Parameter-Uniform Numerical Methods for Singularly perturbed convection-deffusion boundary-value problems using Adaptive Grid

Abstract

This thesis provides some efficient numerical techniques for solving singularly perturbed convectiondiffusion boundary-value problems exhibiting boundary layers. These singular perturbation problems (SPPs) are described by differential equations in which the highest derivative is multiplied by an arbitrarily small parameter "" known as Dsingular perturbation parameterD. This leads to the occurrence of boundary layers, which are basically thin regions in the neighbourhood of the boundary of the domain, where the gradients of the solutions steepen as the perturbation parameter "" tends to zero. Due to this layer phenomena, it is very difficult and also a challenging task to provide ""-uniform numerical methods i.e., methods in which the approximate solution converges to the exact solution independently with respect to the perturbation parameter, measured in the supremum norm. The convergence properties of some ""-uniform numerical methods are developed and analyzed in this thesis for solving SPPs using nonuniform grids. Especially two types of nonuniform grids are discussed here. They are the well-known piecewise-uniform Shishkin mesh and the newly developed adaptive grid which is based on the equidistribution of a strictly positive monitor function depending on the solution. These nonuniform grids are so chosen as to give a numerical solution that is uniformly accurate with respect to the singular perturbation parameter. This thesis consists of eight chapters. Chapter 1 contains the general introduction and it also provides the motivation and objective for solving SPPs. Chapter 2 presents a brief discussion on the generation of the Shishkin mesh and the adaptive grid and proposes an algorithm for the practical implementation of an adaptive remeshing strategy. A model convection-diffusion problem is solved using the upwind scheme on this adaptively generated grid formed by the arc-length monitor function. Here, the first-order accurate global solution via interpolation and the first-order approximation to the normalized flux are found out on this adaptively generated grid and also their uniform convergence analysis is carried out in the whole domain. The monitor function remains the same irrespective of the location of the boundary layer (on left or right) of the domain which shows the advantage of using such kind of adaptive grid over the Shishkin mesh. Again, the same monitor function is used in Chapter 3 for solving a model convection-diffusion problem with Robin boundary conditions and achieves the optimal rate of convergence in the discrete supremum norm. The adaptive grid idea is then extended for solving the singularly perturbed differential-difference equations with the delay and the shift terms, for the first time in the literature in Chapters 4 and 5. The error analysis is carried out for the classical upwind scheme on the adaptive grid and an optimal first-order convergence is obtained. Next, two higher-order methods are discussed for solving the singularly perturbed delay differential equations on the Shishkin mesh namely: the Richardson extrapolation technique (a postprocessing technique) and the defect-correction method in Chapters 6 and 7, respectively, which give almost second-order convergence improving the almost first-order convergence of the upwind scheme. Finally, Chapter 8 summarizes the results obtai...