The modelling of the processes of advection and dispersion-diffusion is the most crucial factor in solute transport simulations. It is generally appreciated that the first order upwind difference scheme gives rise to excessive numerical diffusion, whereas the conventional second order central difference scheme exhibits severe oscillations for advection dominated transport, especially in regions of high solute gradients or discontinuities. Higher order schemes have therefore become increasingly used for improved accuracy and for reducing grid scale oscillations. Two such schemes are the QUICK (Quadratic Upwind Interpolation for Convective Kinematics) and TOASOD (Third Order Advection Second Order Diffusion) schemes, which are similar in formulation but different in accuracy, with the two schemes being second and third order accurate in space respectively for finite difference models. These two schemes can be written in various finite difference forms for transient solute transport models, with the different representations having different numerical properties and computational efficiencies. Although these two schemes are advectively (or convectively) stable, it has been shown that the originally proposed explicit QUICK and TOASOD schemes become numerically unstable for the case of pure advection. The stability constraints have been established for each scheme representation based upon the von Neumann stability analysis. All the derived schemes have been tested for various initial solute distributions and for a number of continuous discharge cases, with both constant and time varying velocity fields. The 1-D QUICKEST (QUICK with Estimated Streaming Term) scheme is third order accurate both in time and space. It has been shown analytically and numerically that a previously derived quasi 2-D explicit QUICKEST scheme, with a reduced accuracy in time, is unstable for the case of pure advection. The modified 2-D explicit QUICKEST, ADI-TOASOD and ADI-QUICK schemes have been developed herein and proved to be numerically stable, with the bility sta- region of each derived 2-D scheme having also been established. All these derived 2-D schemesh ave been tested in a 2-D domain for various initial solute distributions with both uniform and rotational flow fields. They were further tested for a number of 2-D continuous discharge cases, with the corresponding exact solutions having also been derived herein. All the numerical tests in both the 1-D and 2-D cases were compared with the corresponding exact solutions and the results obtained using various other difference schemes, with the higher order schemes generally producing more accurate predictions, except for the characteristic based schemes which failed to conserve mass for the 2-D rotational flow tests. The ADI-TOASOD scheme has also been applied to two water quality studies in the U. K., simulating nitrate and faecal coliform distributions respectively, with the results showing a marked improvement in comparison with the results obtained by the second order central difference scheme. Details are also given of a refined numerical representation of flooding and drying of tidal flood plains for hydrodynamic modelling, with the results showing considerable improvements in comparison with a number of existing models and in good agreement with the field measured data in a natural harbour study.

The objectives of this study are to investigate the dynamic behavior of an apple slab subjected to drying at constant external conditions and under changing in the drying temperatures and to determine the effects of temperature and time combinations at different steps during drying on the process dynamics parameters, time constant and process gain of the system. For this purpose, a semi-batch dryer system was simulated by using integral method of analysis. Initially, the dynamic behavior of the drying temperature was investigated by using first order system dynamic model. Process dynamic parameters, time constant and process gain of the system, for change in drying temperature were determined. Secondly, investigation of the drying kinetics of the apple slab was carried out under constant external conditions in a semi-batch dryer. A mathematical model for diffusion mechanism assumed in one dimensional transient analysis of moisture distribution was solved by using explicit finite difference method of analysis. Thirdly, investigation of the drying kinetics of the apple slab was carried out under change in drying temperature at different time steps during drying. Inverse response system model was used for the representation of the dynamic behavior of drying. Process dynamic parameters, time constant and process gain of the system were determined. Model predicted results for apple slab drying under constant external condition and under step change in the drying temperature were compared with the experimental data.

The time required for inspection and maintenance of piping systems and vessels needs to be reduced to both minimize down time and decrease operational costs of petrochemical plants. Current ultrasonic inspection systems are not suited for on-line monitoring, with the main issues being the resistance of transducers and their coupling to high temperatures and the removal of insulation to access structures. The use of welded cylindrical rods is thus proposed, but measurements are threatened by "trailing echo" generation in waveguides. The introduction of a taper angle is investigated to attenuate these undesired echoes. While clean signals can always be obtained by increasing the taper angle in long rods, that is not always the case for short rods, which are considered here. In addition, temperature variations have a non-negligible impact on the arrival time of the backwall echo when performing measurements with a waveguide, and on-line compensation is essential. Since the interface echo between the rod and the pipe wall may be suppressed after the welding operation, a notch is machined at the end of the rod to create a reflected echo, which can be used for on-line compensation. Finally, the implementation of waveguides is considered for both pulse-echo and pitch-catch modes. In the pitch-catch mode, the backwall echo and the notch echo are received by different transducers and signals of interst are both first arrivals. As a result, trailing echoes do not impede measurements and their attenuation becomes unnecessary. In contrast, pulse-echo measurements are sensitive to trailing echoes and the waveguide's design plays an essential role in the feasibility of measurements. However, the environment also imposes a set of constraints on waveguide dimensions that complicates the implementation of pulse-echo measurements. Being more flexible, the pitch-catch configuration is chosen for final implementation. Experiments are performed to verify the concept feasibility, and the accuracy of measurements with thickness and temperature changes is also confirmed.

Computer simulations have emerged as a cost efficient complement to laboratory experiments, as computers have become increasingly powerful. The aim of the present work is to explore the ideas of some state of the art software development practices, and ways in which these can be useful for developing high performance research codes. The introduction of these practices, and the modular designs that they give rise to, raises issues regarding a potential conflict between runtime efficiency on one hand and development efficiency on the other. Flexible software modules, based on mathematical abstractions, will provide support for convenient implementation and modification of numerical operators. Questions still remain about whether such modules will provide the efficiency which is required for high performance applications. To answer these questions, investigations were performed within two different problem domains. The first domain consisted of modular frameworks for the numerical solution of Partial Differential Equations. Such frameworks proved a suitable setting, since several of my research questions revolved around the issue of modularity. The second problem domain was that of symmetry exploiting algorithms. These algorithms are based on group theory, and make ample use of mathematical abstractions from that field. The domain of symmetry exploiting algorithms gave us opportunities to investigate difficulties in combining modularity based on high level abstractions with low level optimizations using data layout and parallelization. In conclusion, my investigation of software development practices for the area of high performance computing has proved very fruitful indeed. I have found that none of the concerns that were raised should lead us to refrain from the use of the practices that I have considered. On the contrary, in the two case studies presented here, these practices lead to designs that perform well in terms of usability as well as runtime efficiency.

In this study, the development and testing of a finite-difference model for wind-induced flow in shallow lakes, and, in particular, a new technique for improving the land--water boundary representation, are documented. The model solves nonlinear, as well as linear, versions of the two-dimensional depth-integrated shallow water equations. Finite-difference methods on rectangular grids are widely used in numerical models of environmental flows. In these models, land--water boundaries are usually approximated by a series of perpendicular line segments, which enable the impermeability condition to be easily implemented. A disadvantage of this approach is that the actual boundary is often poorly approximated, particularly in regions which have complicated coastlines, and, as a result, currents in these regions cannot be accurately predicted. A technique for improving the land--water boundary representation in finite-difference models is introduced. This technique permits the model boundary to contain diagonal line segments, in addition to the vertical and horizontal line segments used in traditional models. The new technique is based on a simple concept and can easily be included in existing finite-difference models. In order to test the new method, the linearised shallow water equations are solved numerically for oscillatory wind-driven flow in lakes with simple geometry. Predictions obtained using the new approach are compared with predictions from the traditional stepped boundary and known analytic solutions. A significant improvement in the accuracy of results is noticed when the new approach is used, particularly in currents close to shore. The increased accuracy obtained using the improved boundary representation can lead to a significant computational saving, when compared with running the rectangular grid model with smaller grid spacings. A second-order analytic solution to the nonlinear shallow water equations is developed for oscillatory wind-driven flow in a rectangular lake. Comparisons between this solution and numerical results, obtained using the traditional stepped boundary and the improved boundary, verify the finite-difference formulae used in these models, including the approximations used for the cross-advective terms close to shore. Once more, currents are predicted with greater accuracy when the new technique for representing the land--water boundary is implemented. The lake circulation model is applied to the Lower Murray Lakes, South Australia, and predicted water levels at Tauwitchere Barrage are shown to agree very well with observations. The model is then used to examine the effectiveness of two schemes that have been proposed to increase wind-induced circulation, and therefore potentially decrease salinity, in Lake Albert, demonstrating the model's use as an efficient and effective tool for analysing flow behaviour in lakes. / Thesis (Ph.D.)--Mathematical Sciences (Applied Mathematics), 2005.

Neste trabalho é apresentado o desenvolvimento de programa para cálculo de campos eletromagnéticos baseado no método das Diferenças Finitas no Domínio do Tempo (FDTD). O programa é aplicado no cálculo da Taxa de Absorção Específica (“Specific Absortion Rate” – SAR) em modelos de cabeça de adulto e criança usuários de telefone celular. Para a modelagem da cabeça da criança é feita análise detalhada das principais diferenças das características de cabeças de adulto e criança. Também é desenvolvido um programa para o cálculo do aumento de temperatura relacionado com a Taxa de Absorção Especifica. Os resultados são comparados com dados de outros autores, bem como com as normas relativas a níveis de exposição a ondas eletromagnéticas. / The development of software to compute the electromagnetic fields employing the Finite Difference Time Domain (FDTD) method is shown in this work. This software is used to estimate the Specific Absorption Rate (SAR) in head models of child and adult users of cell phones. A detailed analysis of the main differences between child and adult heads is made in order to model the child head. Software to compute the temperature rise related with the calculated SAR is also developed. The results are compared with results obtained by others authors, as well as with electromagnetic field exposure guidelines.

We suppose that the price of a firm follows a nonlinear stochastic delay differential equation. We also assume that any claim whose value depends on firm value and time follows a nonlinear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a random partial differential equation (RPDE) satisfied by any corporate claim whose value is a function of firm value and time. Under specific final and boundary conditions, we solve the RPDE for the debt value and loan guarantees within a single period and homogeneous class of debt. We then analyze the risk structure of a levered firm. We also evaluate loan guarantees in the presence of more than one debt. Furthermore, we perform numerical simulations for specific companies and compare our results with existing models.

Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Optimal control of partial differential equations (PDEs) has tremendous applications in engineering and science, such as shape optimization, image processing, fluid dynamics, and chemical processes. In this thesis, we develop and analyze several efficient numerical methods for the optimal control problems governed by elliptic PDE, parabolic PDE, and wave PDE, respectively. The thesis consists of six chapters. In Chapter 1, we briefly introduce a few motivating applications and summarize some theoretical and computational foundations of our following developed approaches. In Chapter 2, we establish a new multigrid algorithm to accelerate the semi-smooth Newton method that is applied to the first-order necessary optimality system arising from semi-linear control-constrained elliptic optimal control problems. Under suitable assumptions, the discretized Jacobian matrix is proved to have a uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new strategy that leads to a robust multigrid solver is employed to define the coarse grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the popular full approximation storage (FAS) multigrid method. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter. In Chaper 3, we present a new second-order leapfrog finite difference scheme in time for solving the first-order necessary optimality system of the linear parabolic optimal control problems. The new leapfrog scheme is shown to be unconditionally stable and it provides a second-order accuracy, while the classical leapfrog scheme usually is well-known to be unstable. A mathematical proof for the convergence of the proposed scheme is provided under a suitable norm. Moreover, the proposed leapfrog scheme gives a favorable structure that leads to an effective implementation of a fast solver under the multigrid framework. Numerical examples show that the proposed scheme significantly outperforms the widely used second-order backward time differentiation approach, and the resultant fast solver demonstrates a mesh-independent convergence as well as a linear time complexity. In Chapter 4, we develop a new semi-smooth Newton multigrid algorithm for solving the discretized first-order necessary optimality system that characterizes the optimal solution of semi-linear parabolic PDE optimal control problems with control constraints. A new leapfrog discretization scheme in time associated with the standard five-point stencil in space is established to achieve a second-order accuracy. The convergence (or unconditional stability) of the proposed scheme is proved when time-periodic solutions are considered. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of an effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations as well as the optimal linear complexity of computing time. In Chapter 5, we offer a new implicit finite difference scheme in time for solving the first-order necessary optimality system arising in optimal control of wave equations. With a five-point central finite difference scheme in space, the full discretization is proved to be unconditionally convergent with a second-order accuracy, which is not restricted by the classical Courant-Friedrichs-Lewy (CFL) stability condition on the spatial and temporal step sizes. Moreover, based on its advantageous developed structure, an efficient preconditioned Krylov subspace method is provided and analyzed for solving the discretized sparse linear system. Numerical examples are presented to confirm our theoretical conclusions and demonstrate the promising performance of proposed preconditioned iterative solver. Finally, brief summaries and future research perspectives are given in Chapter 6.

Neste trabalho é apresentado o desenvolvimento de programa para cálculo de campos eletromagnéticos baseado no método das Diferenças Finitas no Domínio do Tempo (FDTD). O programa é aplicado no cálculo da Taxa de Absorção Específica (“Specific Absortion Rate” – SAR) em modelos de cabeça de adulto e criança usuários de telefone celular. Para a modelagem da cabeça da criança é feita análise detalhada das principais diferenças das características de cabeças de adulto e criança. Também é desenvolvido um programa para o cálculo do aumento de temperatura relacionado com a Taxa de Absorção Especifica. Os resultados são comparados com dados de outros autores, bem como com as normas relativas a níveis de exposição a ondas eletromagnéticas. / The development of software to compute the electromagnetic fields employing the Finite Difference Time Domain (FDTD) method is shown in this work. This software is used to estimate the Specific Absorption Rate (SAR) in head models of child and adult users of cell phones. A detailed analysis of the main differences between child and adult heads is made in order to model the child head. Software to compute the temperature rise related with the calculated SAR is also developed. The results are compared with results obtained by others authors, as well as with electromagnetic field exposure guidelines.

Neste trabalho é apresentado o desenvolvimento de programa para cálculo de campos eletromagnéticos baseado no método das Diferenças Finitas no Domínio do Tempo (FDTD). O programa é aplicado no cálculo da Taxa de Absorção Específica (“Specific Absortion Rate” – SAR) em modelos de cabeça de adulto e criança usuários de telefone celular. Para a modelagem da cabeça da criança é feita análise detalhada das principais diferenças das características de cabeças de adulto e criança. Também é desenvolvido um programa para o cálculo do aumento de temperatura relacionado com a Taxa de Absorção Especifica. Os resultados são comparados com dados de outros autores, bem como com as normas relativas a níveis de exposição a ondas eletromagnéticas. / The development of software to compute the electromagnetic fields employing the Finite Difference Time Domain (FDTD) method is shown in this work. This software is used to estimate the Specific Absorption Rate (SAR) in head models of child and adult users of cell phones. A detailed analysis of the main differences between child and adult heads is made in order to model the child head. Software to compute the temperature rise related with the calculated SAR is also developed. The results are compared with results obtained by others authors, as well as with electromagnetic field exposure guidelines.