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|Title: ||SPECTRAL METHODS FOR HYPERBOLIC PROBLEMS|
|Authors: ||Tadmor, Eitan|
|Keywords: ||time-dependant problems|
|Issue Date: ||Jan-1994|
|Citation: ||E. Tadmor (1994). 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 the
approximate solutions of hyperbolic systems.
To make these notes self contained, we begin with a very brief overview of Cauchy
problems. Thus, the main focus of the rst part is on hyperbolic systems which are dealt
with two (related) tools: the energy method and Fourier analysis.
The second part deals with spectral approximations. Here we introduce the main ingredients
of spectral accuracy, Fourier and Chebyshev interpolants, aliasing, dierentiation
The third part is devoted to Fourier method for the approximate solution of periodic
systems. The questions of stability and convergence are answered by combining ideas from
the rst two sections. In this context we highlight the role of aliasing and smoothing; in
particular, 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 of
the Chebyshev method, paying special attention to the intricate issue of the CFL stability
restriction on the permitted time-step.|
|Appears in Collections:||Mathematics Research Works|
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