L. Vozovoi et al., ANALYSIS AND APPLICATION OF FOURIER-GEGENBAUER METHOD TO STIFF DIFFERENTIAL-EQUATIONS, SIAM journal on numerical analysis, 33(5), 1996, pp. 1844-1863
The Fourier-Gegenbauer (FG) method, introduced by [Gottlieb, Shu, Solo
monoff, and Vandeven, ICASE Report 92-4, Hampton, VA, 1992] is aimed a
t 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 ps
eudospectral implementation. Tn particular, we analyze the behavior of
the Gegenbauer series with a moderate (several hundred) number of ter
ms suitable for computations. We also demonstrate the ability of the F
G method to get a spectrally accurate approximation on small subinterv
als for rapidly oscillating functions or functions having steep profil
es. 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 con
vergence of the Fourier series and the FG algorithm to evaluate deriva
tives on the boundaries of nonperiodic functions. The present hybrid F
ourier-Gegenbauer (HFG) method possesses better resolution properties
than the original FG method. The precision of this method is demonstra
ted by solving stiff elliptic problems with steep solutions.