Spectral integration and the numerical solution of two-point boundary value problems

Abstract

Spectral integration methods have been introduced for constant-coefficient two-point boundary value problems by Greengard, and pseudospectral integration methods for Volterra integral equations have been investigated by Kauthen. This thesis presents an approach to variable-coefficient two-point boundary value problems which employs pseudospectral integration methods to solve an equivalent integral equation. This thesis covers three topics in the application of spectral integration methods to two-point boundary value problems. The first topic is the development of the spectral integration concept and a derivation of the spectral integration matrices. The derivation utilizes the discrete Chebyshev transform and leads to a stable algorithm for generating the integration matrices. Convergence theory for spectral integration of C[subscript k] and analytic functions is presented. Matrix-free implementations are discussed with an emphasis on computational efficiency. The second topic is the transformation of boundary value problems to equivalent Fredholm integral equations and discretization of the resulting integral equations. The discussion of boundary condition treatments includes Dirichlet, Neumann, and Robin type boundary conditions. The final topic is a numerical comparison of the spectral integration and spectral differentiation approaches to two-point boundary value problems. Numerical results are presented on the accuracy and efficiency of these two methods applied to a set of model problems. The main theoretical result of this thesis is a proof that the error in spectral integration of analytic functions decays exponentially with the number of discretization points N. It is demonstrated that spectrally accurate solutions to variable-coefficient boundary value problems can be obtained in O(NlogN) operations by the spectral integration method. Numerical examples show that spectral integration methods are competitive with spectral differentiation methods in terms of accuracy and efficiency.