Computational and Applied Math Proseminar

Department of Mathematics, Arizona State University

Thurday, February 14, 2002, 4:00 p.m. in PSA Room 106

David Gottlieb

Division of Applied Mathematics, Brown University

Spectral Methods for Discontinuous Problems

Abstract Spectral methods involve approximating the solution by Fourier series or orthogonal polynomials. Those approximations are very accurate for smooth functions, but not for piecewise smooth functions that appear very often in applications.

In this talk we will review the progress made in the last decade, in applying spectral methods to problems where the solution is only piecewise smooth.

The first part of the talk will concentrate on approximation theory: The resolution of the Gibbs phenomenon. We will show that if the first N Fourier coefficients of a piecewise smooth function are known then it is possible to get a rapidly convergent uniform approximation. Applications in computed tomography and splicing of pictures will be shown.

We will then discuss linear hyperbolic equations and show that spectral methods, with pre- and post-processing yield spectral accuracy. The Tadmor theory of spectral approximations to nonlinear hyperbolic equations will be reviewed, together with recent developments concerning the conservation properties of multidomain spectral methods. Simulations of reactive supersonic flows will be presented.

For further information please contact: mittelmann@asu.edu