. The goals of the research are twofold. The first goal is to discover different types of wave patterns that result from the complex interactions of shocks and fluid structures, while the second goal is to develop numerical methods that allow one to follow the evolution in time of the waves generated by the interactions. Many phenomena in fluid dynamics arise out of the interactions of different types of waves with each other and with internal fluid structures such as interfaces. What makes these interactions difficult to study mathematically is that they are described by systems of nonlinear differential equations, and so one must use computers in order to generate solutions. But this is tricky also, because computers can often generate meaningless information, unless the numerical methods used are reliable. The PI will study such problems with "intelligent" numerical methods. These are methods that have sufficient analytical and physical theory built into them that they produce sharp approximations to the correct physical solutions.