Mathematical Sciences: Cartesian Grid Methods for Compressible Flow

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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.