DEFECT CORRECTION METHODS FOR THE SOLUTION OF SINGULARLY PERTURBED CONVECTION-DIFFUSION PROBLEMS

Abstract

This thesis provides efficient and higher-order methods for solving singular convection diffusion perturbation problems. A differential equation is singularly perturbed when some or all of its highest-order derivatives are multiplied by a small parameter, known as the perturbation parameter. When limiting the perturbation parameter to a value of zero, the solution to such problems reveals a multiscale character, which adds stiffness to the problem. This results in the formation of boundary layers. In these layers, the physical variable varies rapidly across small domains. As a result of the presence of layer phenomena, theoretical methods cannot accurately approximate the solution. These methods are only pertinent to a subset of problems, and prior knowledge of the solution’s behaviour is required. Consequently, it is an intriguing task to develop uniform numerical methods for solving such problems. This thesis aims to develop higher-order defect correction methods for the solution of four distinct kinds of convection-diffusion problems. Combining a stable, low-order, precise, and computationally inexpensive upwind difference scheme with a higher-order, less stable modified central difference operator is possible in these methods. In addition to being free of directional bias, the process is also unconditionally stable and converges uniformly. This technique can be utilised in an adaptive procedure to refine the mesh in non-smooth regions. For a convection-diffusion problem, a defect correction method over an adaptive mesh produces uniform second-order convergence. For a convection diffusion problem with a discontinuous coefficient and point source, a defect correction method combining a simple upwind scheme and a central difference scheme at all mesh points over the Bakhvalov Shishkin mesh is studied. A posteriori error estimates are established, yielding second-order convergence at all mesh points. Then, a parabolic convection-diffusion problem with a large shift is solved by utilising an implicit Euler scheme in time variable on a uniform mesh and a defect correction method comprised of the upwind scheme and the modified central difference scheme in space variable on a non-uniform mesh. The second order of spatial convergence and the first order of tem poral convergence are obtained. Finally, a parabolic convection-diffusion problem with xi xii Abstract a discontinuous convection coefficient and a source is solved using an implicit difference scheme in time on a uniform mesh and a defect correction scheme based on a finite dif ference discretisation over an adaptive mesh in space. Estimates of parameter uniform error reveal uniform convergence of first-order in time and second-order in space. Based on the contribution of this study, we suggest future research directions for analysing more complex problems.