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Analysis and Application of FourierGegenbauer Method to Stiff Differential Equations \Lambda

Summary: Analysis and Application of Fourier­Gegenbauer
Method to Stiff Differential Equations \Lambda
appeared in SIAM J. Numerical Analysis, 33:5 (1996), 1844­1863
A. Averbuch y , M. Israeli z , L. Vozovoi z
y School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
z Faculty of Computer Science, Technion, Haifa 32000, Israel
The Fourier­Gegenbauer (FG) method, introduced by [1], is aimed to remove the
Gibbs phenomenon, that is to recover the point values of a non­periodic function from
its Fourier coefficients. In this paper we discuss some numerical aspects of the FG
method related to its pseudospectral implementation. In particular, we analyze the
convergence of finite Gegenbauer series with hundreds terms, suitable for computations.
We also demonstrate the capability of the FG method for getting a spectrally accurate
approximation on small sub­intervals for functions who are rapidly oscillating or have
steep profiles.
Bearing on the previous analysis, we suggest a high­order spectral Fourier method
for the solution of non­periodic differential equations. It includes a polynomial subtrac­
tion technique to accelerate the convergence of the Fourier series and the FG algorithm
to evaluate derivatives on the boundaries of non­periodic functions. The current Hybrid
Fourier­Gegenbauer method possesses better resolution properties than the original FG


Source: Averbuch, Amir - School of Computer Science, Tel Aviv University


Collections: Computer Technologies and Information Sciences