Spectrally Accurate Solution of Nonperiodic Differential Equations by the Fourier--Gegenbauer Method

Abstract

It is well known that the Fourier partial sum of an analytic nonperiodic function, supported on a finite interval, converges slowly inside the interval and exhibits O(1) spurious oscillations near the boundaries (the Gibbs phenomenon). An effective algorithm which allows one to completely overcome the Gibbs phenomenon was developed in [J. Comput. Appl. Math., 43 (1992), pp. 81--92]. The basic concept of this approach consists of the re-expansion of the Fourier partial sums into the rapidly convergent Gegenbauer series. In this paper we extend the Fourier--Gegenbauer (F--G) method of [J. Comput. Appl. Math.}, 43 (1992), pp. 81--92] to the evaluation of the spatial derivatives of a piecewise analytic function. Also, we apply this method to the solution of nonperiodic boundary value problems. Although the derivatives of a discontinuous function are not in L2, the exponential convergence of the truncated Gegenbauer series can be proved, and the rate of convergence can be estimated. The solution of differential equations is accomplished in two steps. First, a particular solution with arbitrary boundary conditions is constructed using the F--G method. This particular solution is then corrected to satisfy the prescribed boundary conditions of the problem by adding a proper linear combination of homogeneous solutions. For boundary layer problems the intermediate (particular) solution has steep profiles near the boundaries. These steep regions introduce a large error into the final solution, which presumably has a smooth profile on the whole interval. A method which compensates for this loss of accuracy by using the appropriately constructed homogeneous solutions is proposed.