RUI: Postprocessing High-Order Approximations of Discontinuous Functions - Algorithms and Software

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In this project the PI develops algorithms and software tofacilitate the removal of Gibbs oscillations from the approximationof discontinuous data by high order approximation methods such aspseudospectral and radial basis function methods. Currentpostprocessing techniques are somewhat successful in one dimension.However, the most powerful methods require that the exact locationof the discontinuities (edges) be pinpointed and that functiondependent parameters be specified in each piecewise smooth region.One focus of the work is in developing Edge Detection free (EDF)postprocessing methods. The EDF methods are easily extended tomultiple dimensions and in some case to irregularly shaped domains.The ultimate goal is a "black box" postprocessing algorithm. Asecond focus of the work is in software development. A Matlabtoolbox which implements both classical and modern postprocessingmethods will be developed. The software will include a user friendlygraphical user interface facilitating use by the wider community.The software will be used on benchmark problems to provideillustration of the methods for both the specialist andnonspecialist and to stimulate further research. The software isdesigned to make it relatively easy for students to learn about themethods and for researchers to apply them.For the most accurate methods available to numerically approximatethe equations that govern natural phenomena, postprocessing methodsare vital. This is because many phenomena in nature are inherentlydiscontinuous, and numerical methods do not perform well when thesolution to the governing equations are discontinuous. Thediscontinuities result in the methods producing solutions withnonphysical oscillations that do not truly represent the phenomenabeing modeled. Areas for which postprocessing methods are importantinclude: gas dynamics, weather forecasting, earthquake and tsunamiprediction, astrophysical modeling, multiphase fluid flow, the flowof glaciers, the propagation of waves in elastic solids, theseparation of chemical species, aerodynamics, detonation waves,acoustics, etc.