In this project the PI develops algorithms and software to facilitate the removal of Gibbs oscillations from the approximation of discontinuous data by high order approximation methods such as pseudospectral and radial basis function methods. Current postprocessing techniques are somewhat successful in one dimension. However, the most powerful methods require that the exact location of the discontinuities (edges) be pinpointed and that function dependent 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 to multiple dimensions and in some case to irregularly shaped domains. The ultimate goal is a "black box" postprocessing algorithm. A second focus of the work is in software development. A Matlab toolbox which implements both classical and modern postprocessing methods will be developed. The software will include a user friendly graphical user interface facilitating use by the wider community. The software will be used on benchmark problems to provide illustration of the methods for both the specialist and nonspecialist and to stimulate further research. The software is designed to make it relatively easy for students to learn about the methods and for researchers to apply them.
For the most accurate methods available to numerically approximate the equations that govern natural phenomena, postprocessing methods are vital. This is because many phenomena in nature are inherently discontinuous, and numerical methods do not perform well when the solution to the governing equations are discontinuous. The discontinuities result in the methods producing solutions with nonphysical oscillations that do not truly represent the phenomena being modeled. Areas for which postprocessing methods are important include: gas dynamics, weather forecasting, earthquake and tsunami prediction, astrophysical modeling, multiphase fluid flow, the flow of glaciers, the propagation of waves in elastic solids, the separation of chemical species, aerodynamics, detonation waves, acoustics, etc.
