Due to the interesting financial moment we are living, my motivations to write this Master thesis has mostly been the behavior of interest rates and models that can be used predict them. Thus, in this dissertation I have presented theHull-White model and the way to calibrate it against market data so it can be used to price interest rate derivatives. The reader can find both theoretical and practical presentations and examples along with the code to program them byhim/herself.

In this thesis we present the Stefan problem with two boundary conditions, one constant and one time-dependent. This problem is a classic example of a free boundary problem in partial differential equations, with a free boundary moving in time. Some properties are being proved for the one-dimensional case and the important Stefan condition is also derived. The importance of the maximum principle, and the existence of a unique solution are being discussed. To numerically solve this problem, an analysis when the time t goes to zero is being done. The approximative solutions are shown graphically with proper error estimates.

Made available in DSpace on 2015-03-04T18:51:08Z (GMT). No. of bitstreams: 1 Apresentacao.pdf: 101300 bytes, checksum: 62de93a7615eeb9b5d3dd49f5ba22f68 (MD5) Previous issue date: 2006-03-28 / Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior / This work deals with the stability analysis of the fully discrete transport problem obtained using a stable finite element method in space and the generalized trapeizoidal family of methods in time. Depeding on the range of parameters the Galerkin and the Streamline Upwind Petrov-Galerkin Methods are introduced. We evaluate the accuracy and stability properties of the methods. The sawtooth pattern in time is observed,caused by spurious higher modes when Crank-Nicolson method is used. We derive a stability analysis of the fully discrete method and investigate the techniques proposed in literature to damp oscillations. We propose a new stability condition to overcome the spurious modes. The proposed methodology is apllied to a one-dimensional contaminant transport problems in a saturated porous media that considers a radioactive contaminant decay at a constant rate. / Nesta dissertação apresenta-se a análise de métodos totalmente discretos, estabilizados espacialmente, para a resolução de problemas de transporte unidimensionais convectivos-difusivos-reativos, lineares e transientes que modelam o transporte de contaminantes radioativos com decaimento a uma taxa constante em um meio poroso saturado. O método de elementos finitos clássico de Galerkin é usado no espaço quando o problema de transporte é pura e/ou predominantemente difusivo, ao passo que quando a convecção domina, o método SUPG (Streamline Upwind Petrov-Galerkin) é utilizado. O método de elementos finitos é combinado a um esquema discreto de integração no tempo, os algoritmos trapezoidais generalizados. Observa-se que a solução aproximada pode apresentar oscilações espúrias quando o método de Crank-Nicolson é utilizado. São consideradas diversas metodologias propostas na literatura para amortecer tais oscilações. Como consequência da análise de estabilidade desenvolvida, uma nova condição de estabilidade é proposta.

No presente trabalho foi estudada a dinâmica de um sistema de muitas partículas no regime de temperaturas ultra-baixas. Realizamos um estudo dinâmico de sistemas condensados bidimensionais em uma rede óptica não-linear em uma direção e também na presença de uma armadilha harmônica assimétrica. Investigamos alguns aspectos sobre a estabilização e propagação de sólitons em condensados de Bose-Einstein. O colapso da função de onda é evitado pela não-linearidade periódica dissipativa, no caso de um meio com campo de fundo positivo (com sistemas atômicos atrativos). A variação adiabática do comprimento de espalhamento de fundo leva a existência de sólitons de onda de matéria metaestáveis. Um sóliton dissipativo pode existir no meio atrativo bidimensional (2D) com uma não-linearidade periódica unidimensional (1D), quando um mecanismo de alimentação atômica é utilizado. Um sóliton estável pode existir no caso de condensados repulsivos, em um campo de fundo negativo, com uma armadilha harmônica em uma direção e uma rede óptica não-linear na outra direção. Os resultados inteiramente numéricos, para a equação de Gross-Pitaevskii 2D, confirmam as simulações da abordagem variacional. / In this work the dynamics of a system of many particles in a ultra-low temperature regime was studied. We performed a dynamic study of two-dimensional condensate systems into a nonlinear optical lattice in one direction and also in the presence of an asymmetrical harmonic trap. We investigated some aspects of the stabilization and spread of solitons in a Bose-Einstein condensate. In the case of positive background field media (with attractive atomic systems), the collapse of the wave-packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in an attractive bidimensional (2D) media with unidimensional (1D) periodic nonlinearity. In the case of repulsive condensates, with a negative background field, a stable soliton may exist when we have an harmonic trap in one direction and a nonlinear optical lattice in the other. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.

Nesta dissertação apresenta-se a análise de métodos totalmente discretos, estabilizados espacialmente, para a resolução de problemas de transporte unidimensionais convectivos-difusivos-reativos, lineares e transientes que modelam o transporte de contaminantes radioativos com decaimento a uma taxa constante em um meio poroso saturado. O método de elementos finitos clássico de Galerkin é usado no espaço quando o problema de transporte é pura e/ou predominantemente difusivo, ao passo que quando a convecção domina, o método SUPG (Streamline Upwind Petrov-Galerkin) é utilizado. O método de elementos finitos é combinado a um esquema discreto de integração no tempo, os algoritmos trapezoidais generalizados. Observa-se que a solução aproximada pode apresentar oscilações espúrias quando o método de Crank-Nicolson é utilizado. São consideradas diversas metodologias propostas na literatura para amortecer tais oscilações. Como consequência da análise de estabilidade desenvolvida, uma nova condição de estabilidade é proposta. / This work deals with the stability analysis of the fully discrete transport problem obtained using a stable finite element method in space and the generalized trapeizoidal family of methods in time. Depeding on the range of parameters the Galerkin and the Streamline Upwind Petrov-Galerkin Methods are introduced. We evaluate the accuracy and stability properties of the methods. The sawtooth pattern in time is observed,caused by spurious higher modes when Crank-Nicolson method is used. We derive a stability analysis of the fully discrete method and investigate the techniques proposed in literature to damp oscillations. We propose a new stability condition to overcome the spurious modes. The proposed methodology is apllied to a one-dimensional contaminant transport problems in a saturated porous media that considers a radioactive contaminant decay at a constant rate.

No presente trabalho foi estudada a dinâmica de um sistema de muitas partículas no regime de temperaturas ultra-baixas. Realizamos um estudo dinâmico de sistemas condensados bidimensionais em uma rede óptica não-linear em uma direção e também na presença de uma armadilha harmônica assimétrica. Investigamos alguns aspectos sobre a estabilização e propagação de sólitons em condensados de Bose-Einstein. O colapso da função de onda é evitado pela não-linearidade periódica dissipativa, no caso de um meio com campo de fundo positivo (com sistemas atômicos atrativos). A variação adiabática do comprimento de espalhamento de fundo leva a existência de sólitons de onda de matéria metaestáveis. Um sóliton dissipativo pode existir no meio atrativo bidimensional (2D) com uma não-linearidade periódica unidimensional (1D), quando um mecanismo de alimentação atômica é utilizado. Um sóliton estável pode existir no caso de condensados repulsivos, em um campo de fundo negativo, com uma armadilha harmônica em uma direção e uma rede óptica não-linear na outra direção. Os resultados inteiramente numéricos, para a equação de Gross-Pitaevskii 2D, confirmam as simulações da abordagem variacional. / In this work the dynamics of a system of many particles in a ultra-low temperature regime was studied. We performed a dynamic study of two-dimensional condensate systems into a nonlinear optical lattice in one direction and also in the presence of an asymmetrical harmonic trap. We investigated some aspects of the stabilization and spread of solitons in a Bose-Einstein condensate. In the case of positive background field media (with attractive atomic systems), the collapse of the wave-packet is arrested by the dissipative periodic nonlinearity. The adiabatic variation of the background scattering length leads to metastable matter-wave solitons. When the atom feeding mechanism is used, a dissipative soliton can exist in an attractive bidimensional (2D) media with unidimensional (1D) periodic nonlinearity. In the case of repulsive condensates, with a negative background field, a stable soliton may exist when we have an harmonic trap in one direction and a nonlinear optical lattice in the other. Variational approach simulations are confirmed by full numerical results for the 2D Gross-Pitaevskii equation.

Orientador: João Frederico da Costa Azevedo Meyer / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T04:55:31Z (GMT). No. of bitstreams: 1 CarmonaTabares_PauloCesar_D.pdf: 24565019 bytes, checksum: 8ebe9aed1d258a0712f49e9711f8d107 (MD5) Previous issue date: 2012 / Resumo: Neste trabalho, construímos um modelo matemático para avaliar as conjecturas existentes acerca do impacto que tem o material inorgânico particulado (sedimento) nas populações bentônicas predominantes na Enseada Potter. Na construção do modelo são utilizadas informações do fenômeno, proporcionadas pelas pesquisas permanentes na região de estudo. Como resultado, logramos comprovar mediante simulações numéricas, o efeito que produz o sedimento na distribuição e abundância das espécies do substrato marinho, constatando neste ecossistema particular as consequências do aquecimento global nessa parte da região antártica. A modelagem é feita com um sistema de equações diferenciais parciais não- lineares sobre um domínio bidimensional irregular (descritiva da região original), o qual é discretizado nas variáveis espaciais por elementos finitos de primeira ordem e na variável temporal pelo Método de Crank-Nicolson. A resolução do sistema não-linear resultante é aproximada através de um método preditor-corretor cuja solução aproximada é visualizada e valorada qualitativamente usando gráficos evolutivos obtidos por simulações em ambiente MATLAB / Abstract: In this work, we built a mathematical model to evaluate existing conjectures about the impact that inorganic particulate material (sediment) has upon predominating benthic populations in Potter Cove. For the mathematical model, phenomena information was that provided by permanent researches in the study area. As a result, by means of numerical simulations, we were able to confirm the effect of sediment over distribution and abundance for species of marine substrate, verifying in this particular ecosystem, the effects of global warming in this specific Antarctic region. Modeling is done with a system of nonlinear partial differential equations over an irregular two-dimensional domain (descriptive of the original region), which is discretized in the spatial variables by first order finite elements and in the time variable by Crank-Nicolson. The resolution of the resulting nonlinear system is approximated by a predictor-corrector method and the solution is displayed and qualitatively valorized using evolutive graphics, obtain in a MATLAB environment / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada

FDS (Fire Dynamics Simulator) använder en version av Navier-Stokes ekvationerna för att göra noggranna beräkningar av värme- och gastransport genom brandbelastade utrymmen. Utvecklarna av programmet arbetar kontinuerligt med att validera det allteftersom nya funktioner tillförs för att öka noggrannheten och bredda tillämpningsområdena. Väldigt lite av detta arbete fokuserar dock på FDS:s konduktionsmodell, den endimensionella Crank-Nicolson metoden. Det här examensarbetet ämnar därför undersöka noggrannheten i FDS:s konduktionsmodell genom jämförelse mot beräkningar med FEM (Finita elementmetoden). En FDS-modell skapades för att tillåta undersökning av en vägg och dess randvillkor med så liten påverkan från andra faktorer som möjligt. Detta för att skapa en kontrollerad omgivning som enkelt kunde replikeras i efterföljande FE-beräkningar av det konduktiva värmeflödet genom den solida obstruktionen. Tre väggar (10 cm betong, 20 cm betong och 1 mm stål) vardera med tre randvillkor (Exposed, Void och Insulated) utsattes för tre temperaturer (100 °C, 500 °C och 1000 °C) vilket ger 27 FDS simuleringar. Den adiabatiska yttemperaturen mättes i varje simulering och användes som indata till motsvarande FE-beräkningar. Resultatet påvisade inga signifikanta motsägelser vad gäller randvillkoren, med tillräcklig tid för termisk penetrering påverkade de den resulterande temperaturen som väntat. Undantaget var en mindre avvikelse i stålväggarna som utsattes för 100 °C och 500 °C med randvillkoren Exposed och Void där FDS aningen underskattade temperaturen relativt FE-beräkningarna. Gastemperaturerna i gridcellerna närmast väggen visade sig vara opålitliga. De tenderade att genomgå substantiella fluktuationer, troligen som ett resultat av hur FDS hanterar diskretiseringen av icke-solida volymer för Navier-Stokes beräkningarna. Dessa fluktuationer påverkade dock inte de resulterande solida temperaturerna eftersom medelgastemperaturen var korrekt. FDS påvisades även ha en tendens att aningen överskatta yttemperaturen under de första minuterna av simuleringarna relativt FE-beräkningarna. Temperaturerna från de två beräkningsmetoderna konvergerade dock efter några få minuter i samtliga tester. Dessa avvikelser ansågs ha för liten påverkan på de solida temperaturerna för att påvisa onoggrannhet i FDS. Därmed drogs slutsatsen att FDS:s beräkningar av temperaturer i solida material är tillräckligt noggranna inom dessa avgränsningar. / FDS (Fire Dynamics Simulator) uses a version of the Navier-Stokes equations to make accurate calculations of heat and gas flow through enclosures exposed to fire. The developers of FDS have, and continue to, validate it as new features get added in an attempt to increase its accuracy and broaden its potential applications. However, little of this effort is focused on FDS’ conductive heat transfer model, based on the one-dimensional Crank-Nicolson method. Thus, this study aims to test the accuracy of FDS’ conduction model by comparing it to calculations using FEM (Finite Element Method). FDS simulations were created so as to facilitate the study of a wall and its boundary conditions with as little interference from other factors as possible. This to create a controlled environment which easily could be replicated in the subsequent FE-calculations of the conductive heat flow through the solid obstructions. Three different walls (10 cm concrete, 20 cm concrete and 1 mm steel), each with the three different boundary conditions for the rear surface (Exposed, Void and Insulated), were exposed to three different temperatures (100 °C, 500 °C and 1000 °C) for a total of 27 FDS simulations. The adiabatic surface temperature was measured in each simulation in FDS and used as input for the corresponding FE-calculations. The results showed no clear inconsistencies in the boundary conditions, given enough time for thermal penetration they affected the resulting temperatures as expected. Save a slight deviation in the steel walls exposed to 100 °C and 500 °C with boundary conditions Exposed and Void where FDS slightly underestimated the temperature relative to the FE-calculations. The gas temperatures in the grid cells closest to the wall were found to be unreliable as they tended to undergo substantial fluctuations, likely as a result of how FDS handles the discretization of non-solid space for the Navier-Stokes calculations. These fluctuations were however not found to affect the solid temperatures as the mean gas temperature was accurate. FDS was also found to have a tendency to slightly overestimate the surface temperature in the first few minutes of the simulations relative to the FE-calculations. Though the resulting temperatures from the two methods converged after a few minutes at most in all tests. These deviations were considered to have too minor an impact on the solid temperature to justify claims of inaccuracy in FDS. Thus, the general conclusion of this study is that FDS’ predictions of solid phase temperatures are sufficiently accurate within these delimitations.

The Manakov system appears in the physics of optical fibers, as well as in quantum mechanics, as multi-component versions of the Nonlinear Schr\"odinger and the Gross-Pitaevskii equations.Although the Manakov system is completely integrable its solutions are far from being explicit in most cases. However, the Inverse Scattering Transform (IST) can be exploited to obtain asymptotic information about solutions.This thesis will describe the IST of the Manakov system, and its asymptotic behavior at short times. I will compare the focusing and defocusing behavior, numerically and analytically, for squared barrier initial potentials. Finally, I will show that the continuous spectrum gives the dominant contribution at short-times.

This work investigates numerically the process of Teflon ablation using a finite-volume discretization, implicit time integration and a domain decomposition method in three-dimensions. The interest in Teflon stems from its use in Pulsed Plasma Thrusters and in thermal protection systems for reentry vehicles. The ablation of Teflon is a complex process that involves phase transition, a receding external boundary where the heat flux is applied, an interface between a crystalline and amorphous (gel) phase and a depolymerization reaction which happens on and beneath the ablating surface. The mathematical model used in this work is based on a two-phase model that accounts for the amorphous and crystalline phases as well as the depolymerization of Teflon in the form of an Arrhenius reaction equation. The model accounts also for temperature-dependent material properties, for unsteady heat inputs and boundary conditions in 3D. The model is implemented in 3D domains of arbitrary geometry with a finite volume discretization on unstructured grids. The numerical solution of the transient reaction-diffusion equation coupled with the Arrhenius-based ablation model advances in time using implicit Crank-Nicolson scheme. For each time step the implicit time advancing is decomposed into multiple sub-problems by a domain decomposition method. Each of the sub-problems is solved in parallel by Newton-Krylov non-linear solver. After each implicit time-advancing step, the rate of ablation and the fraction of depolymerized material are updated explicitly with the Arrhenius-based ablation model. After the computation, the surface of ablation front and the melting surface are recovered from the scalar field of fraction of depolymerized material and the fraction of melted material by post-processing. The code is verified against analytical solutions for the heat diffusion problem and the Stefan problem. The code is validated against experimental data of Teflon ablation. The verification and validation demonstrates the ability of the numerical method in simulating three dimensional ablation of Teflon.