Mathematical Sciences: Advanced Numerical Methods for Modeling Fluid Interfaces

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This research is concerned with the design and development of numerical methods for modeling fluid interfaces and the application of these methods to the study of fundamental problems in fluid mechanics. An important goal of this work is to develop accurate and efficient algorithms for modeling three dimensional problems. The interface tracking algorithms considered are generally of the volume-of-fluid type. Some generalizations of this type of algorithm, based on conservative shock tracking ideas, are also considered. These algorithms will be used, in conjunction with adaptive mesh refinement and high order numerical methods for approximating solutions of the compressible Euler equations, to study shock wave refraction and the Richtmyer-Meshkov instability. In addition, they will also be used with a second order projection method to study the Rayleigh- Taylor instability in incompressible flow. Studies of fluid dynamics have a long and honorable history, with many applications in engineering, earth and ocean sciences, among others.