The Burgers equation, and related partial differential equations (PDEs), can be numerically challenging for small values of the viscosity parameter. For example, these equations can develop discontinuous solutions (or solutions with large gradients) from smooth initial data. Aside from numerical stability issues, standard numerical methods can also give rise to spurious oscillations near these discontinuities. In this study, we consider an equivalent form of the Burgers equation given by Constantin and Iyer, whose solution can be written as the expected value of a stochastic differential equation. This equivalence is used to develop a numerical method for approximating solutions to Burgers equation. Our preliminary analysis of the algorithm reveals that it is a natural generalization of the method of characteristics and that it produces approximate solutions that actually improve as the viscosity parameter vanishes. We present three examples that compare our algorithm to a recently published reference method as well as the vanishing viscosity/entropy solution for decreasing values of the viscosity. / Master of Science / Burgers equation is a Partial Differential Equation (PDE) used to model how fluids evolve in time based on some initial condition and viscosity parameter. This viscosity parameter helps describe how the energy in a fluid dissipates. When studying partial differential equations, it is often hard to find a closed form solution to the problem, so we often approximate the solution with numerical methods. As our viscosity parameter approaches 0, many numerical methods develop problems and may no longer accurately compute the solution. Using random variables, we develop an approximation algorithm and test our numerical method on various types of initial conditions with small viscosity coefficients.

Modeling and simulation have played an essential role in understanding the effects of blast waves. However, a broad area of engineering problems, such as vehicle structures, buildings, bridges, or even the human body, can benefit by accurately predicting the response to blasts with little need for test or event data. This thesis reviews fundamental concepts of blast waves and explosives and discusses research in blast scaling. Blast scaling is a method that reduces the computational costs associated with modeling blasts by using empirical data and numerically calculating blast field parameters over time for various types and sizes of explosives. This computational efficiency is critical in studying blast waves' near and far-field effects. This thesis also reviews research to differentiate between free-air blasts and gun muzzle blasts and the progress of modeling the muzzle blast-structure interaction. The main focus of this thesis covers an investigation of different numerical and analytical solutions to a simple aerospace structure subjected to blast pressure. The thesis finally presents a tool that creates finite element loads utilizing muzzle blast scaling methods. This tool reduces modeling complexity and the need for multiple domains such as coupled computational fluid dynamics and finite element models by coupling blast scaling methods to a finite element model. / Master of Science / {Numerical integration methods have helped solve many complex problems in engineering and science due to their ability to solve non-linear equations that describe many phenomena. These methods are beneficial because of how well they lend to programming into a computer, and their impact has grown with the increases in computing power. In this thesis, ``modeling and simulation" refers to the characterization and prediction of an event's outcome through the use of computers and numerical techniques. Modeling and simulation play important roles in studying the effects of blast waves in many areas of engineering research such as aerospace, biomedical, naval, and civil. Their capability to predict the outcome of the interaction of a blast wave to vehicle structures, buildings, bridges, or even the human body while requiring limited experimental data has the chance to benefit a wide area of engineering problems. This thesis reviews fundamental concepts of blast waves, explosives, and research that has applied blast loading in modeling and simulation. This thesis describes the complexity of modeling an axially symmetric blast wave interaction by comparing the numerical and theoretical response blast loaded structure.

Wave propagation is a one of the most studied phenomenons in history due to the variety of applications such as quantum mechanics, electrodynamics and acoustics. In this thesis, the possibilities of improving numerical methods for solving the wave equation will be studied. More specifically, the dispersion relation will be used as a focal point. Generally there is a difference between the dispersion relation in the numerical solution and the analytic solution and the aim will be to decrease this difference and study the consequences. The numerical method that will be used and improved is the finite difference method (FDM). A dispersion relation for the numerical scheme will be derived including parameters from the spatial discretisation. These parameters will be optimised with the gradient descent method while retaining the second order accuracy of the derivative approximation. Performance is tested with numerical examples and the method of optimising for improved dispersion relation is proved to be successful. The optimised second order accurate schemes outperforms the standard second order accurate method in all simulated examples. When comparing the optimised stencil with the equally computationally expensive fourth order accurate method the optimised stencil performs better for sparse grids, especially when the spatial variation in the solution is high. For finer grids the fourth order accurate method quickly achieves smaller errors and is therefore preferable.

Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided. / South Africa

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston's volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Wie seit schon einigen Jahrzehnten bekannt ist, werden akustische Oszillationen durch stellare Aktivität beeinflusst. Die globalen akustischen Moden in der Sonne weisen eine Variation mit dem 11-jährigen Sonnenzyklus auf. Ein ähnliches Phänomen konnte auch in anderen Sternen mit Hilfe von Asteroseismologie nachgewiesen werden. In dieser Arbeit erforsche ich den Einfluss von großen Sternflecken auf globale Oszillationen mit niedrigem Grad ℓ. Als wichtigstes Werkzeug benutze ich hierfür den GLASS Kode, der die Ausbreitung von linearen akustischen Wellen im Sterninneren in 3D simuliert. Zunächst habe ich das Problem der konvektiven Stabilisierung betrachtet, welches bei jedem linearen Oszillationskode im Zeitbereich auftritt. Ich präsentiere eine allgemeine Methode um konvektiv stabile Hintergrundsmodelle für ein vorgegebenes Sternmodell zu erzeugen. Dabei werden wichtige Eigenschaften des ursprünglichen Modells beibehalten, beispielsweise das hydrostatische Gleichgewicht. Ich schlage einen störungstheoretischen Ansatz vor, um das akustische Wellenfeld in dem ursprünglichen instabilen Sternmodell näherungsweise zu erlangen. Tests zeigen, dass für Moden mit niedrigem Grad ℓ und einer Frequenz um 3 mHz die korrigierten Frequenzen mit einer Genauigkeit von 1 μHz mit den exakten Werten übereinstimmen. Zweitens habe ich mit Hilfe des GLASS Kodes den Einfluss einer am Nordpol des Sterns lokalisierten Störung der Schallgeschwindigkeit auf radiale, dipolare und quadrupolare Oszillationsmoden untersucht. Diese Studie zeigt auf, dass die axialsymmetrischen Moden dadurch am stärksten beeinflusst werden und im Falle von großen Sternflecken können ihre Frequenzen nicht mit der linearen Theorie berechnet werden. Die Form der Eigenfunktionen der Moden weicht von reinen Kugelflächenfunktionen ab und werden mit Kugelflächenfunktionen mit unterschiedlichem Grad ℓ vermischt. Dies könnte die korrekte Identifikation der Moden in der spektralen Leistungsdichte beeinflussen. Drittens habe ich den beobachtbaren Einfluss eines großen Sternflecks auf Moden mit Grad ℓ betrachtet. Im Falle einer aktiven Region, die mit dem Stern rotiert (und sich nicht am Pol befindet), ist die Störung nicht stabil, wenn sie in einem Inertialsystem betrachtet wird. Der kombinierte Einfluss von Rotation und Sternfleck veranlasst jede Mode, in der beobachteten spektralen Leistungsdichte als (2ℓ + 1)² Peaks aufzutreten. Die Einhüllende der spektralen Leistungsdichte eines Multipletts ist also komplex und hängt von dem Breitengrad ab, wo sich die aktive Region befindet, und vom Inklinationswinkel des Sterns. Ich berechne die spektrale Leistungsdichte für einige Beispiele sowohl mit Störungstheorie als auch mit Hilfe von GLASS. Diese Arbeit soll dazu beitragen, die spektrale Leistungsdichte von oszillierenden Sternen, die Sternflecken aufweisen, zu interpretieren.

Since the innovation of computers some 40 years ago and the introduction of microcomputers in 1975, computers are playing an active role in education processes and altering the pattern of interaction between teacher and student in the classroom. Computer assisted learning has been seen as a revolution in education. In this research, the author has studied the impact of using microcomputers on mathematical education, particularly at the Hong Kong tertiary level, in different perspectives. Two computer software packages have been developed on the microcomputer. The consideration of the topic to be used in the computer assisted learning was arrived at in earlier surveys with students who thought that computers could give very accurate solutions to calculations. The two software packages, demonstrating on the spot the error that would be incurred by the computer, have been used by the students. They are both interactive and make use of the advantages of the microcomputer's functions over other teaching media, such as graphics facility and random number generator, to draw to the students' attention awareness of errors that may be obtained using computers in numerical solutions. Much emphasis is put on the significance and effectiveness of using computer packages in learning and teaching. Measurements are based on questionnaires, conversations with students, and tests on content material after the packages have been used. Feedback and subjective opinion of using computers in mathematical education have also been obtained from both students and other teachers. The research then attempts to examine the suitability of applying computer assisted learning in Hong Kong education sectors. Some studies on the comments made by students who participated in the learning process are undertaken. The successes and failures in terms of student accomplishment and interest in the subject area as a result of using a software package is described. Suggestions and recommendations are given in the concluding chapter.

Multiphysics systems with interface coupling are used to model a variety of physical phenomena, such as arterial blood flow, air flow around aeroplane wings, or interactions between surface and ground water flows. Numerical methods enable the practical application of these models through computer simulations. Specifically a high level of detail and accuracy is achieved in finite element methods by discretisations which use extremely large numbers of degrees of freedom, rendering the solution process challenging from the computational perspective. In this thesis we address this challenge by developing a twofold strategy for improving the efficiency of standard finite element coupled solvers. First, we propose to solve a monolithic coupled problem using block-preconditioned GMRES with a new Schur complement approximation. This results in a modular and robust method which significantly reduces the computational cost of solving the system. In particular, numerical tests show mesh-independent convergence of the solver for all the considered problems, suggesting that the method is well-suited to solving large-scale coupled systems. Second, we derive an adjoint-based formula for goal-oriented a posteriori error estimation, which leads to a time-space mesh refinement strategy. The strategy produces a mesh tailored to a given problem and quantity of interest. The monolithic formulation of the coupled problem allows us to obtain expressions for the error in the Lagrange multiplier, which often represents a physically relevant quantity, such as the normal stress on the interface between the problem components. This adaptive refinement technique provides an effective tool for controlling the error in the quantity of interest and/or the size of the discrete system, which may be limited by the available computational resources. The solver and the mesh refinement strategy are both successfully employed to solve a coupled Stokes-Darcy-Stokes problem modelling flow through a cartridge filter.