Dissertations / Theses on the topic 'Finite difference method of analysis'

Thesis (MSc)--Stellenbosch University, 2011.
ENGLISH ABSTRACT: FLAC3D is a three-dimensional explicit nite difference program for solving a variety of solid mechanics problems, both linear and non-linear. The development of the algorithm and its initial implementation were performed by Itasca Consulting Group Inc. The main idea of the algorithm is to discritise the domain of interest into a Lagrangian grid where each cell represents an element of the material. Each cell can then deform according to a prescribed stress/strain law together with the equations of motion. An in-depth study of the algorithm was performed and implemented in Java. During the implementation, it was observed that the type of boundary conditions typically used has a major in uence on the accuracy of the results, especially when boundaries are close to regions with large stress variations, such as in mining excavations. To improve the accuracy of the algorithm, a new type of boundary condition was developed where the FLAC3D domain is embedded in a linear elastic material, named the Boundary Node Shell (BNS). Using the BNS shows a signi cant improvement in results close to excavations. The FLAC algorithm is also quite amendable to paralellization and a multi-threaded version that makes use of multiple Central Processing Unit (CPU) cores was developed to optimize the speed of the algorithm. The nal outcome is new non-commercial Java source code (JFLAC) which includes the Boundary Node Shell (BNS) and shared memory parallelism over and above the basic FLAC3D algorithm.
AFRIKAANSE OPSOMMING: FLAC3D is 'n eksplisiete eindige verskil program wat 'n verskeidenheid liniêre en nieliniêre soliede meganika probleme kan oplos. Die oorspronklike algoritme en die implimentasies daarvan was deur Itasca Consulting Group Inc. toegepas. Die hoo dee van die algoritme is om 'n gebied te diskritiseer deur gebruik te maak van 'n Lagrangese rooster, waar elke sel van die rooster 'n element van die rooster materiaal beskryf. Elke sel kan dan vervorm volgens 'n sekere spannings/vervormings wet. 'n Indiepte ondersoek van die algoritme was uitgevoer en in Java geïmplimenteer. Tydens die implementering was dit waargeneem dat die grense van die rooster 'n groot invloed het op die akkuraatheid van die resultate. Dit het veral voorgekom in areas waar stress konsentrasies hoog is, gewoonlik naby areas waar myn uitgrawings gemaak is. Dit het die ontwikkelling van 'n nuwe tipe rand kondisie tot gevolg gehad, sodat die akkuraatheid van die resultate kon verbeter. Die nuwe rand kondisie, genaamd die Grens Node Omhulsel (GNO), aanvaar dat die gebied omring is deur 'n elastiese materiaal, wat veroorsaak dat die grense van die gebied 'n elastiese reaksie het op die stress binne die gebied. Die GNO het 'n aansienlike verbetering in die resultate getoon, veral in areas naby myn uitgrawings. Daar was ook waargeneem dat die FLAC algoritme parralleliseerbaar is en het gelei tot die implentering van 'n multi-SVE weergawe van die sagteware om die spoed van die algoritme te optimeer. Die nale uitkomste is 'n nuwe nie-kommersiële Java weergawe van die algoritme (JFLAC), wat die implimentering van die nuwe GNO randwaardekondisie insluit, asook toelaat vir die gebruik van multi- Sentrale Verwerkings Eenheid (SVE) as 'n verbetering op die basiese FLAC3D algoritme.

We have constructed a new finite-difference time-domain (FDTD) method in this project. Our new algorithm focuses on the most important and more challenging transverse electric (TE) case. In this case, the electric field is discontinuous across the interface between different dielectric media. We use an electric permittivity that stays as a constant in each medium, and magnetic permittivity that is constant in the whole domain. To handle the interface between different media, we introduce new effective permittivities that incorporates electromagnetic fields boundary conditions. That is, across the interface between two different media, the tangential component, Er(x,y), of the electric field and the normal component, Dn(x,y), of the electric displacement are continuous. Meanwhile, the magnetic field, H(x,y), stays as continuous in the whole domain. Our new algorithm is built based upon the integral version of the Maxwell's equations as well as the above continuity conditions. The theoretical analysis shows that the new algorithm can reach second-order convergence O(∆x2)with mesh size ∆x. The subsequent numerical results demonstrate this algorithm is very stable and its convergence order can reach very close to second order, considering accumulation of some unexpected numerical approximation and truncation errors. In fact, our algorithm has clearly demonstrated significant improvement over all related FDTD methods using effective permittivities reported in the literature. Therefore, our new algorithm turns out to be the most effective and stable FDTD method to solve Maxwell's equations involving multiple media.

The objective of this thesis was to analyze a Coplanar Waveguide (CPW)-fed folded slot antenna using the Finite Difference Time Domain (FDTD) method. Important parameters such as S-parameters and the input impedance of the antenna were simulated using XFDTD software and were analyzed. An important goal of this thesis was to provide design information about the folded slot antenna. For this purpose the effects of antenna layout on the resonant frequency and the bandwidth of the antenna were investigated. First the effect of adding more number of slots to the basic CPW-fed folded slot antenna geometry on the S-parameters, the input impedance and the radiation patterns of the antenna were studied. Next the width of the slot was varied and the effect of changing this design parameter of the antenna was analyzed. Finally the slot separation was varied and its effect on the antenna parameters is studied. This work concluded that, by including additional slots, the input impedance of the antenna can be controlled over a wide range.

The Flux Jacobian matrices, the elements of which are the derivatives of the flux vectors with respect to the flow variables, are needed to be evaluated in implicit flow solutions and in analytical sensitivity analyzing methods. The main motivation behind this thesis study is to explore the accuracy of the numerically evaluated flux Jacobian matrices and the effects of the errors in those matrices on the convergence of the flow solver, on the accuracy of the sensitivities and on the performance of the design optimization cycle. To perform these objectives a flow solver, which uses exact Newton&rsquo
s method with direct sparse matrix solution technique, is developed for the Euler flow equations. Flux Jacobian is evaluated both numerically and analytically for different upwind flux discretization schemes with second order MUSCL face interpolation. Numerical flux Jacobian matrices that are derived with wide range of finite difference perturbation magnitudes were compared with analytically derived ones and the optimum perturbation magnitude, which minimizes the error in the numerical evaluation, is searched. The factors that impede the accuracy are analyzed and a simple formulation for optimum perturbation magnitude is derived. The sensitivity derivatives are evaluated by direct-differentiation method with discrete approach. The reuse of the LU factors of the flux Jacobian that are evaluated in the flow solution enabled efficient sensitivity analysis. The sensitivities calculated by the analytical Jacobian are compared with the ones that are calculated by numerically evaluated Jacobian matrices. Both internal and external flow problems with varying flow speeds, varying grid types and sizes are solved with different discretization schemes. In these problems, when the optimum perturbation magnitude is used for numerical Jacobian evaluation, the errors in Jacobian matrix and the sensitivities are minimized. Finally, the effect of the accuracy of the sensitivities on the design optimization cycle is analyzed for an inverse airfoil design performed with least squares minimization.

This paper provides an overview of the finite difference method and its application to approximating financial partial differential equations (PDEs) in incomplete markets. In particular, we study German’s [6] stochastic volatility PDE derived from indifference pricing. In [6], it is shown that the first order- correction to derivatives valued by indifference pricing can be computed as a function involving the stochastic volatility PDE itself. In this paper, we present three explicit finite difference models to approximate the stochastic volatility PDE and compare the resulting valuations to those generated by an Euler- Maruyama Monte Carlo pricing algorithm. We also discuss the significance of boundary condition choice for explicit finite difference models.

[EN] The present PhD thesis is focused on numerical analysis and computing of finite difference schemes for several relevant option pricing models that generalize the Black-Scholes model. A careful analysis of desirable properties for the numerical solutions of option pricing models as the positivity, stability and consistency, is provided. In order to handle the free boundary that arises in American option pricing problems, various transformation techniques based on front-fixing method are applied and studied. Special attention is paid to multi-asset option pricing, such as exchange or spread option. Appropriate transformation allows eliminating of the cross derivative term. Transformation techniques of partial differential equations to remove convection and reaction terms are studied in order to simplify the models and avoid possible troubles of stability. This thesis consists of six chapters. The first chapter is an introduction containing definitions of option and related terms and derivation of the Black-Scholes equation as well as general aspects of theory of finite difference schemes, including preliminaries on numerical analysis. Chapter 2 is devoted to solve linear Black-Scholes model for American put and call options. A Landau transformation and a new front-fixing transformation are applied to the free boundary value problem. It leads to non-linear partial differential equation (PDE) in a fixed domain. Stable and consistent explicit numerical schemes are proposed preserving positivity and monotonicity of the solution in accordance with the behaviour of the exact solution. Efficiency of the front-fixing method demonstrated in Chapter 2 has motivated us to apply the method to some more complicated nonlinear models. A new change of variables resulting in a time dependent boundary instead of fixed one, is applied to nonlinear Black-Scholes model for American options, such as Barles and Soner and Risk Adjusted Pricing models. Chapter 4 provides a new alternative approach for solving American option pricing problem based on rationality of investor. There exists an intensity function that can be reduced in the simplest case to penalty approach. Chapter 5 deals with multi-asset option pricing. Appropriate transformation allows eliminating of the cross derivative term avoiding computational drawbacks and possible troubles of stability. Concluding remarks are given in Chapter 6. All the considered models and numerical methods are accompanied by several examples and simulations. The convergence rate is computed confirming the theoretical study of consistency. Stability conditions are tested by numerical examples. Results are compared with known relevant methods in the literature showing efficiency of the proposed methods.
[ES] La presente tesis doctoral se centra en la construcción de esquemas en diferencias finitas y el análisis numérico de relevantes modelos de valoración de opciones que generalizan el modelo de Black-Scholes. Se proporciona un análisis cuidadoso de las propiedades de las soluciones numéricas tales como la positividad, la estabilidad y la consistencia. Con el fin de manejar la frontera libre que surge en los problemas de valoración de opciones Americanas, se aplican y se estudian diversas técnicas de transformación basadas en el método de fijación de las fronteras (front-fixing). Se presta especial atención a la valoración de opciones de múltiples activos, como son las opciones ''exchange'' y ''spread''. Esta tesis se compone de seis capítulos. El primer capítulo es una introducción que contiene las definiciones de opción y términos relacionados y la derivación de la ecuación de Black-Scholes, así como aspectos generales de la teoría de los esquemas en diferencias finitas, incluyendo preliminares de análisis numérico. El capítulo 2 está dedicado a resolver el modelo lineal de Black-Scholes para opciones Americanas put y call. Para fijar las fronteras del problema de frontera libre se aplican transformaciones como la de Landau y un nuevo cambio de variable propuesto. La eficiencia del método front-fixing mostrada en el capítulo 2 ha motivado el estudio de su aplicación a algunos modelos no lineales más complicados. En particular, se propone un cambio de variables que lleva a una nueva frontera dependiente del tiempo en lugar de una fija. Este cambio se aplica a modelos no lineales de Black-Scholes para opciones Americanas, como son el de Barles y Soner y el modelo RAPM (Risk Adjusted Pricing Methodology). El capítulo 4 ofrece una nueva técnica para la resolución de problemas de valoración de opciones Americanas basada en la racionalidad de los inversores. Aparece una función de la intensidad que se puede reducir en el caso más simple a la técnica de penalización (penalty method). Este enfoque tiene en cuenta el posible comportamiento irracional de los inversores. En la sección 4.2 se aplica esta técnica al modelo de cambio de regímenes lo que lleva a un nuevo modelo que tiene en cuenta el posible ejercicio irracional, así como varios estados del mercado. El enfoque del parámetro de racionalidad junto con una transformación logarítmica permiten construir un esquema numérico eficiente sin aplicar el método front-fixing o la conocida formulación de LCP (Linear Complementarity Problem). El capítulo 5 se dedica a la valoración de opciones de activos múltiples. Una transformación apropiada permite la eliminación del término de derivadas cruzadas evitando inconvenientes computacionales y posibles problemas de estabilidad. Las conclusiones se muestran en el capítulo 6. Se pone en relieve varios aspectos de la presente tesis. Todos los modelos considerados y los métodos numéricos van acompañados de varios ejemplos y simulaciones. Se estudia la convergencia numérica que confirma el estudio teórico de la consistencia. Las condiciones de estabilidad son corroboradas con ejemplos numéricos. Los resultados se comparan con métodos relevantes de la bibliografía mostrando la eficiencia de los métodos propuestos.
[CAT] La present tesi doctoral se centra en la construcció d'esquemes en diferències finites i l'anàlisi numèrica de rellevants models de valoració d'opcions que generalitzen el model de Black-Scholes. Es proporciona una anàlisi cuidadosa de les propietats de les solucions numèri-ques com ara la positivitat, l'estabilitat i la consistència. A fi de manejar la frontera lliure que sorgix en els problemes de valoració d'opcions Americanes, s'apliquen i s'estudien diverses tècniques de transformació basades en el mètode de fixació de les fronteres (front-fixing). Es presta especial atenció a la valoració d'opcions de múltiples actius, com són les opcions ''exchange'' i ''spread''. Esta tesi es compon de sis capítols. El primer capítol és una introducció que conté les definicions d'opció i termes relacionats i la derivació de l'equació de Black-Scholes, així com aspectes generals de la teoria dels esquemes en diferències finites, incloent aspectes preliminars d'anàlisi numèrica. El 2n capítol està dedicat a resoldre el model lineal de Black-Scholes per a opcions Americanes ''put'' i ''call''. Per a fixar les fronteres del problema de frontera lliure s'apliquen transformacions com la de Landau i s'ha proposat un nou canvi de variable proposat. Açò porta a una equació diferencial en derivades parcials no lineal en un domini fix. L'eficiència del mètode front-fixing mostrada en el 2n capítol ha motivat l'estudi de la seua aplicació a alguns models no lineals més complicats. En particular, es proposa un canvi de variables que porta a una nova frontera dependent del temps en compte d'una fixa. Este canvi s'aplica a models no lineals de Black-Scholes per a opcions Americanes, com són el de Barles i Soner i el model RAPM (Risk Adjusted Pricing Methodology). El 4t capítol oferix una nova tècnica per a la resolució de problemes de valoració d'opcions Americanes basada en la racionalitat dels inversors. Apareix una funció de la intensitat que es pot reduir en el cas més simple a la tècnica de penalització (penal method) . Este enfocament té en compte el possible comportament irracional dels inversors. En la secció 4.2 s'aplica esta tècnica al model de canvi de règims el que porta a un nou model que té en compte el possible exercici irracional, així com diversos estats del mercat. L'enfocament del paràmetre de racionalitat junt amb una transformació logarítmica permeten construir un esquema numèric eficient sense aplicar el mètode front-fixing o la coneguda formulació de LCP (Linear Complementarity Problem). El 5é capítol es dedica a la valoració d'opcions d'actius múltiples. Una transformació apropiada permet l'eliminació del terme de derivades mixtes evitant inconvenients computacionals i possibles problemes d' estabilitat. Les conclusions es mostren al 6é capítol. Es posa en relleu diversos aspectes de la present tesi. Tots els models considerats i els mètodes numèrics van acompanyats de diversos exemples i simulacions. S'estu-dia la convergència numèrica que confirma l'estudi teòric de la consistència. Les condicions d'estabilitat són corroborades amb exemples numèrics. Els resultats es comparen amb mètodes rellevants de la bibliografia mostrant l'eficiència dels mètodes proposats.
Egorova, V. (2016). Finite Difference Methods for nonlinear American Option Pricing models: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/68501
TESIS
Premiado

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.

Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method. The finite difference method is conceptually simple and easy to implement, but has difficulties in handling complex geometries of the computational domain. We construct high order finite difference methods for wave propagation in heterogeneous media with complex geometries. In addition, we derive error estimates to a class of finite difference operators applied to the acoustic wave equation. The discontinuous Galerkin method is flexible with complex geometries. Moreover, the discontinuous nature between elements makes the method suitable for multiphysics problems. We use an energy based discontinuous Galerkin method to solve a coupled acoustic-elastic problem.

Computer simulation of electromagnetic behaviour of a device is a common practice in modern engineering. Maxwell's equations are solved on a computer with help of numerical methods. Contemporary devices constantly grow in size and complexity. Therefore, new numerical methods should be highly efficient. Many industrial and research applications of numerical methods need to account for the frequency dependent materials. The Finite-Difference Time-Domain (FDTD) method is one of the most widely adopted algorithms for the numerical solution of Maxwell's equations. A major drawback of the FDTD method is the interdependence of the spatial and temporal discretisation steps, known as the Courant-Friedrichs-Lewy (CFL) stability criterion. Due to the CFL condition the simulation of a large object with delicate geometry will require a high spatio-temporal resolution everywhere in the FDTD grid. Application of subgridding increases the efficiency of the FDTD method. Subgridding decomposes the simulation domain into several subdomains with different spatio-temporal resolutions. The research project described in this dissertation uses the Huygens Subgridding (HSG) method. The frequency dependence is included with the Auxiliary Differential Equation (ADE) approach based on the one-pole Debye relaxation model. The main contributions of this work are (i) extension of the one-dimensional (1D) frequency-dependent HSG method to three dimensions (3D), (ii) implementation of the frequency-dependent HSG method, termed the dispersive HSG, in Fortran 90, (iii) implementation of the radio environment setting from the PGM-files, (iv) simulation of the electromagnetic wave propagating from the defibrillator through the human torso and (v) analysis of the computational requirements of the dispersive HSG program.

The fractional Laplacian is a promising mathematical tool due to its ability to capture the anomalous diffusion and model the complex physical phenomenon with long-range interaction, such as fractional quantum mechanics, image processing, jump process, etc. One of the important applications of fractional Laplacian is a turbulence intermittency model of fractional Navier-Stokes equation which is derived from Boltzmann's theory. However, the efficient computation of this model on bounded domains is challenging as highly accurate and efficient numerical methods are not yet available. The bottleneck for efficient computation lies in the low accuracy and high computational cost of discretizing the fractional Laplacian operator. Although many state-of-the-art numerical methods have been proposed and some progress has been made for the existing numerical methods to achieve quasi-optimal complexity, some issues are still fully unresolved: i) Due to nonlocal nature of the fractional Laplacian, the implementation of the algorithm is still complicated and the computational cost for preparation of algorithms is still high, e.g., as pointed out by Acosta et al \cite{AcostaBB17} 'Over 99\% of the CPU time is devoted to assembly routine' for finite element method; ii) Due to the intrinsic singularity of the fractional Laplacian, the convergence orders in the literature are still unsatisfactory for many applications including turbulence intermittency simulations. To reduce the complexity and computational cost, we consider two numerical methods, finite difference and spectral method with quasi-linear complexity, which are summarized as follows. We develop spectral Galerkin methods to accurately solve the fractional advection-diffusion-reaction equations and apply the method to fractional Navier-Stokes equations. In spectral methods on a ball, the evaluation of fractional Laplacian operator can be straightforward thanks to the pseudo-eigen relation. For general smooth computational domains, we propose the use of spectral methods enriched by singular functions which characterize the inherent boundary singularity of the fractional Laplacian. We develop a simple and easy-to-implement fractional centered difference approximation to the fractional Laplacian on a uniform mesh using generating functions. The weights or coefficients of the fractional centered formula can be readily computed using the fast Fourier transform. Together with singularity subtraction, we propose high-order finite difference methods without any graded mesh. With the use of the presented results, it may be possible to solve fractional Navier-Stokes equations, fractional quantum Schrodinger equations, and stochastic fractional equations with high accuracy. All numerical simulations will be accompanied by stability and convergence analysis.

The objective of this thesis was to analyze a Coaxial-Fed U-slot Rectangular Patch Antenna using Finite Difference Time Domain (FDTD) method. Important parameters such as S-parameters and Input Impedance of the antenna were simulated using XFDTD software and were analyzed. An important goal of this thesis was to provide design information about the U-slot antenna. For this purpose the effects of antenna layout on the resonant frequency and the bandwidth of the antenna were investigated. First the effect of varying slot width on the S-parameters, Voltage Standing Wave Ratio (VSWR) and Input Impedance were studied. Next the length of the slot was varied and effect of changing this design parameter of the antenna was analyzed. Finally the substrate thickness was varied and its effect on the antenna parameters is studied. This work concluded that by varying antenna dimensions such as slot width, slot length and substrate thickness, higher bandwidth could be achieved with required impedance matching.

The knowledge of the behavior of pager antennas and the magnetic field within 2 cm of lossy dielectric cylinders is important in applications involving terrestrial and satellite communications. The distribution of the magnetic fields close to the lossy dielectric cylinders affects the performance of these personal communication devices. By analyzing the magnetic fields close to these dielectric cylinders, information on the magnetic field effects may be obtained that may allow us to improve on the design and performance of these close-to-body communication devices. Finite-Difference Time-Domain (FD-TD) simulations were applied to lossy dielectric cylinders, “SALTY” and SALTY-LITE”, and to a microstrip antenna in the presence of these structures. The results obtained were compared to measurements. The frequency-dependent scattering parameters, the inputimpedance, and the radiation pattern of this antenna are calculated and validated. The results clearly indicate that FD-TD is an efficient and accurate tool to model and analyze complex electromagnetic structures with the unique ability to display the transient response of the electromagnetic fields in time domain.

Financial engineering problems are of great importance in the academic community and BlackScholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks and derivatives can be evaluated using this model. Option evaluation, is extremely important to trade in the stocks. The numerical solutions of the Black-Scholes equation are used to simulate these options. In this thesis, the explicit and the implicit Euler methods are used for the approximation of Black-scholes partial differential equation and a second order finite difference scheme is used for the spatial derivatives. These temporal and spatial discretizations are used to gain an insight about the stability properties of the explicit and the implicit methods in general. The numerical results show that the explicit methods have some constraints on the stability, whereas, the implicit Euler method is unconditionally stable. It is also demostrated that both the explicit and the implicit Euler methods are only first order convergent in time and this implies too small step-sizes to achieve a good accuracy.

Induced seismicity (earthquakes caused by injection or extraction of fluids in Earth's subsurface) is a major, new hazard in the United States, the Netherlands, and other countries, with vast economic consequences if not properly managed. Addressing this problem requires development of predictive simulations of how fluid-saturated solids containing frictional faults respond to fluid injection/extraction. Here we present a numerical method for linear poroelasticity with rate-and-state friction faults. A numerical method for approximating the fully coupled linear poroelastic equations is derived using the summation-by-parts-simultaneous-approximation-term (SBP-SAT) framework. Well-posedness is shown for a set of physical boundary conditions in 1D and in 2D. The SBP-SAT technique is used to discretize the governing equations and show semi-discrete stability and the correctness of the implementation is verified by rigorous convergence tests using the method of manufactured solutions, which shows that the expected convergence rates are obtained for a problem with spatially variable material parameters. Mandel's problem and a line source problem are studied, where simulation results and convergence studies show satisfactory numerical properties. Furthermore, two problem setups involving fault dynamics and slip on faults triggered by fluid injection are studied, where the simulation results show that fluid injection can trigger earthquakes, having implications for induced seismicity. In addition, the results show that the scheme used for solving the fully coupled problem, captures dynamics that would not be seen in an uncoupled model. Future improvements involve imposing Dirichlet boundary conditions using a different technique, extending the scheme to handle curvilinear coordinates and three spatial dimensions, as well as improving the high-performance code and extending the study of the fault dynamics.

Thesis (Ph.D)--Electrical and Computer Engineering, Georgia Institute of Technology, 2009.
Committee Chair: Smith, Glenn; Committee Member: Buck, John; Committee Member: Goldsztein, Guillermo; Committee Member: Peterson, Andrew; Committee Member: Scott, Waymond. Part of the SMARTech Electronic Thesis and Dissertation Collection.

The main objective of this research is to model and analyse complex electromagnetic problems by means of a new hybridised computational technique combining the frequency domain Method of Moments (MoM), Finite-Difference Time-Domain (FDTD) method and a subgridded Finite-Difference Time-Domain (SGFDTD) method. This facilitates a significant advance in the ability to predict electromagnetic absorption in inhomogeneous, anisotropic and lossy dielectric materials irradiated by geometrically intricate sources. The Method of Moments modelling employed a two-dimensional electric surface patch integral formulation solved by independent linear basis function methods in the circumferential and axial directions of the antenna wires. A similar orthogonal basis function is used on the end surface and appropriate attachments with the wire surface are employed to satisfy the requirements of current continuity. The surface current distributions on structures which may include closely spaced parallel wires, such as dipoles, loops and helical antennas are computed. The results are found to be stable and showed good agreement with less comprehensive earlier work by others. The work also investigated the interaction between overhead high voltage transmission lines and underground utility pipelines using the FDTD technique for the whole structure, combined with a subgridding method at points of interest, particularly the pipeline. The induced fields above the pipeline are investigated and analysed. FDTD is based on the solution of Maxwell¿s equations in differential form. It is very useful for modelling complex, inhomogeneous structures. Problems arise when open-region geometries are modelled. However, the Perfectly Matched Layer (PML) concept has been employed to circumvent this difficulty. The establishment of edge elements has greatly improved the performance of this method and the computational burden due to huge numbers of time steps, in the order of tens of millions, has been eased to tens of thousands by employing quasi-static methods. This thesis also illustrates the principle of the equivalent surface boundary employed close to the antenna for MoM-FDTD-SGFDTD hybridisation. It depicts the advantage of using hybrid techniques due to their ability to analyse a system of multiple discrete regions by employing the principle of equivalent sources to excite the coupling surfaces. The method has been applied for modelling human body interaction with a short range RFID antenna to investigate and analyse the near field and far field radiation pattern for which the cumulative distribution function of antenna radiation efficiency is presented. The field distributions of the simulated structures show reasonable and stable results at 900 MHz. This method facilitates deeper investigation of the phenomena in the interaction between electromagnetic fields and human tissues.
Ministry of Higher Education Malaysia and Universiti Tun Hussein Onn Malaysia (UTHM)

The main objective of this research is to model and analyse complex electromagnetic problems by means of a new hybridised computational technique combining the frequency domain Method of Moments (MoM), Finite-Difference Time-Domain (FDTD) method and a subgridded Finite-Difference Time-Domain (SGFDTD) method. This facilitates a significant advance in the ability to predict electromagnetic absorption in inhomogeneous, anisotropic and lossy dielectric materials irradiated by geometrically intricate sources. The Method of Moments modelling employed a two-dimensional electric surface patch integral formulation solved by independent linear basis function methods in the circumferential and axial directions of the antenna wires. A similar orthogonal basis function is used on the end surface and appropriate attachments with the wire surface are employed to satisfy the requirements of current continuity. The surface current distributions on structures which may include closely spaced parallel wires, such as dipoles, loops and helical antennas are computed. The results are found to be stable and showed good agreement with less comprehensive earlier work by others. The work also investigated the interaction between overhead high voltage transmission lines and underground utility pipelines using the FDTD technique for the whole structure, combined with a subgridding method at points of interest, particularly the pipeline. The induced fields above the pipeline are investigated and analysed. FDTD is based on the solution of Maxwell's equations in differential form. It is very useful for modelling complex, inhomogeneous structures. Problems arise when open-region geometries are modelled. However, the Perfectly Matched Layer (PML) concept has been employed to circumvent this difficulty. The establishment of edge elements has greatly improved the performance of this method and the computational burden due to huge numbers of time steps, in the order of tens of millions, has been eased to tens of thousands by employing quasi-static methods. This thesis also illustrates the principle of the equivalent surface boundary employed close to the antenna for MoM-FDTD-SGFDTD hybridisation. It depicts the advantage of using hybrid techniques due to their ability to analyse a system of multiple discrete regions by employing the principle of equivalent sources to excite the coupling surfaces. The method has been applied for modelling human body interaction with a short range RFID antenna to investigate and analyse the near field and far field radiation pattern for which the cumulative distribution function of antenna radiation efficiency is presented. The field distributions of the simulated structures show reasonable and stable results at 900 MHz. This method facilitates deeper investigation of the phenomena in the interaction between electromagnetic fields and human tissues.

The sound generated by high speed trains can be exacerbated by the presence of trackside structures. Tunnels are the principal structures that have a strong influence on the noise produced by trains. A train entering a tunnel causes air to flow in and out of the tunnel portal, forming a monopole source of low frequency sound ["infrasound"] whose wavelength is large compared to the tunnel diameter. For the compact case, when the tunnel diameter is small, incompressible flow theory can be used to compute the Green's function that determines the monopole sound. However, when the infrasound is "shielded" from the far field by a large "flange" at the tunnel portal, the problem of calculating the sound produced in the far field is more complex. In this case, the monopole contribution can be calculated in a first approximation in terms of a modified Compact Green's function, whose properties are determined by the value at the center of a. disk (modelling the flange) of a diffracted potential produced by a thin circular disk. In this thesis this potential is calculated numerically. The scattering of sound by a thin circular disk is investigated using the Finite Difference Method applied to the three dimensional Helmholtz equation subject to appropriate boundary conditions on the disk. The solution is also used to examine the unsteady force acting on the disk.

Orientadores: Joerg Schleicher, Amelia Novais
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Este trabalho aborda a questão de como resolver a equação da onda imagem para o problema de remigração na profundidade através de métodos numéricos. O objetivo deste problema é a reconstrução de uma imagem das camadas geológicas do subsolo a partir de uma imagem previamente migrada com um modelo de velocidade, geralmente, incorreto. Nosso principal objetivo neste trabalho é a investigação de possíveis métodos que possam resolver os problemas que surgiram ao usarmos esquemas explícitos do método de diferenças _nitas na solução da equação da onda imagem em trabalhos anteriores, como, por exemplo, a dispersão numérica. Para isso, estudamos aqui o método de volumes _nitos, assim como esquemas implícitos do método de diferenças _nitas. O método de volumes _nitos possui como característica principal propagar as médias das células da malha ao invés de simplesmente os dados pontuais como é feito no método de diferenças _nitas. As outras tentativas para solucionar o problema da dispersão foram dois tipos de implementação de esquemas implícitos do método de diferenças _nitas, isto é, implementações implícitas de esquemas convencionais avaliados em pontos da malha e um esquema avaliado nos centros das células. A qualidade dos algoritmos estudados foi testada numericamente. Estes testes numéricos mostram que o método de volumes _nitos não é adequado para resolver o problema da dispersão, uma vez que a média calculada a cada passo aumenta o estiramento do pulso. Além disso, as implementações implícitas dos esquemas convencionais mostram o mesmo comportamento de dispersão que as implementações explícitas. Unicamente o esquema centrado foi capaz de melhorar a dispersão numérica em comparação com as implementações anteriores,porém somente para dados contendo exclusivamente baixas freqüências
Abstract: This work approaches the question of how to solve the image-wave equation for depth remigration by numerical methods. The objective is the reconstruction of an image of the geologic layers of the subsoil from a previously migrated image with a different velocity model. Our main objective in this work is the investigation of possible methods that can solve the problems that appeared when using explicit _nite-difference schemes for the solution of the image-wave equation in previous works, particularly numerical dispersion. For this purpose, we study the method of _nite volumes, as well as implicit _nite-difference schemes. The main characteristic of the _nite-volume method is to simply propagate the averages in the cells of the mesh instead of the discretized data themselves as it is done in the _nitedifference method. As another attempt to solve the problem of the dispersion, we study two types of implementation of implicit _nite-difference schemes, that is, implicit implementations of conventional schemes evaluated out the edge of the cell and a scheme evaluated in the center of the cell. The quality of the studied algoritms has been tested numerically. These numerical tests show that the method of _nite volumes is not adequate to solve the problem of dispersion, for the average calculated in each step additionally increases the pulse stretch. Moreover, the implicit implementations of the conventional schemes show the same dispersion behavior as the explicit implementations. Solely the centered scheme was capable to improve the numerical dispersion in comparison with the previous implementations, however only for data containing
Mestrado
Geofisica Computacional
Mestre em Matemática Aplicada

Approved for public release; distribution is unlimited
A FORTRAN 77 computer code employing an adaptation of the finite differencing algorithm proposed by Brian was developed for the solution of transient heat conduction problems in cylindrical geometries. Validation of code was accomplished by comparison with an ana­lytic solution derived for a model with symmetric, linear boundary conditions. Accuracy of results for asymmetric and non-linear boundary conditions was determined by comparison with a similarly vali­dated code employing the explicit method. Code effectiveness was then demonstrated by conducting a transient temperature analysis for a simulated earth-orbiting satellite. Brian's method demonstrated unconditional stability with associated significant reductions in execu­tion time compared to the explicit method. The effects of discretization error on the accuracy of results require further investigation.
http://archive.org/details/applicationofbri00hein
Lieutenant Commander, United States Navy

Thesis (Ph. D.)--West Virginia University, 2000.
Title from document title page. Document formatted into pages; contains xiii, 325 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 138-144).

This thesis has proposed a simplified model of corona discharge from an overhead wire struck by lightning for surge computations using the FDTD method. In the corona model, the progression of corona streamers from the wire is represented as the radial expansion of cylindrical conducting region around the wire. The validity of this corona model has been tested against experimental data. Then, its applications to lightning electromagnetic pulse computations have been reviewed.
博士(工学)
Doctor of Philosophy in Engineering
同志社大学
Doshisha University

We consider high order finite difference methods for the wave equations in the second order form, where the finite difference operators satisfy the summation-by-parts principle. Boundary conditions and interface conditions are imposed weakly by the simultaneous-approximation-term method, and non-conforming grid interfaces are handled by an interface operator that is based on either interpolating directly between the grids or on projecting to piecewise continuous polynomials on an intermediate grid. Stability and accuracy are two important aspects of a numerical method. For accuracy, we prove the convergence rate of the summation-by-parts finite difference schemes for the wave equation. Our approach is based on Laplace transforming the error equation in time, and analyzing the solution to the boundary system in the Laplace space. In contrast to first order equations, we have found that the determinant condition for the second order equation is less often satisfied for a stable numerical scheme. If the determinant condition is satisfied uniformly in the right half plane, two orders are recovered from the boundary truncation error; otherwise we perform a detailed analysis of the solution to the boundary system in the Laplace space to obtain an error estimate. Numerical experiments demonstrate that our analysis gives a sharp error estimate. For stability, we study the numerical treatment of non-conforming grid interfaces. In particular, we have explored two interface operators: the interpolation operators and projection operators applied to the wave equation. A norm-compatible condition involving the interface operator and the norm related to the SBP operator is essential to prove stability by the energy method for first order equations. In the analysis, we have found that in contrast to first order equations, besides the norm-compatibility condition an extra condition must be imposed on the interface operators to prove stability by the energy method. Furthermore, accuracy and efficiency studies are carried out for the numerical schemes.

This thesis concerns the analytical and practical aspects of applying the Closest Point Method to solve elliptic partial differential equations (PDEs) on smooth surfaces and domains with smooth boundaries. A new numerical scheme is proposed to solve surface elliptic PDEs and a novel geometric multigrid solver is constructed to solve the resulting linear system. The method is also applied to coupled bulk-surface problems. A new embedding equation in a narrow band surrounding the surface is formulated so that it agrees with the original surface PDE on the surface and has a unique solution which is constant along the normals to the surface. The embedding equation is then discretized using standard finite difference scheme and barycentric Lagrange interpolation. The resulting scheme has 2nd-order accuracy in practice and is provably 2nd-order convergent for curves without boundary embedded in ℝ². To apply the method to solve elliptic equations on surfaces and domains with boundaries, the "ghost" point approach is adopted to handle Dirichlet, Neumann and Robin boundary conditions. A systematic method is proposed to represent values of ghost points by values of interior points according to boundary conditions. A novel geometric multigrid method based on the closest point representation of the surface is constructed to solve the resulting large sparse linear systems. Multigrid solvers are designed for surfaces with or without boundaries and domains with smooth boundaries. Numerical results indicate that the convergence rate of the multigrid solver stays roughly the same as we refine the mesh, as is desired of a multigrid algorithm. Finally the above methods are combined to solve coupled bulk-surface PDEs with some applications to biology.

MODELING AND NUMERICAL ANALYSIS OF SINGLE DROPLET DRYING DALMAZ, Nesip M.Sc., Department of Chemical Engineering Supervisor: Prof. Dr. H. Ö
nder Ö
ZBELGE Co-Supervisor: Asst. Prof. Dr. Yusuf ULUDAg August 2005, 120 pages A new single droplet drying model is developed that can be used as a part of computational modeling of a typical spray drier. It is aimed to describe the drying behavior of a single droplet both in constant and falling rate periods using receding evaporation front approach coupled with the utilization of heat and mass transfer equations. A special attention is addressed to develop two different numerical solution methods, namely the Variable Grid Network (VGN) algorithm for constant rate period and the Variable Time Step (VTS) algorithm for falling rate period, with the requirement of moving boundary analysis. For the assessment of the validity of the model, experimental weight and temperature histories of colloidal silica (SiO2), skimmed milk and sodium sulfate decahydrate (Na2SO4&
#8901
10H2O) droplets are compared with the model predictions. Further, proper choices of the numerical parameters are sought in order to have successful iteration loops. The model successfully estimated the weight and temperature histories of colloidal silica, dried at air temperatures of 101oC and 178oC, and skimmed milk, dried at air temperatures of 50oC and 90oC, droplets. However, the model failed to predict both the weight and the temperature histories of Na2SO4&
#8901
10H2O droplets dried at air temperatures of 90oC and 110oC. Using the vapor pressure expression of pure water, which neglects the non-idealities introduced by solid-liquid interactions, in model calculations is addressed to be the main reason of the model resulting poor estimations. However, the developed model gives the flexibility to use a proper vapor pressure expression without much effort for estimation of the drying history of droplets having highly soluble solids with strong solid-liquid interactions. Initial droplet diameters, which were calculated based on the estimations of the critical droplet weights, were predicted in the range of 1.5-2.0 mm, which are in good agreement with the experimental measurements. It is concluded that the study has resulted a new reliable drying model that can be used to predict the drying histories of different materials.

The 3D Finite-Difference Time-Domain (FDTD) method simulates structures in the time-domain using a direct form of Maxwell's curl equations. This method has the advantage over other simulation methods in that it does not use empirical approximations. Unfortunately, it requires large amounts of memory and long simulation times. This thesis applies parallel processing to the method so that simulation times are greatly reduced. Parallel processing, though, has the disadvantage in that simulation programs require to be segmented so that each processor processes a separate part of the simulation. Another disadvantage of parallel processing is that each processor communicates with neighbouring processors to report their conditions. For large processor arrays this can result in a large overhead in simulation time. Two main methods of parallel processing discussed: Transputer arrays and clustered workstations over a local area network (LAN). These have been chosen because of their relatively cheapness to use, and their widespread availability. The results presented apply to the simulation of a microstrip antenna and to propagation of electrical signals in a printed circuit board (PCB). Microstrip antennas are relatively difficult to simulate in the time-domain because they have resonant pulses. Methods that reduce this problem are discussed in the thesis. The thesis contains a novel analysis of the parallel processing showing, using equations, tables and graphs, the optimum array size for a given inter-processor communication speed and for a given iteration time. This can be easily applied to any processing system. Background material on the 3D FDTD method and microstrip antennas is also provided. From the work on the parallel processing of the 3D FDTD a novel technique for the simulation of the Finite-element (FE) method is also discussed.

We address in this thesis the numerical solution of state constrained optimal control problems for systems modeled by linear parabolic equations. For the unconstrained or control-constrained optimal control problem, the first order optimality condition can be obtained in a general way and the associated Lagrange multiplier has low regularity, such as in the L²(Ω). However, for state-constrained optimal control problems, additional assumptions are required in general to guarantee the existence and regularity of Lagrange multipliers. The resulting optimality system leads to difficulties for both the numerical solution and the theoretical analysis. The approach discussed here combines the alternating direction of multipliers (ADMM) with a conjugate gradient (CG) algorithm, both operating in well-chosen Hilbert spaces. The ADMM approach allows the decoupling of the state constraints and the parabolic equation, in which we need solve an unconstrained parabolic optimal control problem and a projection onto the admissible set in each iteration. It has been shown that the CG method applied to the unconstrained optimal control problem modeled by linear parabolic equation is very efficient in the literature. To tackle the issue about the associated Lagrange multiplier, we prove the convergence of our proposed algorithm without assuming the existence and regularity of Lagrange multipliers. Furthermore, a worst case O(1/k) convergence rate in the ergodic sense is established. For numerical purposes, we employ the finite difference method combined with finite element method to implement the time-space discretization. After full discretization, the numerical results we obtain validate the methodology discussed in this thesis.

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Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq
Propomos um método numérico de diferenças finitas para substituir os esquemas clássicos utilizados para solucionar EDPs em engenharia financeira. A motivação para desenvolvê-lo advém da perda de precisão na tentativa de estabilizar a solução via up-wind no termo convectivo bem como o fato de que oscilações espúrias ocorrem quando a volatilidade é baixa, o que é comumente observado nos mercados de taxas de juros. Ao contrário dos esquemas clássicos, nosso método cobre todo o espectro de volatilidade da dinâmica das taxas de juros. Nós comparamos resultados analíticos e numéricos precificando e realizando o hedge de uma variedade de contratos financeiros de renda fixa para mostrar que o método que desenvolvemos é confiável e altamente competitivo. O método se adapta bem a derivativos exóticos de taxas de juros, incluindo um derivativo dependente da trajetória denominado Opção IDI (índice brasileiro de depósito interbancário). O método dá ênfase à abordagem realística da capitalização discreta do índice em detrimento da capitalização contínua explorada frequentemente na literatura.
We propose a second order accurate numerical finite difference method to replace the classical schemes used to solving PDEs in financial engineering. The motivation for doing so stems from the accuracy loss while trying to stabilize the solution via the up-wind trick in the convective term as well as the fact that spurious oscillation solutions occur when volatilities are low. This is actually the range that we commonly observe in the interest rate markets. Unlike the classical schemes, our method covers the whole spectrum of volatilities in the interest rate dynamics. We compare the analytical and numerical results by both pricing and hedging a variety of fixed income financial contracts to show that the method we developed is reliable and highly competitive. The method adapts well to exotic interest rate derivative securities, including a path-dependent derivative named IDI (the Brazilian Interbank Deposit Rate Index) option. The method highlights the use of the realistic discretely compounding interest rate scheme, in detriment of the continuously compounding case often exploited in the literature.

This thesis discusses properties arising when finite differences are implemented forsolving the two dimensional wave equation on media with various properties. Both homogeneous and heterogeneous surfaces are considered. The time derivative of the wave equation is discretised using a weighted central difference scheme, dependenton a variable parameter gamma. Stability and convergence properties are studied forsome different values of gamma. The report furthermore features an introduction to solving large sparse linear systems of equations, using so-called multigrid methods.The linear systems emerge from the finite difference discretisation scheme. Aconclusion is drawn stating that values of gamma in the unconditionally stable region provides the best computational efficiency. This holds true as the multigrid based numerical solver exhibits optimal or near optimal scaling properties.

New Quasi-optical sensor technology, based on the millimetre and submillimetre band of the electromagnetic spectrum, is actually being implemented for many commercial and scientific applications such as remote sensing, astronomy, collision avoidance radar, etc. These novel devices make use of integrated active and passive structures usually as planar arrays. The electromagnetic design and computer simulation of these new structures requires novel numerical techniques. The Finite Difference Time Domain method (FDTD) is well suited for the electromagnetic analysis of integrated devices using active non-linear elements, but is difficult to use for large and/or periodic structures. A rigorous revision of this popular numerical technique is performed in order to permit FDTD to model practical quasi-optical devices. The system impulse response or discrete Green's function (DGF) for FDTD is determined as a polynomial then the FDTD technique is reformulated as a convolution sum. This new alternative algorithm avoids Absorbing Boundary Conditions (ABC's) and can save large amounts of memory to model wire or slot structures. Many applications for the DGF can be foreseen, going beyond quasi-optical components. As an example, the exact ABC based on the DGF for FDTD is implemented for a single grid wall is presented. The problem of time domain analysis of planar periodic structures modelling only one periodic cell is also investigated. Simple Periodic Boundary Conditions (PBC) can be implemented for FDTD, but they can not handle periodic devices (such as phased shift arrays or dichroic screens) which produce fields periodic in a 4D basis (three spatial dimensions plus time). An extended FDTD scheme is presented which uses Lorentz type coordinate transformations to reduce the problem to 3D. The analysis of non-linear devices using FDTD is also considered in the thesis. In this case, the non linear devices are always model using an equivalent lumped element circuit. These circuits are introduced into the FDTD grid by means of the current density following an iterative implicit algorithm. As a demonstration of the technique a quasi-optically feed slot ring mixer with integral lens is designed for operation at 650 GHz.

Thesis (Ph. D.)--Ohio State University, 2004.
Title from first page of PDF file. Document formatted into pages; contains xvii, 165 p.; also includes graphics (some col.). Includes bibliographical references (p. 158-165).

[EN] In the stock markets, the process of estimating a fair price for a stock, option or commodity is consider the corner stone for this trade. There are several attempts to obtain a suitable mathematical model in order to enhance the estimation process for evaluating the options for short or long periods. The Black-Scholes partial differential equation (PDE) and its analytical solution, 1973, are considered a breakthrough in the mathematical modeling for the stock markets. Because of the ideal assumptions of Black-Scholes several alternatives have been developed to adequate the models to the real markets. Two strategies have been done to capture these behaviors; the first modification is to add jumps into the asset following Lévy processes, leading to a partial integro-differential equation (PIDE); the second is to allow the volatility to evolve stochastically leading to a PDE with two spatial variables. Here in this work, we solve numerically PIDEs for a wide class of Lévy processes using finite difference schemes for European options and also, the associated linear complementarity problem (LCP) for American option. Moreover, the models for options under stochastic volatility incorporated with jump-diffusion are considered. Numerical analysis for the proposed schemes is studied since it is the efficient and practical way to guarantee the convergence and accuracy of numerical solutions. In fact, without numerical analysis, careless computations may waste good mathematical models. This thesis consists of four chapters; the first chapter is an introduction containing historically review for stochastic processes, Black-Scholes equation and preliminaries on numerical analysis. Chapter two is devoted to solve the PIDE for European option under CGMY process. The PIDE for this model is solved numerically using two distinct discretization approximations; the first approximation guarantees unconditionally consistency while the second approximation provides unconditional positivity and stability. In the first approximation, the differential part is approximated using the explicit scheme and the integral part is approximated using the trapezoidal rule. In the second approximation, the differential part is approximated using the Patankar-scheme and the integral part is approximated using the four-point open type formula. Chapter three provides a unified treatment for European and American options under a wide class of Lévy processes as CGMY, Meixner and Generalized Hyperbolic. First, the reaction and convection terms of the differential part of the PIDE are removed using appropriate mathematical transformation. The differential part for European case is explicitly discretized , while the integral part is approximated using Laguerre-Gauss quadrature formula. Numerical properties such as positivity, stability and consistency for this scheme are studied. For the American case, the differential part of the LCP is discretized using a three-time level approximation with the same integration technique. Next, the Projected successive over relaxation and multigrid techniques have been implemented to obtain the numerical solution. Several numerical examples are given including discussion of the errors and computational cost. Finally in Chapter four, the PIDE for European option under Bates model is considered. Bates model combines both stochastic volatility and jump diffusion approaches resulting in a PIDE with a mixed derivative term. Since the presence of cross derivative terms involves the existence of negative coefficient terms in the numerical scheme deteriorating the quality of the numerical solution, the mixed derivative is eliminated using suitable mathematical transformation. The new PIDE is solved numerically and the numerical analysis is provided. Moreover, the LCP for American option under Bates model is studied.
[ES] El proceso de estimación del precio de una acción, opción u otro derivado en los mercados de valores es objeto clave de estudio de las matemáticas financieras. Se pueden encontrar diversas técnicas para obtener un modelo matemático adecuado con el fin de mejorar el proceso de valoración de las opciones para periodos cortos o largos. Históricamente, la ecuación de Black-Scholes (1973) fue un gran avance en la elaboración de modelos matemáticos para los mercados de valores. Es un modelo práctico para estimar el valor razonable de una opción. Sobre unos supuestos determinados, F. Black y M. Scholes obtuvieron una ecuación diferencial parcial lineal y su solución analítica. Desde entonces se han desarrollado modelos más complejos para adecuarse a la realidad de los mercados. Un tipo son los modelos con volatilidad estocástica que vienen descritos por una ecuación en derivadas parciales con dos variables espaciales. Otro enfoque consiste en añadir saltos en el precio del subyacente por medio de modelos de Lévy lo que lleva a resolver una ecuación integro-diferencial parcial (EIDP). En esta memoria se aborda la resolución numérica de una amplia clase de modelos con procesos de Lévy. Se desarrollan esquemas en diferencias finitas para opciones europeas y también para opciones americanas con su problema de complementariedad lineal (PCL) asociado. Además se tratan modelos con volatilidad estocástica incorporando difusión con saltos. Se plantea el análisis numérico ya que es el camino eficiente y práctico para garantizar la convergencia y precisión de las soluciones numéricas. De hecho, la ausencia de análisis numérico debilita un buen modelo matemático. Esta memoria está organizada en cuatro capítulos. El primero es una introducción con un breve repaso de los procesos estocásticos, el modelo de Black-Scholes así como nociones preliminares de análisis numérico. En el segundo capítulo se trata la EIDP para las opciones europeas según el modelo CGMY. Se proponen dos esquemas en diferencias finitas; el primero garantiza consistencia incondicional de la solución mientras que el segundo proporciona estabilidad y positividad incondicionales. Con el primer enfoque, la parte diferencial se discretiza por medio de un esquema explícito y para la parte integral se usa la regla del trapecio. En la segunda aproximación, para la parte diferencial se usa un esquema tipo Patankar y la parte integral se aproxima por medio de la fórmula de tipo abierto con cuatro puntos. En el capítulo tercero se propone un tratamiento unificado para una amplia clase de modelos de opciones en procesos de Lévy como CGMY, Meixner e hiperbólico generalizado. Se eliminan los términos de reacción y convección por medio de un apropiado cambio de variables. Después la parte diferencial se aproxima por un esquema explícito mientras que para la parte integral se usa la fórmula de cuadratura de Laguerre-Gauss. Se analizan positividad, estabilidad y consistencia. Para las opciones americanas, la parte diferencial del LCP se discretiza con tres niveles temporales mediante cuadratura de Laguerre-Gauss para la integración numérica. Finalmente se implementan métodos iterativos de proyección y relajación sucesiva y la técnica de multimalla. Se muestran varios ejemplos incluyendo estudio de errores y coste computacional. El capítulo 4 está dedicado al modelo de Bates que combina los enfoques de volatilidad estocástica y de difusión con saltos derivando en una EIDP con un término con derivadas cruzadas. Ya que la discretización de una derivada cruzada comporta la existencia de coeficientes negativos en el esquema que deterioran la calidad de la solución numérica, se propone un cambio de variables que elimina dicha derivada cruzada. La EIDP transformada se resuelve numéricamente y se muestra el análisis numérico. Por otra parte se estudia el LCP para opciones americanas con el modelo de Bates.
[CAT] El procés d'estimació del preu d'una acció, opció o un altre derivat en els mercats de valors és objecte clau d'estudi de les matemàtiques financeres . Es poden trobar diverses tècniques per a obtindre un model matemàtic adequat a fi de millorar el procés de valoració de les opcions per a períodes curts o llargs. Històricament, l'equació Black-Scholes (1973) va ser un gran avanç en l'elaboració de models matemàtics per als mercats de valors. És un model matemàtic pràctic per a estimar un valor raonable per a una opció. Sobre uns suposats F. Black i M. Scholes van obtindre una equació diferencial parcial lineal amb solució analítica. Des de llavors s'han desenrotllat models més complexos per a adequar-se a la realitat dels mercats. Un tipus és els models amb volatilitat estocástica que ve descrits per una equació en derivades parcials amb dos variables espacials. Un altre enfocament consistix a afegir bots en el preu del subjacent per mitjà de models de Lévy el que porta a resoldre una equació integre-diferencial parcial (EIDP) . En esta memòria s'aborda la resolució numèrica d'una àmplia classe de models baix processos de Lévy. Es desenrotllen esquemes en diferències finites per a opcions europees i també per a opcions americanes amb el seu problema de complementarietat lineal (PCL) associat. A més es tracten models amb volatilitat estocástica incorporant difusió amb bots. Es planteja l'anàlisi numèrica ja que és el camí eficient i pràctic per a garantir la convergència i precisió de les solucions numèriques. De fet, l'absència d'anàlisi numèrica debilita un bon model matemàtic. Esta memòria està organitzada en quatre capítols. El primer és una introducció amb un breu repàs dels processos estocásticos, el model de Black-Scholes així com nocions preliminars d'anàlisi numèrica. En el segon capítol es tracta l'EIDP per a les opcions europees segons el model CGMY. Es proposen dos esquemes en diferències finites; el primer garantix consistència incondicional de la solució mentres que el segon proporciona estabilitat i positivitat incondicionals. Amb el primer enfocament, la part diferencial es discretiza per mitjà d'un esquema explícit i per a la part integral s'empra la regla del trapezi. En la segona aproximació, per a la part diferencial s'usa l'esquema tipus Patankar i la part integral s'aproxima per mitjà de la fórmula de tipus obert amb quatre punts. En el capítol tercer es proposa un tractament unificat per a una àmplia classe de models d'opcions en processos de Lévy com ara CGMY, Meixner i hiperbòlic generalitzat. S'eliminen els termes de reacció i convecció per mitjà d'un apropiat canvi de variables. Després la part diferencial s'aproxima per un esquema explícit mentres que per a la part integral s'usa la fórmula de quadratura de Laguerre-Gauss. S'analitzen positivitat, estabilitat i consistència. Per a les opcions americanes, la part diferencial del LCP es discretiza amb tres nivells temporals amb quadratura de Laguerre-Gauss per a la integració numèrica. Finalment s'implementen mètodes iteratius de projecció i relaxació successiva i la tècnica de multimalla. Es mostren diversos exemples incloent estudi d'errors i cost computacional. El capítol 4 està dedicat al model de Bates que combina els enfocaments de volatilitat estocástica i de difusió amb bots derivant en una EIDP amb un terme amb derivades croades. Ja que la discretización d'una derivada croada comporta l'existència de coeficients negatius en l'esquema que deterioren la qualitat de la solució numèrica, es proposa un canvi de variables que elimina dita derivada croada. La EIDP transformada es resol numèricament i es mostra l'anàlisi numèrica. D'altra banda s'estudia el LCP per a opcions americanes en el model de Bates.
El-Fakharany, MMR. (2015). Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/53917
TESIS

In a structure, a joint is considered as the weakest part, and it should not get separated when subjected to loading, so that an unstable collapse of structure can be avoided. It is important to investigate the failure in joint before it is used in a structure. Failure of a joint depends on various factors such as the geometry of joint configuration, sheet strength that are joined, rivet material used, cracks developed during joining, and many other. Self-Piercing riveting process is a new technology for joining sheet metals in automobile and aircraft industries. This process has many advantages over conventional joining processes. In this thesis, the failure of a self-piercing riveted joint is investigated. Failure of three different riveted configurations under 35m/s and 60m/s velocities were predicted using the general purpose non-linear finite element software LS-DYNA. This research is divided into three stages of work. In the first stage, a 2D simulation of riveting process is carried out over two Aluminum sheets. An r-adaptive methodology is utilized to acquire a higher accuracy of results and to avoid high element distortion. A parametrical study is then conducted to study the effect of rivet penetration velocity and adaptive mesh size varies the quality of the joint. In the second stage of work, a spring back analysis of joint is conducted to study the deformations of work piece after the riveting process. In the third stage, a Peel specimen, a U-shaped single riveted connection, and a U-shaped double riveted connection were investigated for failure under 35 m/s and 60 m/s velocities in both shear and tension testing conditions. Three different loading conditions were used for testing. The results from this study will show how process parameters can influence the quality of riveted joint, amount of deformations that occur in the work piece after the removal of rigid bodies, and failure load of SPR joint in different configurations.
Thesis (M.S.)--Wichita State University, College of Engineering, Dept. of Mechanical Engineering.

In this thesis we will investigate how the SBP-SAT finite difference method behave with and without an interface. As model problem, we consider the heat equation with piecewise constant coefficients. The thesis is split in two main parts. In the first part we look at the heat equation in one-dimension, and in the second part we expand the problem to a two-dimensional domain. We show how the SAT-parameters are chosen such that the scheme is dual consistent and stable. Then, we perform numerical experiments, now looking at the static case. In the one-dimensional case we see that the second order SBP-SAT method with an interface converge with an order of two, while the second order SBP-SAT method without an interface converge with an order of one.