| Over the past twenty years, spline collocation methods have evolved as valuable techniques for the solution of a broad class of problems covering ordinary and partial differential equations, integral equations and integro-differential equations. The popularity of such methods is due in part to their conceptual simplicity, wide applicability, and ease of implementation. In Part I of this thesis, we describe the efficacy of parallel processing applied to orthogonal spline collocation for ordinary differential equations (ODEs). Considerable attention has been given in recent years to the development of efficient methods for the solution of the almost block diagonal (ABD) linear systems which arise in this method, and some of these methods have been designed to exploit parallelism. However, in COLNEW, a widely used package implementing orthogonal spline collocation for solving ODEs with multi-point boundary conditions, the solution of the ABD systems contributes only a modest portion of the overall execution time, this limiting the potential benefit of these new solvers. We describe a modification to the COLNEW package which incorporates coarse-grained parallelism to minimize the cost of the linear system setup. With this modification in place, the ABD solver becomes more significant in terms of relative execution time, and we examine in this thesis the implementation of new ABD solvers in this modified version of the code.;The approximate solution of Poisson's equation on a rectangle subject to various standard boundary conditions (Dirichlet, Neumann, periodic) is required in many physical problems, especially in computational fluid dynamics. Recently, fast direct methods have been developed for the solution of linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to the Dirichlet problem for Poisson's equation on a rectangular region. In Part II of this thesis, we describe these algorithms and their implementations on various parallel architectures. We also extend these methods to problems with Neumann and periodic boundary conditions. Results are presented which demonstrate the high degree of parallelism inherent in these algorithms. Also, convergence results are presented, and demonstrate superconvergence properties of these collocation methods which have not been previously reported in the literature. |
