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.

Details are given herein of the development, refinement and application of a higher-order accurate 3-D finite difference model for non-cohesive suspended sediment transport processes, in non-stratified estuarine and coastal waters. The velocity fields are computed using a 2-D horizontal depth-integrated model, in combination with either an assumed logarithmic velocity profile or a velocity profile obtained from field data. Also, for convenience in handling variable bed topographies and for better vertical resolution, a δ-stretching co-ordinate system has been used. In order to gain insight into the relative merits of various numerical schemes for modelling the convection of high concentration gradients, in terms of both accuracy and efficiency, thirty six existing finite difference schemes and two splitting techniques have been reviewed and compared by applying them to the following cases: i) 1-D and 2-D pure convection, ii) 1-D and 2-D convection and diffusion, and iii) 1-D non-linear Burger's equation. Modifications to some of the considered schemes have also been proposed, together with two new higher-order accurate finite difference schemes for modelling the convection of high concentration gradients. The schemes were derived using a piecewise cubic interpolation and an universal limiter (proposed scheme 1) or a modified form of the TVD filter (proposed scheme 2). The schemes have been tested for: i) 1-D and 2-D pure convection, and ii) 2-D convection and diffusion problems. The schemes have produced accurate, oscillation-free and non-clipped solutions, comparable with the ULTIMATE fifth- and sixth-order schemes. However, the proposed schemes need only three (proposed scheme 1) or five cell stencils. Hence, they are very attractive and can be easily implemented to solve convection dominated problems for complex bathymetries with flooding and drying. The 3-D sediment transport equation was solved using a splitting technique, with two different techniques being considered. With this technique the 3-D convective-diffusion equation for suspended sediment fluxes was split into consecutive 1-D convection, diffusion and convective-diffusion equations. The modified and proposed higher-order accurate finite difference schemes mentioned above were then used to solve the consecutive 1-D equations. The model has been calibrated and verified by applying it to predict the development of suspended sediment concentration profiles under non-equilibrium conditions in three test flumes. The results of numerical predictions were compared with existing analytical solutions and experimental data. The numerical results were in excellent agreement with the analytical solutions and were in reasonable agreement with the experimental data. Finally, the model has also been applied to predict sediment concentration and velocity profiles in the Humber Estuary, UK. Reasonable agreement was obtained between the model predictions and the corresponding field measurements, particularly when considered in the light of usual sediment transport predictions. The model is therefore thought to be a potentially useful tool for hydraulic engineers involved in practical case studies

The aerodynamics of modern rotorcraft is highly complex and has proven to be an arduous challenge for computational fluid dynamics (CFD). Flow features such as massively separated boundary layers or transition to turbulence are common in engineering applications and need to be accurately captured in order to predict the vehicle performance. The recent advances in numerical methods and turbulence modeling have resolved each of these issues independent of the other. First, state-of-the-art hybrid RANS-LES turbulence closures have shown great promise in capturing the unsteady flow details and integrated performance quantities for stalled flows. Similarly, the correlation-based transition model of Langtry and Menter has been successfully applied to a wide range of applications involving attached or mildly separated flows. However, there still lacks a unified approach that can tackle massively separated flows in the transitional flow region. In this effort, the two approaches have been combined and expended to yield a methodology capable of accurately predicting the features in these highly complex unsteady turbulent flows at a reasonable computational cost. Comparisons are evaluated on several cases, including a transitional flat plate, circular cylinder in crossflow and NACA 63-415 wing. Cost and accuracy correlations with URANS and prior hybrid URANS-LES approaches with and without transition modeling indicate that this new method can capture both separation and transition more accurately and cost effectively. This new turbulence approach has been applied to the study of wings in the reverse flow regime. The flight envelope of modern helicopters has increased significantly over the last few decades, with design concepts now reaching advance ratios up to μ = 1. In these extreme conditions, the freestream velocity exceeds the rotational speed of the blades, and a large region of the retreating side of the rotor disk experiences reverse flow. For a conventional airfoil with a sharp trailing edge, the reverse flow regime is generally characterized by massive boundary layer separation and bluff body vortex shedding. This complex aerodynamic environment has been utilized to evaluate the new hybrid transitional approach. The assessment has proven the efficiency of the new hybrid model, and it has provided a transformative advancement to the modeling of dynamic stall.

Computing reliable numerical approximations of invariant sets for nonlinear systems is the core problem for computer assisted study of dynamical systems. In the case of conservative systems the problem is complicated by the fact that there is no phase space dissipation to drive orbits onto attractors. In this dissertation we discuss several contributions to the field of computer assisted study of invariant dynamics in conservative systems. / text

A numerical reservoir model was created to optimize CO₂ Huff-and-Puff operations in the Bakken Shale. Huff-and-Puff is an enhanced oil recovery treatment in which a well alternates between injection, soaking, and production. Injecting CO₂ into the formation and allowing it to “soak” re-pressurizes the reservoir and improves oil mobility, boosting production from the well. A compositional reservoir simulator was used to study the various design components of the Huff-and-Puff process in order to identify the parameters with the largest impact on recovery and understand the reservoir’s response to cyclical CO₂ injection. It was found that starting Huff-and-Puff too early in the life of the well diminishes its effectiveness, and that shorter soaking periods are preferable over longer waiting times. Huff-and-Puff works best in reservoirs with highly-conductive natural fracture networks, which allow CO₂ to migrate deep into the formation and mix with the reservoir fluids. The discretization of the computational domain has a large impact on the simulation results, with coarser gridding corresponding to larger projected recoveries. Doubling the number of hydraulic fractures per stage results in considerably greater CO₂ injection requirements without proportionally larger incremental recovery factors. Incremental recovery from CO₂ Huff-and-Puff appears to be insufficient to make the process commercially feasible under current economic conditions. However, re-injecting mixtures of CO₂ and produced hydrocarbon gases was proven to be technically and economically viable, which could significantly improve profit margins of Huff-and-Puff operations. A substantial portion of this project involved studying alternative numerical methods for modeling hydraulically-fractured reservoir models. A domain decomposition technique known as mortar coupling was used to model the reservoir system as two individually-solved subdomains: fracture and matrix. A mortar-based numerical reservoir simulator was developed and its results compared to a tradition full-domain finite difference model for the Cinco-Ley et al. (1978) finite-conductivity vertical fracture problem. Despite some numerical issues, mortar coupling closely matched Cinco-Ley et al.'s (1978) solution and has potential applications in complex problems where decoupling the fracture-matrix system might be advantageous. / text

Stability analysis methods may be categorized by two major stability analysis methods: small-signal stability and transient stability analyses. Transient stability methods are further categorized into two major categories: numerical methods based on numerical integration, and direct methods. The purpose of this thesis is to study and investigate transient stability analysis using a combination of step-by-step and direct methods using Equal Area Criterion. The proposed method is extended for transient stability analysis of multi machine power systems. The proposed method calculates the potential and kinetic energies for all machines in a power system and then compares the largest group of kinetic energies to the smallest groups of potential energies. A decision based on the comparison can be made to determine stability of the power system. The proposed method is used to simulate the IEEE 39 Bus system to verify its effectiveness by comparison to the results obtained by pure numerical methods.

<div>A strongly implicit, nonlinear Crank-Nicolson-based finite-difference scheme was constructed for the numerical study of the self-similar behavior of viscous gravity currents. Viscous gravity currents are low Reynolds number flow phenomena in which a dense, viscous fluid displaces a lighter (usually immiscible) fluid. Under the lubrication approximation, the mathematical description of the spreading of these fluids is reduced to solving a nonlinear parabolic partial differential equation for the shape of the fluid interface. This thesis focuses on the finite-speed propagation of a power-law non-Newtonian current in a variable width channel-like geometry (a "Hele-Shaw cell'') subject to a given mass conservation/balance constraint. The proposed numerical scheme was implemented on a uniform but staggered grid. It is shown to be strongly stable, while possessing formal truncation error that is of second-order in space and it time. The accuracy of the scheme was verified by benchmarking it against established analytical solutions, which were obtained via a first-kind self-similarity transformation. A series of numerical simulations confirmed that the proposed scheme accurately respects the mass conservation/balance constraint. Next, the numerical scheme was used to study the second-kind self-similar behaviour of Newtonian viscous gravity currents flowing towards the end of a converging channel. Second-kind self-similar transformations are not fully specified without further information from simulation or experiment. Thus, using the proposed numerical scheme, the self-similar spreading and leveling leveling of the current was definitively addressed. The numerical results showed favorable comparison with experimental data.</div>

ESS neutrino Super Beam (ESSnuSB) is a project that aim to create ahigh energy beam of neutrinos and anti-neutrinos to study thephenomenon neutrino oscillation and learn more about symmetryviolations in quantum mechanics. To create the neutrino beam, negativeHydrogen ions must be transported from the ESS linear accelerator at2.5 GeV, to a proton accumulation ring. This is done through a transferline, that shall direct the ion beam while preserve the beam as much aspossible. In thisproject, there was an attempt at finding a design for this transferline. Preferably, the line consists of a long main line of FODO cellsand two matching sections at each end. A simulation of the beam wasdone that gives the progression beta and dispersion functions,statistical measurements of the particle distribution, through a partof the transfer line. A design for the main line was found. For tuningthe quadrupole magnets, an iterative method using the system's responsematrix was used. However, it could not match more than four parametersat the time, while six was required for complete matching. Because ofthis, it is not able to match thedispersion.