Numerical methods for the solution of two-point boundary value problems
The numerical approximation of solutions of ordinary differential equations played an important role in Numerical Analysis and still continues to be an active field of research. This is mainly due to the pressure of needs to model mathematically real world phenomena. In this thesis we are mainly concerned with the numerical solution of the first-order system of nonlinear two-point boundary value problems dy dx = f (x, y), a≤ x ≤ b, g(y(a), y(b)) = 0, where y ∈ Rn, f : R × Rn → Rn, and g : Rn × Rn → Rn. We will focus on the problem of solving singular perturbation problems since this has for many years been a source of difficulty to applied mathematicians and numerical analysts alike. We consider first deferred correction schemes based on Mono-Implicit Runge- Kutta (MIRK) and Lobatto formulae. As is to be expected, the scheme based on Lobatto formulae, which are implicit, is more stable than the scheme based on MIRK formulae which are explicit. Another deferred correction scheme, which uses the idea of the superconvergent deferred correction schemes, is also derived, and is shown to be highly stable compared to MIRK deferred correction schemes. To provide the continuous extension of the discrete solution, we construct high order interpolants based on an approach of using the already computed discrete solutions obtained on the final mesh. We will consider the construction of both explicit and implicit interpolants. An interpolation using a quasi-uniform grid is also introduced. This grid is naturally obtained in the mesh doubling which is a part of Richardson extrapolation. The estimation of conditioning numbers is discussed and used to develop mesh selection algorithms which will be appropriate for solving stiff linear and nonlinear boundary value problems. The algorithms are implemented in codes using deferred correction schemes based on MIRK and Lobatto formulae and the performance of codes which take account of the conditioning is compared with the performance of codes which use accuracy alone. Most of problems discussed in this thesis are two-point boundary value problems with separated boundary conditions. To complete our discussion, we explain numerical methods for solving two-point boundary value problems with nonseparated conditions, problems which contain parameters and those where the boundary conditions are given as integral constraints. We implement QR decomposition based on Householder transformations in the numerical experiments and discuss the results compared with Gaussian elimination.