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

This thesis focuses on the incorporation of carbon nanoparticles within continuous fibre reinforcements by liquid composite moulding processes, in order to provide enhanced electrical and delamination properties to the multiscale composites. The mechanisms controlling the flow and filtration of these nanoparticles during liquid composite moulding are studied, in order to develop a predictive 1-D model which allows design of the processing of these composite materials. Five different carbon nanoparticles at 0.25 wt% loading, three unmodified and one surface modified carbon nanotube systems and one carbon nanofibre system, were utilised to modify a commercial two-component epoxy resin utilised to impregnate carbon and glass reinforcements at high fibre volume fraction by resin transfer moulding. The dispersion of the nanofillers in the prepolymer was carried out by ultrasonication, high shear mixing or triple roll milling or a combination of the three. Electrical conductivity measurements of the carbon nanoparticle liquid suspensions during dispersion, alongside optical microscopy imaging and rheological analysis of these allowed the selection of the concentration of nanofiller and the appropriate dispersion technique for each nanoparticle system. The resin transfer moulding process required adaptation to incorporate the dispersion and modify degassing steps, especially when utilising unmodified carbon nanoparticles suspensions, due to their higher viscosity and tendency to be filtered. Nanoparticle filtration was identified by electrical conductivity measurements and microscopy of specimens cut at increasing distances from the inlet. Cake filtration was observed for some of the unmodified systems, whereas deep bed filtration occurred for the surface modified CNT material. Property graded composites were obtained due to filtration, where the average electrical conductivity of the carbon and glass composites produced increased by a factor of two or one order of magnitude respectively. The effect of filler on the delamination properties of the carbon fibre composites was tested under mode I. The results do not show a statistically significant improvement of delamination resistance with the presence of nanoparticles, although localised toughening mechanisms such as nanoparticle pull-out and crack bridging as well as inelastic deformation have been observed on fracture surfaces. Particle filtration and gradients in concentration resulted in non-linear flow behaviour. An 1-D analytical and a finite difference model, based on Darcy’s law accompanied by particle mass conservation and filtration kinetics were developed to describe the flow and filtration of carbon nanoparticle filled thermosets. The numerical model describes the non-linear problem by incorporating material property update laws, i.e. permeability, porosity and viscosity variations on concentration of retained and suspended particles with location and time. The finite difference model is consistent and converges to the analytical solution. The range of applicability of the analytical model is limited to lower filtration coefficients and shorter filling lengths, providing an approximate solution for through thickness infusion; whereas the numerical model presents a solution outside this range, i.e. in-plane filling processes. These models allow process design, with specified carbon nanoparticle concentration distributions achieved via modifying the nanofiller loading at the inlet as a function of time.

by Ng Chi-Keung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 127-134). / Abstract also in Chinese. / Abstract --- p.2 / 摘要 --- p.3 / Acknowledgements --- p.4 / Chapter Chapter 1 --- Introduction --- p.7 / Chapter Chapter 2 --- Packaging Effects of Integrated Circuits --- p.9 / Chapter 2.1 --- The Structure of the IC Package --- p.9 / Chapter 2.2 --- Microstrip Discontinuities --- p.11 / Chapter Chapter 3 --- The Finite-Difference Time-Domain Method --- p.19 / Chapter 3.1 --- Basic Theory --- p.19 / Chapter 3.2 --- Stability Criterion --- p.25 / Chapter 3.3 --- Formulation of Source --- p.30 / Chapter A. --- Source Function --- p.30 / Chapter (i) --- Sinusoidal --- p.30 / Chapter (ii) --- Gaussian Pulse --- p.31 / Chapter B. --- Source Realization --- p.36 / Chapter (i) --- Electric Field Source --- p.36 / Chapter (ii) --- Lumped Source --- p.38 / Chapter (iii) --- Current Source --- p.40 / Chapter C. --- Source Placement --- p.41 / Chapter 3.4 --- Parameter Extraction --- p.42 / Chapter A. --- Voltage and Current --- p.42 / Chapter B. --- Characteristic Impedance --- p.44 / Chapter C. --- Effective Dielectric Constant --- p.45 / Chapter D. --- Scattering Parameters --- p.46 / Chapter 3.5 --- Termination and Boundary Treatment --- p.48 / Chapter A. --- Perfect Electric Conductor (PEC) --- p.48 / Chapter B. --- Perfect Magnetic Conductor (PMC) --- p.49 / Chapter C. --- Interface between Two Materials --- p.50 / Chapter 3.6 --- Perfectly Matched Layer (PML) --- p.54 / Chapter A. --- Theory of PML in Three Dimensions --- p.56 / Chapter B. --- Incorporation of PML as Absorbing Boundary Condition (ABC) --- p.65 / Chapter C. --- Discretization of Maxwell's Equations in PML --- p.73 / Chapter 3.7 --- Flowcharts --- p.75 / Chapter A. --- Free Space Radiation by a Dipole Antenna --- p.77 / Chapter B. --- Parameters of a Microstrip Line --- p.79 / Chapter C. --- Scattering Parameters of Planar Network --- p.85 / Chapter 3.8 --- Summary --- p.87 / Chapter Chapter 4 --- Effects of Ground Via Allocation --- p.88 / Chapter 4.1 --- Introduction --- p.88 / Chapter 4.2 --- Simulation and Experimental Results --- p.91 / Chapter 4.3 --- Equivalent Circuit Modelling --- p.108 / Chapter 4.4 --- Summary --- p.124 / Chapter Chapter 5 --- Conclusions --- p.125 / Chapter Chapter 6 --- Recommendation for Future Work --- p.126 / References --- p.127 / Publication --- p.134

In this project,European Call and Put options,and also American Call and Put options have been priced by some finite difference methods using the C++ programming language.The report describes the following:The theory behind the pricing of options,some pricing methods,and how some finite difference pricing methods have been implemented in C++.

O presente trabalho aborda um problema inverso associado a difus~ao de calor em uma barra unidimensional. Esse fen^omeno e modelado por meio da equac~ao diferencial par- cial parabolica ut = uxx, conhecida como equac~ao de difus~ao do calor. O problema classico (problema direto) envolve essa equac~ao e um conjunto de restric~oes { as condic~oes inicial e de contorno {, o que permite garantir a exist^encia de uma soluc~ao unica. No problema inverso que estudamos, o valor da temperatura em um dos extremos da barra n~ao esta disponvel. Entretanto, conhecemos o valor da temperatura em um ponto x0 xo no interior da barra. Para aproximar o valor da temperatura no intervalo a direita de x0, propomos e testamos tr^es algoritmos de diferencas nitas: diferencas regressivas, leap-frog e diferencas regressivas maquiadas. / This work deals with an inverse problem for the heat diusion in a bar of size L. This one-dimensional phenomenum is modeled by the parabolic partial dierential equation ut = uxx, known as the heat diusion equation. The classic problem (Direct Problem) involves this equation coupled to a set of constraints { initial and boundary conditions { in such a way as to guarantee a unique solution for it. The inverse problem hereby considered may be described in the following way: at one bar extreme point the temperature is un- known, but it is given at a xed interior point for all time. Three nite dierence algorithms (backward dierences, leap-frog, disguised backward dierences) are proposed and tested to approximate solutions for this problem. Keywords: Diusion equation. Finite dierences. Inverse problem.

No description available.

Bibliography : leaves 211-216 / 216 leaves : ill ; 20 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Applied Mathematics, 1994?

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 161-164).

This thesis is concerned with the development of numerical methods using finite difference techniques for the discretization of initial value problems (IVPs) and initial boundary value problems (IBVPs) of certain hyperbolic systems which are first order in time and second order in space. This type of system appears in some formulations of Einstein equations, such as ADM, BSSN, NOR, and the generalized harmonic formulation. For IVP, the stability method proposed in [14] is extended from second and fourth order centered schemes, to 2n-order accuracy, including also the case when some first order derivatives are approximated with off-centered finite difference operators (FDO) and dissipation is added to the right-hand sides of the equations. For the model problem of the wave equation, special attention is paid to the analysis of Courant limits and numerical speeds. Although off-centered FDOs have larger truncation errors than centered FDOs, it is shown that in certain situations, off-centering by just one point can be beneficial for the overall accuracy of the numerical scheme. The wave equation is also analyzed in respect to its initial boundary value problem. All three types of boundaries - outflow, inflow and completely inflow that can appear in this case, are investigated. Using the ghost-point method, 2n-accurate (n = 1, 4) numerical prescriptions are prescribed for each type of boundary. The inflow boundary is also approached using the SAT-SBP method. In the end of the thesis, a 1-D variant of BSSN formulation is derived and some of its IBVPs are considered. The boundary procedures, based on the ghost-point method, are intended to preserve the interior 2n-accuracy. Numerical tests show that this is the case if sufficient dissipation is added to the rhs of the equations. / Diese Doktorarbeit beschäftigt sich mit der Entwicklung numerischer Verfahren für die Diskretisierung des Anfangswertproblems und des Anfangs-Randwertproblems unter Einsatz von finite-Differenzen-Techniken für bestimmte hyperbolischer Systeme erster Ordnung in der Zeit und zweiter Ordnung im Raum. Diese Art von Systemen erscheinen in einigen Formulierungen der Einstein'schen-Feldgleichungen, wie zB. den ADM, BSSN oder NOR Formulierungen, oder der sogenanten verallgemeinerten harmonischen Darstellung. Im Hinblick auf das Anfangswertproblem untersuche ich zunächst tiefgehend die mathematischen Eigenschaften von finite-Differenzen-Operatoren (FDO) erster und zweiter Ordnung mit 2n-facher Genaugigkeit. Anschließend erweitere ich eine in der Literatur beschriebene Methode zur Stabilitätsanalyse für Systeme mit zentrierten FDOs in zweiter und vierter Genauigkeitsordung auf Systeme mit gemischten zentrierten und nicht zentrierten Ableitungsoperatoren 2n-facher Genauigkeit, eingeschlossen zusätzlicher Dämpfungsterme, wie sie bei numerischen Simulationen der allgemeinen Relativitätstheorie üblich sind. Bei der Untersuchung der einfachen Wellengleichung als Fallbeispiel wird besonderes Augenmerk auf die Analyse der Courant-Grenzen und numerischen Geschwindigkeiten gelegt. Obwohl unzentrierte, diskrete Ableitungsoperatoren größere Diskretisierungs-Fehler besitzen als zentrierte Ableitungsoperatoren, wird gezeigt, daß man in bestimmten Situationen eine Dezentrierung des numerischen Moleküls von nur einem Punkt bezüglich des zentrierten FDO eine höhere Genauigkeit des numerischen Systems erzielen kann. Die Wellen-Gleichung in einer Dimension wurde ebenfalls im Hinblick auf das Anfangswertproblem untersucht. In Abhängigkeit des Wertes des sogenannten Shift-Vektors, müssen entweder zwei (vollständig eingehende Welle), eine (eingehende Welle) oder keine Randbedingung (ausgehende Welle) definiert werden. In dieser Arbeit wurden alle drei Fälle mit Hilfe der 'Ghost-point-methode' numerisch simuliert und untersucht, und zwar auf eine Weise, daß alle diese Algorithmen stabil sind und eine 2n-Genauigkeit besitzen. In der 'ghost-point-methode' werden die Evolutionsgleichungen bis zum letzen Punkt im Gitter diskretisiert unter Verwendung von zentrierten FDOs und die zusätzlichen Punkte die am Rand benötigt werden ('Ghost-points') werden unter Benutzung von Randwertbedingungen und Extrapolationen abgeschätzt. Für den Zufluß-Randwert wurde zusätzlich noch eine andere Implementierung entwickelt, welche auf der sogenannten SBP-SAT (Summation by parts-simulatanous approximation term) basiert. In dieser Methode werden die diskreten Ableitungen durch Operatoren angenähert, welche die 'Summation-by-parts' Regeln erfüllen. Die Randwertbedingungen selber werden in zusätzlichen Termen integriert, welche zu den Evolutionsgleichnungen der Punkte nahe des Randes hinzuaddiert werden und zwar auf eine Weise, daß die 'summation-by-parts' Eigenschaften erhalten bleiben. Am Ende dieser Arbeit wurde noch eine eindimensionale (kugelsymmetrische) Version der BSSN Formulierung abgeleitet und einige physikalisch relevanten Anfangs-Randwertprobleme werden diskutiert. Die Randwert-Algorithmen, welche für diesen Fall ausgearbeitet wurden, basieren auf der 'Ghost-point-Methode' and erfüllen die innere 2n-Genauigkeit solange genügend Reibung in den Gleichungen zugefügt wird.

A general finite difference discretization of the time-dependent radiation transport equation is developed around the framework of an existing steady-state three dimensional radiation transport solver based on the slice-balance approach. Three related algorithms are outlined within the general finite difference scheme: an explicit, an implicit, and a semi-implicit approach. The three algorithms are analyzed with respect to the discretizations of each element of the phase space in the transport solver. The explicit method, despite its small computational cost per time step, is found to be unsuitable for many purposes due to its inability to accurately handle rapidly varying solutions. The semi-implicit method is shown to produce results nearly as reliable as the fully implicit solver, while requiring significantly less computational effort.