Fast Spectral Methods and their Applications

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The PI will develop and analyze fast and accurate numerical schemes for liquid crystal flows and for PDEs in unbounded domains, and produce a publicly available high performance software package of fast spectral elliptic solvers which will be a valuable tool for computational scientists and engineers. The PI will also implement the proposed numerical schemes to investigate several important problems of current interest, including in particular the coupling and defect motion of liquid crystal flows, and the dynamic control and parametric resonance in rotating flows. The accuracy and efficiency of the proposed schemes will allow us to simulate these three-dimensional time-dependent flows with a reasonable turn-over time.Computer simulations are playing an ever increasing role in many branches of science and engineering and are rapidly replacing much of the expensive prototyping and testing phase required before any product is brought to market. Consequently, fast and reliable numerical methods/software are becoming an indispensable tool for many scientists and engineers, especially for computational scientists in fluid dynamics and materials science. It is expected that our numerical simulations will contribute towards better understandings of the complex physical and mathematical problems, and provide valuable information for the design of advanced materials and of turbo-machinery.