ANALYSIS AND APPLICATION OF FOURIER-GEGENBAUER METHOD TO STIFF DIFFERENTIAL-EQUATIONS

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.