Computational and Applied Math Proseminar

Thursday, March 24, 2005, 12:15 p.m. GWC 604

Jae-Hun Jung

PIMS and IAM, University of British Columbia

Numerical approximation of nonsmooth solutions with spectral methods

Abstract Spectral methods have been proved to be powerful computational methods solving a wide range of physical and engineering problems. As the spectral method is global, it provides the so-called spectral convergence if the functions of interest are globally smooth. If they are discontinuous, however, spectral accuracy is deteriorated and the overall convergence is only O(1) in the maximum norm. Several reconstruction methods have been developed to deal with this problem. These methods roughly fall into two different theories: projection theory and direct-inverse theory. Projection methods seek the reconstruction by means of projection of the given spectral data into the new spectral space while the inverse methods use the projection of the reconstruction onto the given spectral space instead.

In this talk, we will briefly explain these theories and discuss the spectral filtering methods as one of the projection methods. Then we introduce a new inverse method. The proposed method seeks a reconstruction in a polynomial space by making the residue of the given spectral data and the reconstruction orthogonal to the polynomial space. This method referred as to the "inverse polynomial reconstruction method is unique, exact and spectrally convergent. As the inverse method involves a matrix inversion and it is known to be ill-posed, the maximum error grows exponentially once the round-off errors become dominant. We show that there exist polynomial basis sets which do not exhibit such exponential growth. One such basis set is the Jacobi polynomials. We also provide some numerical results including 1) the spectral simulation of highly supersonic reactive cavity flows of scramjet engine and 2) the image reconstructions based on the wiggly Fourier images.

For further information please contact: mittelmann@asu.edu