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

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 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.