Computational methods for singularly perturbed two point boundary value problems.

摘要

The goal of this dissertation is to automate the singular perturbation of two point boundary value problems (BVPs) and use the results of this analysis in conjunction with numeric BVP solvers to improve their efficiency. The analysis provides both an approximate solution and the location and behavior of any inner or boundary layers. These layers represent regions where the solution is changing rapidly. It is the presence of these regions in the solution that make the BVPs difficult to solve numerically. In this research, we first outline the general singular perturbation methods used to analyze a particular class of problems. We consider both the well-documented examples and the special cases of the linear singularly perturbed BVP. In most cases, the analysis yields both an approximate solution and the location of potential inner and boundary layers. When more then one method of analysis is available, only one method is automated. In every case, the analysis is described in detail in order to make clear the process of automating.; The perturbation code used to automate the analysis is then described along with cases where it encounters difficulty. A catalogue of the problems tested and the results are presented in an appendix along with the perturbation code itself. The information produced by the perturbation code is then used to design an initial mesh of Bakhvalov type which can be provided to the numeric solvers. The efficiency of the solvers are tested by providing both a Bakhvalov initial mesh and an evenly spaced initial mesh and examining the diagnostic information provided by the solver.; In addition to examining the efficiency of the numeric solver, we also check the numeric solution to ensure it has the correct character. Since the perturbation code produces an approximate solution, this approximate solution is compared to the numeric in order to check that the numeric solution has boundary or inner layers in the correct locations. This comparison is of particular relevance for ill-conditioned problems where the numeric solution may not be correct.