Magister Scientiae - MSc / This thesis deals with the design and implementation of some novel numerical methods for non-linear singular pertubations problems (NSPPs). It provide a survey of asymptotic and numerical methods for some NSPPs in the past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information. / South Africa

>Magister Scientiae - MSc / This thesis deals with the design and implementation of some novel numerical methods for nonlinear singular perturbations problems (NSPPs). We provide a survey of asymptotic and numerical methods for some NSPPs in past decade. By considering two test problems, rigorous asymptotic analysis is carried out. Based on this analysis, suitable numerical methods are designed, analyzed and implemented in order to have some relevant results of physical importance. Since the asymptotic analysis provides only qualitative information, the focus is more on the numerical analysis of the problem which provides the quantitative information.

The present research focuses on the development and computer implementation of a novel threedimensional, anisotropic turbulence model not only capable of handling complex geometries but also the turbulence driven secondary currents. The model equations comprise advanced algebraic Reynolds stress models in conjunction with Reynolds Averaged Navier-Stokes equations. In order to tackle the complex geometry of compound meandering channels, the body-fitted orthogonal coordinate system is used. The finite volume method with collocated arrangement of variables is used for discretization of the governing equations. Pressurevelocity coupling is achieved by the standard iterative SIMPLE algorithm. A central differencing scheme and upwind differencing scheme are implemented for approximation of diffusive and convective fluxes on the control volume faces respectively. A set of algebraic equations, derived after discretization, are solved with help of Stones implicit matrix solver. The model is validated against standard benchmarks on simple and compound straight channels. For the case of compound meandering channels with varying sinuosity and floodplain height, the model results are compared with the published experimental data. It is found that the present method is able to predict the mean velocity distribution, pressure and secondary flow circulations with reasonably good accuracy. In terms of engineering applications, the model is also tested to understand the importance of turbulence driven secondary currents in slightly curved channel. The development of this unique model has opened many avenues of future research such as flood risk management, the effects of trees near the bank on the flow mechanisms and prediction of pollutant transport.

No description available.

The work presented in this thesis attempts to provide an understanding of the physics behind the dynamic stall process by simulating the flow past pitching NACA-0012 airfoil at 100,000 and 1 million Reynolds number based on the chord length of the airfoil and at different reduced frequencies of 0.188 and 0.25 respectively in a three dimensional flow field. The mean angles of attack are 12 deg. and 15 deg. and the amplitudes of pitching are 6 deg. and 10 deg. respectively. The turbulence in the flow field is resolved using large eddy simulations with dynamic Smagorinsky model at the sub grid scale. The lift hysteresis plots of this simulation for both the configurations are compared with the corresponding experiments. The development of dynamic stall vortex, vortex shedding and reattachment as predicted by the present study are discussed in detail. There is a fairly good match between the predicted and experimentally measured lift coefficient during the upstroke for both cases. The net lift coefficient for the Re = 100,000 case during downstroke matches with the corresponding experimental data, the present study under-predicts the lift coefficient as compared to the experimental values at the start of downstroke and over-estimates for the remaining part of the downstroke. The trend of the lift coefficient hysteresis plot with the experimental data for the Re = 1 million case is also similar. This present simulations have shown that the downstroke phase of the pitching motion is strongly three dimensional and is highly complex, whereas the flow is practically two dimensional during the upstroke. / Master of Science

Philosophiae Doctor - PhD / Fitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape. This thesis deals with the design and analysis of tted numerical methods for some delay di erential models that arise in biology. Very often such di erential equations are very complex in nature and hence the well-known standard numerical methods seldom produce reliable numerical solutions to these problems. Ine ciencies of these methods are mostly accumulated due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are initially designed to solve a class of general problems without considering the structure of any individual problems. In this thesis, issues like these are resolved for a set of delay di erential equations. Though the developed approaches are very simplistic in nature, they could solve very complex problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better. Finally, wherever appropriate, we have indicated possible extensions of our approaches to cater for other classes of problems. May 2009.

Submitted by Josirene Barbosa (josirene.henrique@ufcg.edu.br) on 2018-01-22T14:06:12Z No. of bitstreams: 1 BRAUNER GONÇALVES COUTINHO - TESE PPGEP 2013.pdf: 5662839 bytes, checksum: 38e4233d72065d1a5e3baa86c6f3f108 (MD5) / Made available in DSpace on 2018-01-22T14:06:12Z (GMT). No. of bitstreams: 1 BRAUNER GONÇALVES COUTINHO - TESE PPGEP 2013.pdf: 5662839 bytes, checksum: 38e4233d72065d1a5e3baa86c6f3f108 (MD5) Previous issue date: 2013-08-29 / CNPq / O processo RTM é amplamente utilizado para a produção de materiais compósitos de alta qualidade. As simulações computacionais podem desempenhar um importante papel na otimização deste processo, reduzindo custos e tornando-o mais eficiente. Neste trabalho, foi desenvolvida uma modelagem matemática bidimensional transiente para a etapa de preenchimento do molde em processos RTM que prediz o escoamento de duas fases (ar-resina) em meios porosos. O conjunto de equações diferenciais parciais, escritas em coordenadas generalizadas, é discretizado utilizando o método de volumes finitos e resolvido por meio de uma abordagem totalmente implícita via método de Newton com um esquema de passo de tempo variável. Foi desenvolvido um simulador computacional com ferramentas de pré e pós-processamento para ajudar na definição de parâmetros de simulações e na visualização dos resultados obtidos. Para validar a metodologia matemática proposta, resultados numéricos de tempo de preenchimento, posição da frente de fluxo, pressão e vazão de injeção para escoamentos radiais e retilíneos foram comparados com resultados analíticos conhecidos. Como aplicação, o modelo foi usado para descrever o escoamento de óleo vegetal em uma pré-forma de fibra de vidro no interior de uma cavidade retangular e os resultados comparados com dados oriundos de estudos experimentais. Também foram simulados casos envolvendo fronteiras irregulares arbitrárias. O modelo proposto e o simulador mostraram-se consistentes produzindo resultados fisicamente coerentes para as variáveis do processo. / The RTM process is widely used for the production of high quality composite materials. Computer simulations can play an important role in the optimization of this process, reducing costs and increasing efficiency. In this work, it was developed a 2D transient mathematical model for the mold filling stage in RTM process which predicts the twophase flow (air-resin) through porous media. The set of partial differential equations, written in boundary fitted coordinates, is discretized using the finite volume method in a fully implicit approach and solved by using the Newton’s method. It was developed a computational simulator with pre- and post-processing tools to help the definition of simulations parameters and the visualization of the results. To validate the mathematical methodology, numerical results for filling time, flow front position, injection pressure and injection flow rate for rectilinear and radial flows were compared to analytical results from known models. The model was employed to describe the fluid flow of a vegetable oil in a glass fiber preform within a rectangular cavity and the results were compared to experimental data. Some cases involving arbitrary irregular boundaries were also simulated. The proposed model and the simulator generated physically consistent results of the process variables.

Magister Scientiae - MSc / Efficient numerical approaches for parameter dependent problems have been an inter- esting subject to numerical analysts and engineers over the past decades. This is due to the prominent role that these problems play in modeling many real life situations in applied sciences. Often, the choice and the e ciency of the approaches depend on the nature of the problem to solve. In this work, we consider the general linear first-order singularly perturbed Volterra integro-differential equations (SPVIDEs). These singularly perturbed problems (SPPs) are governed by integro-differential equations in which the derivative term is multiplied by a small parameter, known as "perturbation parameter". It is known that when this perturbation parameter approaches zero, the solution undergoes fast transitions across narrow regions of the domain (termed boundary or interior layer) thus affecting the convergence of the standard numerical methods. Therefore one often seeks for numerical approaches which preserve stability for all the values of the perturbation parameter, that is "numerical methods. This work seeks to investigate some "numerical methods that have been used to solve SPVIDEs. It also proposes alternative ones. The various numerical methods are composed of a fitted finite difference scheme used along with suitably chosen interpolating quadrature rules. For each method investigated or designed, we analyse its stability and convergence. Finally, numerical computations are carried out on some test examples to con rm the robustness and competitiveness of the proposed methods.

L’avancée des infrastructures informatiques a permis l’émergence de la modélisation moléculaire. À cet effet, une multitude de modèles mathématiques sont aujourd’hui disponibles pour simuler différents systèmes chimiques. À l’aide de la modélisation moléculaire, différents types d’interactions chimiques ont été observés. À partir des systèmes les plus simples permettant l’utilisation de modèles quantiques rigoureux, une série d’approximations a été considérée pour rendre envisageable la simulation de systèmes moléculaires de plus en plus complexes. En premier lieu, la théorie de la fonctionnelle de densité dépendante du temps a été utilisée pour simuler les énergies d’excitation de molécules photoactives. De manière similaire, la DFT indépendante du temps a permis la simulation du pont hydrogène intramoléculaire de structures analogues au 1,3,5-triazapentadiène et la rationalisation de la stabilité des états de transition. Par la suite, la dynamique moléculaire et la mécanique moléculaire ont permis de simuler les interactions d’un trimère d’acide cholique et d’un pyrène dans différents solvants. Cette même méthodologie a été utilisée pour simuler les interactions d’un rotaxane-parapluie à l’interface d’un système biphasique. Finalement, l’arrimage moléculaire et les fonctions de score ont été utilisés pour simuler les interactions intermoléculaires entre une protéine et des milliers de candidats moléculaires. Les résultats ont permis de mettre en place une stratégie de développement d’un nouvel inhibiteur enzymatique. / The evolution of computer systems has led to the emergence of molecular modeling. To this end, a variety of mathematical models are now available to simulate various chemical systems. Using molecular modeling, different types of chemical interactions were observed. From the simplest systems allowing the use of rigorous quantum models, a series of approximations were considered in order to make possible the simulation of increasingly complex molecular systems. First, time-dependent density fonctional theory has been used to simulate the excitation energies of photoactive molecules. Similarly, time-independent DFT has enabled the simulation of intramolecular hydrogen bonding in the 1,3,5-triazapentadiene system and the rationalization of the stability of the transition states. Subsequently, molecular dynamics and molecular mechanics were used to simulate the interactions of a trimer of cholic acid with a pyrene in different solvents. This methodology was then used to simulate the interactions of an umbrella-rotaxane at the interface of a biphasic system. Finally, molecular docking and the concept of scoring functions were used to simulate the intermolecular interactions between a protein molecule and thousands of potential ligands. The results were then used to create a strategy for the development of a new enzyme inhibitor.

<p>In 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 &macr / nd 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</p>