The non-hydrostatic dynamical core ENDGame (Even Newer Dynamics for the General Atmospheric Modelling of the Environment) is extended into the thermosphere to test its feasability as a whole-atmosphere dynamical core that can simulate the large scale fluid dynamics of the whole atmosphere from the surface to the top of the thermosphere at 600km. This research may have applications in the development of a Sun-to-Earth modelling system involving the Met Office Unified Model, which will be useful for space weather forecasting and chemical climate modelling. Initial attempts to raise the top boundary of ENDGame above ∼100km give rise to instabilities. To explore the potential causes of these instabilities, a one dimensional column version of ENDGame: ENDGame1D, is developed to study the effects of vertically propagating acoustic waves in the dynamical core. A 2D ray-tracing scheme is also developed, which accounts for the numerical effects on wave propagation. It is found that ENDGame’s numerics have a tendency towards the excessive focussing of wave energy towards vertical propagation, and have poor handling of large amplitude waves, also being unable to handle shocks. A key finding is that the physical processes of vertical molecular viscosity and diffusion prevent the excessive growth of wave amplitudes in the thermosphere in ENDGame, which may be crucial to improving ENDGame’s stability as it is extended upwards. Therefore, a fully implicit-in-time implementation of vertical molecular viscosity and diffusion is developed in both ENDGame1D and the full three-dimensional version of ENDGame: ENDGame3D. A new scheme is developed to deal with the viscous and diffusive terms with the dynamics terms in a fully coupled way to avoid time-splitting errors that may arise. The combination of a small amount of off-centring of ENDGame’s semi-implicit formulation and the inclusion of vertical molecular viscosity and diffusion act to make ENDGame significantly more stable, as long as the simulation is able to remain stable up to the molecularly diffused region above an altitude of ∼130km.

Ce manuscrit de thèse traite de l'analyse asymptotique de problèmes paraboliques possédant des termes raides. Dans un premier temps, on fait l'analyse asymptotique d'un système parabolique possédant des termes de transport raide. Une analyse à deux échelles, basée sur des résultats de théorie ergodique, nous permet de dériver un système limite effectif. Ce système effectif se trouve être, de nouveau, un système parabolique dont le champ de diffusion peut être explicité par une moyenne du champ de diffusion initial le long d'un groupe d'opérateurs unitaires. L'introduction d'un correcteur nous permet d'obtenir un résultat de convergence forte, avec un ordre de convergence, pour des données initiales non nécessairement bien préparées. On propose dans un second temps une méthode numérique permettant de calculer le champ de diffusion effectif. Celle-ci est basée sur la combinaison d'un schéma Runge-Kutta et d'un schéma de type semi-Lagrangien. L'ordre de convergence obtenu théoriquement est mis en évidence de manière numérique. On propose une méthode numérique basée sur un splitting d'opérateur pour la résolution du système parabolique avec termes de transport raide. Enfin, on effectue l'analyse asymptotique d'un système parabolique fortement anisotrope. Sous de bonnes hypothèses de régularité, un système variationnel effectif est proposé et l'introduction d'un correcteur adapté permet d'obtenir un résultat de convergence forte avec un ordre de convergence. Les arguments utilisés relèvent une nouvelle fois de l'analyse à deux échelles et de la théorie ergodique. / This manuscript is devoted to the asymptotic analysis of parabolic equations with stiff terms. First, we perform the asymptotic analysis of a parabolic equation with stiff transport terms. An effective limit model is obtained by a two-scale analysis based on ergodic theory results. This effective system is again a parabolic system whose diffusion field is an average of the initial diffusion field along a group of unitary operators. The introduction of a corrector allows us to obtain a strong convergence result, with an order of convergence, for initial data not necessarily well prepared. We propose a numerical method to compute the effective diffusion field. This method is based on a Runge-Kutta scheme and a semi-Lagrangian scheme. The theoretically order of convergence is obtained numerically. We propose a numerical method based on operator splitting for the resolution of the parabolic system with stiff transport terms. Finally, we perform the asymptotic analysis of a strongly anisotropic parabolic problem. Under suitable smoothness hypotheses, an effective variational system is proposed. By using a suitable corrector, we obtain a strong convergence result and we are able to perform the error analysis. The arguments relate again to the two-scale analysis and the ergodic theory.

Les instabilités de type Weibel naissent si la distribution des vitesses du plasma présente une anisotropie. Elles entraînent la génération d’un champ magnétique dû à la formation de filaments de courant ainsi qu’une activité électrostatique importante. Ces phénomènes de base apparaissent dans de nombreuses situations, naturelles (vent solaire, jets relativistes) ou expérimentales (interaction laser-plasma) : les plasmas dans lesquels ils naissent peuvent être relativistes ou non, magnétisés ou non, collisionnels ou non, ce qui pose la question du choix du modèle à utiliser pour les décrire. La théorie cinétique est le cadre le plus complexe dans lequel nous travaillerons. De par sa complexité, il est intéressant de développer des modèles réduits. Un premier travail mené au cours de cette thèse est l’utilisation d’un modèle fluide incluant la dynamique du tenseur de pression pour modéliser la phase linéaire des instabilités de type Weibel. On discute le rôle essentiel joué par les composantes hors diagonale du tenseur dans la génération du champ magnétique, puis la capacité du modèle à reproduire quantitativement ou qualitativement les résultats cinétiques en introduisant la notion de limite hydrodynamique. La seconde partie de la thèse est ciblée sur le développement du code semi-lagrangien relativiste VLEM utilisant une méthode de décomposition de domaine : on présente les principales méthodes mathématiques utilisées dans le code, puis on aborde la problématique de la conservation de la charge à laquelle on apporte une réponse reposant sur une adaptation de la méthode d’Esirkepov. Le code est enfin validé grâce à plusieurs simulations d’instabilités de type Weibel / Weibel-type instabilities occurs when the velocity distribution function of the charged particles displays a pronounced anisotropy. A long-lasting magnetic field is generated due to the formation of current filaments, and it is accompanied by an important electrostatic activity. These ``basic’’ phenomena have been greatly investigated because of their involvement in many physical problems, natural (solar wind, relativistic jets) or experimental (laser-plasma interaction) : they occurs in plasmas which can be collisional or not, magnetised or not, relativistic or not. One needs to choose a suitable model for their description. The kinetic theory is the most complete and somewhat complex theoretical framework which we will consider. Due to its complexity, it may be interesting to develop reduced models. The first work realised during this thesis is the utilisation of a non-relativistic fluid description, including the dynamics of the pressure tensor, in order to model the linear Weibel-type instabilities. We put in evidence the effect of the non-diagonal components of the tensor on the magnetic field generation. We discuss the ability of the model to reproduce quantitatively or qualitatively the kinetic results by introducing the hydrodynamics limit. The second part of this thesis work is dedicated to the development of the relativistic semi-lagrangian code VLEM, using a domain decomposition scheme : we present the main mathematical tools used in the code, then we deal with the problem of the charge conservation and propose a solution for VLEM, based on an adaptation of the Esirkepov method. Finally, we validate the code through simulations of Weibel-type

Cette thèse a pour contexte la résolution numérique du système de Vlasov–Poisson (modèle utilisé en physique des plasmas, par exemple dans le cadre du projet ITER) par les méthodes classiques particulaires (PIC pour "Particle-in-Cell") et semi-Lagrangiennes. La contribution principale de notre thèse est une implémentation efficace de la méthode PIC pour architectures multi-coeurs, écrite dans le langage C, dont le nom est Pic-Vert. Notre implémentation (a) atteint un nombre quasi-minimal de transferts mémoires avec la mémoire principale, (b) exploite les instructions vectorielles (SIMD) pour les calculs numériques, et (c) expose une quantité suffisante de parallélisme, en mémoire partagée. Pour mettre notre travail en perspective avec l'état de l'art, nous proposons une métrique permettant de comparer différentes implémentations sur différentes architectures. Notre implémentation est 3 fois plus rapide que d'autres implémentations récentes sur la même architecture (Intel Haswell). / In this thesis, we are interested in solving the Vlasov–Poisson system of equations (useful in the domain of plasma physics, for example within the ITER project), thanks to classical Particle-in-Cell (PIC) and semi-Lagrangian methods. The main contribution of our thesis is an efficient implementation of the PIC method on multi-core architectures, written in C, called Pic-Vert. Our implementation (a) achieves close-to-minimal number of memory transfers with the main memory, (b) exploits SIMD instructions for numerical computations, and (c) exhibits a high degree of shared memory parallelism. To put our work in perspective with respect to the state-of-the-art, we propose a metric to compare the efficiency of different PIC implementations when using different multi-core architectures. Our implementation is 3 times faster than other recent implementations on the same architecture (Intel Haswell).

A esfera é utilizada como domínio computacional na modelagem de diversos fenômenos físicos, como em previsão numérica do tempo. Sua discretização pode ser feita de diversas formas, sendo comum o uso de malha regulares em latitude/longitude. Recentemente, também para melhor uso de computação paralela, há uma tendência ao uso de malhas mais isotrópicas, dentre as quais a icosaédrica. Apesar de já existirem modelos atmosféricos que usam malhas icosaédricas, não há consenso sobre as metodologias mais adequadas a esse tipo de malha. Nos propusemos, portanto, a estudar em detalhe diversos fatores envolvidos no desenvolvimento de modelos atmosféricos globais usando malhas geodésicas icosaédricas. A discretização usual por volumes finitos para divergente de um campo vetorial utiliza como base o Teorema da Divergência e a regra do ponto médio nas arestas das células computacionais. A distribuição do erro obtida com esse método apresenta uma forte relação com características geométricas da malha. Definimos o conceito geométrico de alinhamento de células computacionais e desenvolvemos uma teoria que serve de base para explicar interferências de malha na discretização usual do divergente. Destacamos os impactos de certas relações de alinhamento das células na ordem da discretização do método. A teoria desenvolvida se aplica a qualquer malha geodésica e também pode ser usada para os operadores rotacional e laplaciano. Investigamos diversos métodos de interpolação na esfera adequados a malhas icosaédricas, e abordamos o problema de interpolação e reconstrução vetorial na esfera em malhas deslocadas. Usamos métodos alternativos de reconstrução vetorial aos usados na literatura, em particular, desenvolvemos um método híbrido de baixo custo e boa precisão. Por fim, utilizamos as técnicas de discretização, interpolação e reconstrução vetorial analisadas em um método semi-lagrangiano para o transporte na esfera em malhas geodésicas icosaédricas. Realizamos experimentos computacionais de transporte, incluindo testes de deformações na distribuição do campo transportado, que mostraram a adequação da metodologia para uso em modelos atmosféricos globais. A plataforma computacional desenvolvida nesta tese, incluindo geração de malhas, interpolações, reconstruções vetoriais e um modelo de transporte, fornece uma base para o futuro desenvolvimento de um modelo atmosférico global em malhas icosaédricas. / Spherical domains are used to model many physical phenomena, as, for instance, global numerical weather prediction. The sphere can be discretized in several ways, as for example a regular latitude-longitude grid. Recently, also motivated by a better use of parallel computers, more isotropic grids have been adopted in atmospheric global circulation models. Among those, the icosahedral grids are promising. Which kind of discretization methods and interpolation schemes are the best to use on those grids are still a research subject. Discretization of the sphere may be done in many ways and, recently, to make better use of computational resources, researchers are adopting more isotropic grids, such as the icosahedral one. In this thesis, we investigate in detail the numerical methodology to be used in atmospheric models on icosahedral grids. The usual finite volume method of discretization of the divergence of a vector field is based on the divergence theorem and makes use of the midpoint rule for integration on the edges of computational cells. The error distribution obtained with this method usually presents a strong correlation with some characteristics of the icosahedral grid. We introduced the concept of cell alignment and developed a theory which explains the grid imprinting patterns observed with the usual divergence discretization. We show how grid alignment impacts in the order of the divergence discretization. The theory developed applies to any geodesic grid and can also be used for other operators such as curl and Laplacian. Several interpolation schemes suitable for icosahedral grids were analysed, including the vector interpolation and reconstruction problem on staggered grids. We considered alternative vector reconstruction methods, in particular, we developed a hybrid low cost and good precision method. Finally, employing the discretizations and interpolations previously analysed, we developed a semi-Lagrangian transport method for geodesic icosahedral grids. Several tests were carried out, including deformational test cases, which demonstrated that the methodology is suitable to use in global atmospheric models. The computational platform developed in this thesis, including mesh generation, interpolation, vector reconstruction and the transport model, provides a basis for future development of global atmospheric models on icosahedral grids.

Atualmente diversos produtos são fabricados por meio de injeção de polímeros, num processo denominado moldagem por injeção: material fundido é injetado em um molde no qual resfria e endurece. Contudo, ao contrário de outros processos de produção, a qualidade da peça criada por meio de moldagem por injeção não depende apenas do material e da sua forma geométrica, mas também da maneira na qual o material é processado durante a moldagem. Por esse motivo, o uso de modelagem matemática e simulações numéricas tem aumentado consideravelmente como maneira de auxiliar o processo de produção e tem-se tornado uma ferramenta indispensável. Desta forma, este projeto tem o propósito de simular o escoamento de fluidos durante a fase de preenchimento do processo de moldagem por injeção, utilizando o modelo 21/2-dimensional, composto por uma equação bidimensional para a pressão, conhecida como equação de Hele-Shaw, e uma equação tridimensional para a temperatura do fluido. Um modelo bidimensional para a temperatura é também desenvolvido e apresentado. Este projeto de doutorado propõe duas estratégias numéricas para a solução da equação de Hele-Shaw. A primeira delas é baseada em uma formulação euleriana do método Smoothed Particle Hydrodynamics, onde os pontos utilizados na discretização não se movem, e não há utilização de malhas. A segunda estratégia é baseada na criação de malhas dinamicamente construídas na região do molde que já encontra-se parcialmente cheio de fluido e subseqüente aplicação do método Control Volume Finite Element Method. Uma estratégia dinâmica do método semi lagrangeano é apresentada e aplicada à solução da equação bidimensional da temperatura. O projeto também pretende investigar três novas abordagens para o tratamento da superfície livre. Duas delas são baseadas na técnica Volume of Fluid e uma delas é uma adaptação meshless do método Front-Tracking. O comportamento não newtoniano do fluido é caracterizado por uma família de modelos de viscosidade. Testes numéricos indicando a confiabilidade das metodologias propostas são conduzidos / Currently, several plastic products are manufactured by polymer injection, in a process named injection molding: molten material is injected into a thin mold where it cools and solidifies. However, unlike other manufacturing processes, the quality of injection-molded parts depends not only on the material and shape of the part, but also on how the material is processed throughout the molding. For this reason, the use of mathematical modelling and numerical simulations has been increasing in order to assist in the manufacturing process, and it has become an essential tool. Therefore, this Sc.D. project has the purpose of simulating the fluid flow during the filling stage of the injection molding process, using the 21/2-dimensional model, compounded by a two-dimensional equation for the pressure field (also known as Hele-Shaw equation) and a three-dimensional equation for the temperature of the fluid. A simpler two-dimensional model for the temperature field is also derived and presented. This project proposes two novel numerical strategies for the solution of Hele-Shaw equation. The first one is based on an Eulerian formulation of the Smoothed Particle Hydrodynamics method, where the particles used in the discretization do not move along as the simulation evolves, thereby avoing the use of meshes. In the second strategy, local active dual patches are constructed on-the-fly for each active point to form a dynamic virtual mesh of active elements that evolves with the moving interface, then the Control Volume Finite Element Method is applied for the pressure field approximation. A dynamic approach of the semi-Lagrangian scheme is applied to the solution of the two-dimensional temperature equation. The project also assesses three new approaches for the treatment of the free surface of the fluid flow. Two of them are based on the Volume of Fluid technique and one of them is a meshless adaptation of the Front-Tracking method. The non-Newtonian behavior is characterized by a family of generalized viscosity models. Supporting numerical tests and performance studies, which assess the accuracy and the reliability of the proposed methodologies, are conducted

In this thesis, we develop partial differential equation (PDE) based numerical methods to solve certain optimal stochastic control problems in finance. The value of a stochastic control problem is normally identical to the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation or an HJB variational inequality. The HJB equation corresponds to the case when the controls are bounded while the HJB variational inequality corresponds to the unbounded control case. As a result, the solution to the stochastic control problem can be computed by solving the corresponding HJB equation/variational inequality as long as the convergence to the viscosity solution is guaranteed. We develop a unified numerical scheme based on a semi-Lagrangian timestepping for solving both the bounded and unbounded stochastic control problems as well as the discrete cases where the controls are allowed only at discrete times. Our scheme has the following useful properties: it is unconditionally stable; it can be shown rigorously to converge to the viscosity solution; it can easily handle various stochastic models such as jump diffusion and regime-switching models; it avoids Policy type iterations at each mesh node at each timestep which is required by the standard implicit finite difference methods. In this thesis, we demonstrate the properties of our scheme by valuing natural gas storage facilities---a bounded stochastic control problem, and pricing variable annuities with guaranteed minimum withdrawal benefits (GMWBs)---an unbounded stochastic control problem. In particular, we use an impulse control formulation for the unbounded stochastic control problem and show that the impulse control formulation is more general than the singular control formulation previously used to price GMWB contracts.

In this thesis, we develop partial differential equation (PDE) based numerical methods to solve certain optimal stochastic control problems in finance. The value of a stochastic control problem is normally identical to the viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation or an HJB variational inequality. The HJB equation corresponds to the case when the controls are bounded while the HJB variational inequality corresponds to the unbounded control case. As a result, the solution to the stochastic control problem can be computed by solving the corresponding HJB equation/variational inequality as long as the convergence to the viscosity solution is guaranteed. We develop a unified numerical scheme based on a semi-Lagrangian timestepping for solving both the bounded and unbounded stochastic control problems as well as the discrete cases where the controls are allowed only at discrete times. Our scheme has the following useful properties: it is unconditionally stable; it can be shown rigorously to converge to the viscosity solution; it can easily handle various stochastic models such as jump diffusion and regime-switching models; it avoids Policy type iterations at each mesh node at each timestep which is required by the standard implicit finite difference methods. In this thesis, we demonstrate the properties of our scheme by valuing natural gas storage facilities---a bounded stochastic control problem, and pricing variable annuities with guaranteed minimum withdrawal benefits (GMWBs)---an unbounded stochastic control problem. In particular, we use an impulse control formulation for the unbounded stochastic control problem and show that the impulse control formulation is more general than the singular control formulation previously used to price GMWB contracts.

Fundação Carlos Chagas Filho de Amparo a Pesquisa do Estado do Rio de Janeiro / Em suma, esta tese propõe uma metodologia de acompanhamento de interfaces móveis que baseia-se no método dos conjuntos de nível aqui chamado de método das curvas de nível, uma denominação baseada nas aplicações em que as interfaces são representadas por curvas acoplado a uma implementação semi-Lagrangeana, para problemas em meios porosos. Embora esta técnica possa, em princípio, ser aplicada a qualquer problema físico que apresente uma interface móvel, nesta tese são focados escoamentos em meios porosos consolidados e saturados por um ou dois fluidos imiscíveis e incompressíveis. Adicionalmente, um método iterativo paralelizável para a resolução de sistemas de equações lineares definidos em redes, que podem ser reduzidos à forma das equações fundamentais de equilíbrio, é empregado na determinação dos campos de velocidade associados aos escoamentos em meios porosos. O cenário semi-Lagrangeano acoplado ao método das curvas de nível é comparado com a implementação utilizando o bem conhecido esquema up-wind. Um exaustivo estudo realizado revela a superioridade da metodologia proposta frente à concorrente utilizando o up-wind. Finalmente, o método das curvas de nível com implementação semi-Lagrangeana (método semi-Lagrangeano das curvas de nível), e o método iterativo para a determinação do campo de velocidades são aplicados no estudo de problemas transientes em meios porosos que apresentam instabilidades dos tipos Saffman-Taylor e Rayleigh-Taylor. Este estudo envolve uma análise de estabilidade linear, a introdução de diversas perturbações trigonométricas na interface e a sua evolução não-linear. / Briefy, this thesis proposes a method for capturing moving interfaces based on the level set method coupled to a Semi-Lagrangian implementation for problems in porous media. Although this method could, in principle, be applied to any physical problem with moving interfaces, we foccus, in this thesis, on flows inside a consolidated porous media saturated by one or two imiscible and incompressible fluids. Besides, a parallelizable iterative method for solving linear systems defined on a network that can be reduced to the fundamental equilibrium equations, is employed to determine the velocity field associated with the flow in a porous medium. The semi-Lagrangian scheme coupled with the level set method is compared with the well-known implementation with the up-wind scheme. An exhaustive study is performed and reveals the superiority of the proposed scheme in relation to the competing one using the up-wind method. Finally, the level set method with semi-Lagrangian implementation and the iterative method for determining the velocity field are applied to the study of transient problems in porous media which present Saffman-Taylor and Rayleigh-Taylor instabilities. This study involves the application of a linear stability analysis, the introduction of several trigonometric perturbations to the interface and its non-linear evolution.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Os escoamentos altamente convectivos representam um desafio na simulação pelo método de elementos finitos. Com a solução de elementos finitos de Galerkin para escoamentos incompressíveis, a matriz associada ao termo convectivo é não simétrica, e portanto, a propiedade de aproximação ótima é perdida. Na prática as soluções apresentam oscilações espúrias. Muitos métodos foram desenvolvidos com o fim de resolver esse problema. Neste trabalho apresentamos um método semi- Lagrangeano, o qual é implicitamente um método do tipo upwind, que portanto resolve o problema anterior, e comparamos o desempenho do método na solução das equações de convecção-difusão e Navier-Stokes incompressível com o Streamline Upwind Petrov Galerkin (SUPG), um método estabilizador de reconhecido desempenho. No SUPG, as funções de forma e de teste são tomadas em espaços diferentes, criando um efeito tal que as oscilações espúrias são drasticamente atenuadas. O método semi-Lagrangeano é um método de fator de integração, no qual o fator é um operador de convecção que se desloca para um sistema de coordenadas móveis no fluido, mas restabelece o sistema de coordenadas Lagrangeanas depois de cada passo de tempo. Isto prevê estabilidade e a possibilidade de utilizar passos de tempo maiores.Existem muitos trabalhos na literatura analisando métodos estabilizadores, mas não assim com o método semi-Lagrangeano, o que representa a contribuição principal deste trabalho: reconhecer as virtudes e as fraquezas do método semi-Lagrangeano em escoamentos dominados pelo fenômeno de convecção. / Convection dominated flows represent a challenge for finite element method simulation. Many methods have been developed to address this problem. In this work we compare the performance of two methods in the solution of the convectiondiffusion and Navier-Stokes equations on environmental flow problems: the Streamline Upwind Petrov Galerkin (SUPG) and the semi-Lagrangian method. In Galerkin finite element methods for fluid flows, the matrix associated with the convective term is non-symmetric, and as a result, the best approximation property is lost. In practice, solutions are often corrupted by espurious oscillations. In this work, we present a semi- Lagrangian method, which is implicitly an upwind method, therefore solving the spurious oscillations problem, and a comparison between this semi-Lagrangian method and the Streamline Upwind Petrov Galerkin (SUPG), an stabilizing method of recognized performance. The SUPG method takes the interpolation and the weighting functions in different spaces, creating an effect so that the spurious oscillations are drastically attenuated. The semi-Lagrangean method is a integration factor method, in which the factor is an operator that shifts to a coordinate system that moves with the fluid, but it resets the Lagrangian coordinate system after each time step. This provides stability and the possibility to take bigger time steps. There are many works in the literature analyzing stabilized methods, but they do not analyze the semi-Lagrangian method, which represents the main contribution of this work: to recognize the strengths and weaknesses of the semi-Lagrangian method in convection dominated flows.