On the design and implementation of a hybrid numerical method for singularly perturbed two-point boundary value problems

>Magister Scientiae - MScWith the development of technology seen in the last few decades, numerous solvers have been developed to provide adequate solutions to the problems that model different aspects of science and engineering. Quite often, these solvers are tailor-made for specific classes of problems. Therefore, more of such must be developed to accompany the growing need for mathematical models that help in the understanding of the contemporary world. This thesis treats two point boundary value singularly perturbed problems. The solution to this type of problem undergoes steep changes in narrow regions (called boundary or internal layer regions) thus rendering the classical numerical procedures inappropriate. To this end, robust numerical methods such as finite difference methods, in particular fitted mesh and fitted operator methods have extensively been used. While the former consists of transforming the continuous problem into a discrete one on a non-uniform mesh, the latter involves a special discretisation of the problem on a uniform mesh and are known to be more accurate. Both classes of methods are suitably designed to accommodate the rapid change(s) in the solution. Quite often, finite difference methods on piece-wise uniform meshes (of Shishkin-type) are adopted. However, methods based on such non-uniform meshes, though layer-resolving, are not easily extendable to higher dimensions. This work aims at investigating the possibility of capitalising on the advantages of both fitted mesh and fitted operator methods. Theoretical results are confirmed by extensive numerical simulations

Defektkorrekturverfahren für singulär gestörte Randwertaufgaben

Wir untersuchen ein Defektkorrekturverfahren, das ein einfaches Upwind-Differenzenverfahren erster Ordnung mit einem zentralen Differenzenverfahren kombiniert, für ein- und zweidimensionale singulär gestörte Konvektions-Diffusions-Probleme auf einer Klasse von Shishkin-Typ-Gittern. Im eindimensionalen Fall wird nachgewiesen, dass das Verfahren von (fast) zweiter Ordnung, gleichmäßig bezüglich des Diffusionsparameters $\epsilon$ konvergiert. Zur Konvergenzanalyse für das zweidimensionale Modellproblem werden verschiedene Techniken diskutiert. In einem Spezialfall kann auf einem stückweise uniformen Shishkin-Gitter die $\epsilon$-gleichmäßige Konvergenz des Verfahrens von fast zweiter Ordnung gezeigt werden. Ferner sind die bisher bekannten Stabilitätsaussagen und ihre Verwendung zur Konvergenzanalysis der betrachteten Differenzenverfahren sowie Methoden zur Analyse von Defektkorrekturverfahren zusammengestellt. Einige Bemerkungen zu Defektkorrekturverfahren und Finite-Elemente-Methoden schließen die Arbeit ab. Numerische Experimente untermauern die theoretischen Resultate.We consider a defect correction method that combines a first-order upwinded difference scheme with a second-order central difference scheme for model singularly perturbed convection-diffusion problems in one and two dimensions on a class of Shishkin-Type meshes. In one dimension, the method is shown to be convergent uniformly in the diffusion parameter $\epsilon$ of second order in the discrete maximum norm. To analyze the two-dimensional case, we discuss several proof techniques for defect correction methods. For a special problem with constant coefficients on a piecewise uniform Shishkin-mesh we can show the second order convergence of the considered scheme, uniformly with respect to the diffusion parameter. Moreover the known stability properties and their impact on the convergence analysis of the considered differnce schemes are compiled. Some remarks on defect correction and finite elements conclude the theses. Numerical experiments support our theoretical results

Maximum-norm a posteriori error bounds for an extrapolated upwind scheme applied to a singularly perturbed convection-diffusion problem

Richardson extrapolation is applied to a simple first-order upwind difference scheme for the approximation of solutions of singularly perturbed convection-diffusion problems in one dimension. Robust a posteriori error bounds are derived for the proposed method on arbitrary meshes. It is shown that the resulting error estimator can be used to stear an adaptive mesh algorithm that generates meshes resolving layers and singularities. Numerical results are presented that illustrate the theoretical findings

Layer-adapted meshes for convection-diffusion problems

This is a book on numerical methods for singular perturbation problems - in particular stationary convection-dominated convection-diffusion problems. More precisely it is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. An early important contribution towards the optimization of numerical methods by means of special meshes was made by N.S. Bakhvalov in 1969. His paper spawned a lively discussion in the literature with a number of further meshes being proposed and applied to various singular perturbation problems. However, in the mid 1980s this development stalled, but was enlivend again by G.I. Shishkin's proposal of piecewise- equidistant meshes in the early 1990s. Because of their very simple structure they are often much easier to analyse than other meshes, although they give numerical approximations that are inferior to solutions on competing meshes. Shishkin meshes for numerous problems and numerical methods have been studied since and they are still very much in vogue. With this contribution we try to counter this development and lay the emphasis on more general meshes that - apart from performing better than piecewise-uniform meshes - provide a much deeper insight in the course of their analysis. In this monograph a classification and a survey are given of layer-adapted meshes for convection-diffusion problems. It tries to give a comprehensive review of state-of-the art techniques used in the convergence analysis for various numerical methods: finite differences, finite elements and finite volumes. While for finite difference schemes applied to one-dimensional problems a rather complete convergence theory for arbitrary meshes is developed, the theory is more fragmentary for other methods and problems and still requires the restriction to certain classes of meshes

Higher order numerical methods for singular perturbation problems

Philosophiae Doctor - PhDIn recent years, there has been a great interest towards the higher order numerical methods for singularly perturbed problems. As compared to their lower order counterparts, they provide better accuracy with fewer mesh points. Construction and/or implementation of direct higher order methods is usually very complicated. Thus a natural choice is to use some convergence acceleration techniques, e.g., Richardson extrapolation, defect correction, etc. In this thesis, we will consider various classes of problems described by singularly perturbed ordinary and partial differential equations. For these problems, we design some novel numerical methods and attempt to increase their accuracy as well as the order of convergence. We also do the same for existing numerical methods in some instances. We find that, even though the Richardson extrapolation technique always improves the accuracy, it does not perform equally well when applied to different methods for certain classes of problems. Moreover, while in some cases it improves the order of convergence, in other cases it does not. These issues are discussed in this thesis for linear and nonlinear singularly perturbed ODEs as well as PDEs. Extrapolation techniques are analyzed thoroughly in all the cases, whereas the limitations of the defect correction approach for certain problems is indicated at the end of the thesis.South Afric

Robust computational methods for two-parameter singular perturbation problems

Magister Scientiae - MScThis thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.South Afric

A new parameter-uniform discretization of semilinear singularly perturbed problems

In this paper, we present a numerical approach to solving singularly perturbed semilinear convection-diffusion problems. The nonlinear part of the problem is linearized via the quasilinearization technique. We then design and implement a fitted operator finite difference method to solve the sequence of linear singularly perturbed problems that emerges from the quasilinearization process. We carry out a rigorous analysis to attest to the convergence of the proposed procedure and notice that the method is first-order uniformly convergent. Some numerical evaluations are implemented on model examples to confirm the proposed theoretical results and to show the efficiency of the method