SPECTRAL METHODS FOR HYPERBOLIC PROBLEMS

Abstract

We review several topics concerning spectral approximations of time-dependent problems,primarily | the accuracy and stability of Fourier and Chebyshev methods for theapproximate solutions of hyperbolic systems.To make these notes self contained, we begin with a very brief overview of Cauchyproblems. Thus, the main focus of the rst part is on hyperbolic systems which are dealtwith two (related) tools: the energy method and Fourier analysis.The second part deals with spectral approximations. Here we introduce the main ingredientsof spectral accuracy, Fourier and Chebyshev interpolants, aliasing, dierentiationmatrices ...The third part is devoted to Fourier method for the approximate solution of periodicsystems. The questions of stability and convergence are answered by combining ideas fromthe rst two sections. In this context we highlight the role of aliasing and smoothing; inparticular, we explain how the lack of resolution might excite small scales weak instability,which is avoided by high modes smoothing.The forth and nal part deals with non-periodic problems. We study the stability ofthe Chebyshev method, paying special attention to the intricate issue of the CFL stabilityrestriction on the permitted time-step.