Mathematical Sciences: Cartesian Grid Methods for Compressible Flow

本研究涉及在一个以上的空间维度上发展随时间变化的气体动力学计算的数值方法。目标是开发基于底层均匀笛卡尔网格的方法，即使几何结构很复杂，被建模的现象涉及相对于网格的任意方向的复杂特征。具体方面包括开发更好的多维有限体积方法，用于处理由笛卡尔网格单元的界面或边界切割产生的不规则单元，以及将这些方法与冲击跟踪和复合网格结合使用。流体流动的计算机模拟对于理解和模拟科学技术中的各种重要过程至关重要。仅举一个例子，为了确定飞机的飞行性能和设计安全和节能的车辆，需要对飞行中的飞机周围的气流进行建模。计算机方法通过近似求解一组复杂的微分方程，产生飞机周围有限网格点处空气速度和压力的近似值。需要数以百万计的网格点来获得复杂物体周围流动的良好近似，即使是最强大的计算机也会达到极限。这项工作的目标是开发更有效和准确的方法来解决这些问题，特别强调处理具有复杂形状的物体（如飞机）和具有复杂行为的流模型的更好方法。

This research involves the development of numerical methods for time-dependent gas dynamics calculations in more than one space dimension. The goal is to develop methods that are based on an underlying uniform Cartesian grid even when the geometry is complicated and the phenomena being modeled involves complex features with arbitrary orientation relative to the grid. Specific aspects include the development of better multi- dimensional finite volume methods on irregular cells generated by an interface or boundary cutting through a Cartesian grid cell, and the use of such methods in conjunction with shock tracking and composite grids. Computer simulation of fluid flow is critical in understanding and modeling a wide variety of important processes in science and technology. As just one example, modeling the flow of air around an aircraft in flight is required in order to determine how well it flies and to design safe and fuel-efficient vehicles. Computer methods produce approximations to the velocity and pressure of the air at a finite set of grid points around the aircraft, by approximately solving a complicated set of differential equations. Millions of grid points are required to obtain a good approximation to the flow around a complicated object, stretching even the most powerful computers to their limits. The goal of this work is to develop more efficient and accurate methods to solve such problems, with particular emphasis on better ways to handle objects with complicated shapes (such as aircraft) and to model flows with complicated behavior.