Analysis and Application of Fourier--Gegenbauer Method to Stiff Differential Equations

Authors:
L. Vozovoi;M. Israeli;A. Averbuch
Affiliations:
-;-;-
Venue:
SIAM Journal on Numerical Analysis
Year:
1996

Citing 0
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Solving boundary value problems, integral, and integro-differential equations using Gegenbauer integration matrices

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Abstract

The Fourier--Gegenbauer (FG) method, introduced by [Gottlieb, Shu, Solomonoff, and Vandeven, ICASE Report 92-4, Hampton, VA, 1992] is aimed at removing the Gibbs phenomenon; that is, recovering 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 {\em pseudospectral\/} implementation. In particular, we analyze the behavior of the Gegenbauer series with a moderate (several hundred) number of terms suitable for computations. We also demonstrate the ability of the FG method to get a spectrally accurate approximation on small subintervals for rapidly oscillating functions or functions having steep profiles.Bearing on the previous analysis, we suggest a high-order spectral Fourier method for the solution of nonperiodic differential equations. It includes a polynomial subtraction technique to accelerate the convergence of the Fourier series and the FG algorithm to evaluate derivatives on the boundaries of nonperiodic functions. The present hybrid Fourier--Gegenbauer (HFG) method possesses better resolution properties than the original FG method. The precision of this method is demonstrated by solving stiff elliptic problems with steep solutions.