Neste trabalho desenvolveu-se e implementou-se um modelo computacional paralelo multifísica para a simulação do transporte de substâncias e do escoamento hidrodinâmico, bidimensional (2D) e tridimensional (3D), em corpos de água. Sua motivação está centrada no fato de que as margens e zonas costeiras de rios, lagos, estuários, mares e oceanos são locais de aglomerações de seres humanos, dada a sua importância para as atividades econômica, de transporte e de lazer, causando desequilíbrios a esses ecossistemas. Esse fato impulsiona o desenvolvimento de pesquisas relativas a esta temática. Portanto, o objetivo deste trabalho é o de construir um modelo computacional com alta qualidade numérica, que possibilite simular os comportamentos da hidrodinâmica e do transporte escalar de substâncias em corpos de água com complexa configuração geométrica, visando a contribuir para seu manejo racional. Visto que a ênfase nessa tese são os aspectos numéricos e computacionais dos algoritmos, analisaram-se as características e propriedades numérico-computacionais que as soluções devem contemplar, tais como a estabilidade, a monotonicidade, a positividade e a conservação da massa. As estratégias de soluções enfocam os termos advectivos e difusivos, horizontais e verticais, da equação do transporte. Desse modo, a advecção horizontal é resolvida empregando o método da limitação dos fluxos de Sweby, e o transporte vertical (advecção e difusão) é resolvido com os métodos beta de Gross e de Crank-Nicolson. São empregadas malhas com distintas resoluções para a solução do problema multifísica. O esquema numérico resultante é semi-implícito, computacionalmente eficiente, estável e fornece acurácia espacial e temporal de segunda ordem. Os sistemas de equações resultantes da discretização, em diferenças finitas, das equações do escoamento e do transporte 3D, são de grande porte, lineares, esparsos e simétricos definidos-positivos (SDP). No caso 2D os sistemas são lineares, mas os sistemas de equações para a equação do transporte não são simétricos. Assim, para a solução de sistemas de equações SDP e dos sistemas não simétricos empregam-se, respectivamente, os métodos do subespaço de Krylov do gradiente conjugado e do resíduo mínimo generalizado. No caso da solução dos sistemas 3-diagonal, utiliza-se o algoritmo de Thomas e o algoritmo de Cholesky. A solução paralela foi obtida sob duas abordagens. A decomposição ou particionamento de dados, onde as operações e os dados são distribuídos entre os processos disponíveis e são resolvidos em paralelo. E, a decomposição de domínio, onde obtém-se a solução do problema global combinando as soluções de subproblemas locais. Em particular, emprega-se neste trabalho, o método de decomposição de domínio aditivo de Schwarz, como método de solução, e como pré-condicionador. Para maximizar a relação computação/comunicação, visto que a eficiência computacional da solução paralela depende diretamente do balanceamento de carga e da minimização da comunicação entre os processos, empregou-se algoritmos de particionamento de grafos para obter localmente os subproblemas, ou as partes dos dados. O modelo computacional paralelo resultante mostrou-se computacionalmente eficiente e com alta qualidade numérica. / A multi-physics parallel computational model was developed and implemented for the simulation of substance transport and for the two-dimensional (2D) and threedimensional (3D) hydrodynamic flow in water bodies. The motivation for this work is focused in the fact that the margins and coastal zones of rivers, lakes, estuaries, seas and oceans are places of human agglomeration, because of their importance for economic, transport, and leisure activities causing ecosystem disequilibrium. This fact stimulates the researches related to this topic. Therefore, the goal of this work is to build a computational model of high numerical quality, that allows the simulation of hydrodynamics and of scalar transport of substances behavior in water bodies of complex configuration, aiming at their rational management. Since the focuses of this thesis are the numerical and computational aspects of the algorithms, the main numerical-computational characteristics and properties that the solutions need to fulfill were analyzed. That is: stability, monotonicity, positivity and mass conservation. Solution strategies focus on advective and diffusive terms, horizontal and vertical terms of the transport equation. In this way, horizontal advection is solved using Sweby’s flow limiting method; and the vertical transport (advection and diffusion) is solved with Gross and Crank-Nicolson’s beta methods. Meshes of different resolutions are employed in the solution of the multi-physics problem. The resulting numerical scheme is semi-implicit, computationally efficient, stable and provides second order accuracy in space and in time. The equation systems resulting of the discretization, in finite differences, of the flow and 3D transport are of large scale, linear, sparse and symmetric positive definite (SPD). In the 2D case, the systems are linear, but the equation systems for the transport equation are not symmetric. Therefore, for the solution of SPD equation systems and of the non-symmetric systems we employ, respectively, the methods of Krylov’s sub-space of the conjugate gradient and of the generalized minimum residue. In the case of the solution of 3-diagonal systems, Thomas algorithm and Cholesky algorithm are used. The parallel solution was obtained through two approaches. In data decomposition or partitioning, operation and data are distributed among the processes available and are solved in parallel. In domain decomposition the solution of the global problem is obtained combining the solutions of the local sub-problems. In particular, in this work, Schwarz additive domain decomposition method is used as solution method and as preconditioner. In order to maximize the computation/communication relation, since the computational efficiency of the parallel solution depends directly of the load balancing and of the minimization of the communication between processes, graph-partitioning algorithms were used to obtain the sub-problems or part of the data locally. The resulting parallel computational model is computationally efficient and of high numerical quality.

Estimativas de erro são estabelecidas em termos do número de Rossby para a aproximação de Galerkin não linear nas soluções do modelo atmosférico balanceado de Lorenz com massa forçante. Desse modo a aproximação espectral da aproximação de Galerkin não linear é ligada ao número de Rossby. / Error estimates are established in terms of the Rossby number for a nonlinear Galerkin approximation to the solutions of the balanced atmosphere model of Lorenz with mass forcing. Thereby, the approximation spectral dimension of the nonlinear Galerkin approximation is linked to the Rossby number.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-11T21:36:37Z No. of bitstreams: 1 HALLYSON GUSTAVO GUEDES DE MORAIS LIMA - DISSERTAÇÃO PPGMAT 2007..pdf: 2962280 bytes, checksum: 027c0c4dc68684f41c7b168cacb0b228 (MD5) / Made available in DSpace on 2018-07-11T21:36:37Z (GMT). No. of bitstreams: 1 HALLYSON GUSTAVO GUEDES DE MORAIS LIMA - DISSERTAÇÃO PPGMAT 2007..pdf: 2962280 bytes, checksum: 027c0c4dc68684f41c7b168cacb0b228 (MD5) Previous issue date: 2007-03 / CNPq / Neste trabalho deduzimos o sistema de Equações de Águas Rasas na forma Lagrangeana e obtemos a sua solução analítica. Aplicamos o Método Espectral na análise numérica deste sistema e mostramos que a propagação de ondas de águas rasas não depende do meio em que ela se propaga. / In this work we deduce the system of Shallow Water Equations in the Lagrangian form and we obtain its analytical solution. We have applied the spectral method in the numerical analysis of this system and we have shown that the propagation of the shallow water waves doesn't depend on the medium in which it spreads.

Neste trabalho desenvolveu-se e implementou-se um modelo computacional paralelo multifísica para a simulação do transporte de substâncias e do escoamento hidrodinâmico, bidimensional (2D) e tridimensional (3D), em corpos de água. Sua motivação está centrada no fato de que as margens e zonas costeiras de rios, lagos, estuários, mares e oceanos são locais de aglomerações de seres humanos, dada a sua importância para as atividades econômica, de transporte e de lazer, causando desequilíbrios a esses ecossistemas. Esse fato impulsiona o desenvolvimento de pesquisas relativas a esta temática. Portanto, o objetivo deste trabalho é o de construir um modelo computacional com alta qualidade numérica, que possibilite simular os comportamentos da hidrodinâmica e do transporte escalar de substâncias em corpos de água com complexa configuração geométrica, visando a contribuir para seu manejo racional. Visto que a ênfase nessa tese são os aspectos numéricos e computacionais dos algoritmos, analisaram-se as características e propriedades numérico-computacionais que as soluções devem contemplar, tais como a estabilidade, a monotonicidade, a positividade e a conservação da massa. As estratégias de soluções enfocam os termos advectivos e difusivos, horizontais e verticais, da equação do transporte. Desse modo, a advecção horizontal é resolvida empregando o método da limitação dos fluxos de Sweby, e o transporte vertical (advecção e difusão) é resolvido com os métodos beta de Gross e de Crank-Nicolson. São empregadas malhas com distintas resoluções para a solução do problema multifísica. O esquema numérico resultante é semi-implícito, computacionalmente eficiente, estável e fornece acurácia espacial e temporal de segunda ordem. Os sistemas de equações resultantes da discretização, em diferenças finitas, das equações do escoamento e do transporte 3D, são de grande porte, lineares, esparsos e simétricos definidos-positivos (SDP). No caso 2D os sistemas são lineares, mas os sistemas de equações para a equação do transporte não são simétricos. Assim, para a solução de sistemas de equações SDP e dos sistemas não simétricos empregam-se, respectivamente, os métodos do subespaço de Krylov do gradiente conjugado e do resíduo mínimo generalizado. No caso da solução dos sistemas 3-diagonal, utiliza-se o algoritmo de Thomas e o algoritmo de Cholesky. A solução paralela foi obtida sob duas abordagens. A decomposição ou particionamento de dados, onde as operações e os dados são distribuídos entre os processos disponíveis e são resolvidos em paralelo. E, a decomposição de domínio, onde obtém-se a solução do problema global combinando as soluções de subproblemas locais. Em particular, emprega-se neste trabalho, o método de decomposição de domínio aditivo de Schwarz, como método de solução, e como pré-condicionador. Para maximizar a relação computação/comunicação, visto que a eficiência computacional da solução paralela depende diretamente do balanceamento de carga e da minimização da comunicação entre os processos, empregou-se algoritmos de particionamento de grafos para obter localmente os subproblemas, ou as partes dos dados. O modelo computacional paralelo resultante mostrou-se computacionalmente eficiente e com alta qualidade numérica. / A multi-physics parallel computational model was developed and implemented for the simulation of substance transport and for the two-dimensional (2D) and threedimensional (3D) hydrodynamic flow in water bodies. The motivation for this work is focused in the fact that the margins and coastal zones of rivers, lakes, estuaries, seas and oceans are places of human agglomeration, because of their importance for economic, transport, and leisure activities causing ecosystem disequilibrium. This fact stimulates the researches related to this topic. Therefore, the goal of this work is to build a computational model of high numerical quality, that allows the simulation of hydrodynamics and of scalar transport of substances behavior in water bodies of complex configuration, aiming at their rational management. Since the focuses of this thesis are the numerical and computational aspects of the algorithms, the main numerical-computational characteristics and properties that the solutions need to fulfill were analyzed. That is: stability, monotonicity, positivity and mass conservation. Solution strategies focus on advective and diffusive terms, horizontal and vertical terms of the transport equation. In this way, horizontal advection is solved using Sweby’s flow limiting method; and the vertical transport (advection and diffusion) is solved with Gross and Crank-Nicolson’s beta methods. Meshes of different resolutions are employed in the solution of the multi-physics problem. The resulting numerical scheme is semi-implicit, computationally efficient, stable and provides second order accuracy in space and in time. The equation systems resulting of the discretization, in finite differences, of the flow and 3D transport are of large scale, linear, sparse and symmetric positive definite (SPD). In the 2D case, the systems are linear, but the equation systems for the transport equation are not symmetric. Therefore, for the solution of SPD equation systems and of the non-symmetric systems we employ, respectively, the methods of Krylov’s sub-space of the conjugate gradient and of the generalized minimum residue. In the case of the solution of 3-diagonal systems, Thomas algorithm and Cholesky algorithm are used. The parallel solution was obtained through two approaches. In data decomposition or partitioning, operation and data are distributed among the processes available and are solved in parallel. In domain decomposition the solution of the global problem is obtained combining the solutions of the local sub-problems. In particular, in this work, Schwarz additive domain decomposition method is used as solution method and as preconditioner. In order to maximize the computation/communication relation, since the computational efficiency of the parallel solution depends directly of the load balancing and of the minimization of the communication between processes, graph-partitioning algorithms were used to obtain the sub-problems or part of the data locally. The resulting parallel computational model is computationally efficient and of high numerical quality.

Estimativas de erro são estabelecidas em termos do número de Rossby para a aproximação de Galerkin não linear nas soluções do modelo atmosférico balanceado de Lorenz com massa forçante. Desse modo a aproximação espectral da aproximação de Galerkin não linear é ligada ao número de Rossby. / Error estimates are established in terms of the Rossby number for a nonlinear Galerkin approximation to the solutions of the balanced atmosphere model of Lorenz with mass forcing. Thereby, the approximation spectral dimension of the nonlinear Galerkin approximation is linked to the Rossby number.

Neste trabalho desenvolveu-se e implementou-se um modelo computacional paralelo multifísica para a simulação do transporte de substâncias e do escoamento hidrodinâmico, bidimensional (2D) e tridimensional (3D), em corpos de água. Sua motivação está centrada no fato de que as margens e zonas costeiras de rios, lagos, estuários, mares e oceanos são locais de aglomerações de seres humanos, dada a sua importância para as atividades econômica, de transporte e de lazer, causando desequilíbrios a esses ecossistemas. Esse fato impulsiona o desenvolvimento de pesquisas relativas a esta temática. Portanto, o objetivo deste trabalho é o de construir um modelo computacional com alta qualidade numérica, que possibilite simular os comportamentos da hidrodinâmica e do transporte escalar de substâncias em corpos de água com complexa configuração geométrica, visando a contribuir para seu manejo racional. Visto que a ênfase nessa tese são os aspectos numéricos e computacionais dos algoritmos, analisaram-se as características e propriedades numérico-computacionais que as soluções devem contemplar, tais como a estabilidade, a monotonicidade, a positividade e a conservação da massa. As estratégias de soluções enfocam os termos advectivos e difusivos, horizontais e verticais, da equação do transporte. Desse modo, a advecção horizontal é resolvida empregando o método da limitação dos fluxos de Sweby, e o transporte vertical (advecção e difusão) é resolvido com os métodos beta de Gross e de Crank-Nicolson. São empregadas malhas com distintas resoluções para a solução do problema multifísica. O esquema numérico resultante é semi-implícito, computacionalmente eficiente, estável e fornece acurácia espacial e temporal de segunda ordem. Os sistemas de equações resultantes da discretização, em diferenças finitas, das equações do escoamento e do transporte 3D, são de grande porte, lineares, esparsos e simétricos definidos-positivos (SDP). No caso 2D os sistemas são lineares, mas os sistemas de equações para a equação do transporte não são simétricos. Assim, para a solução de sistemas de equações SDP e dos sistemas não simétricos empregam-se, respectivamente, os métodos do subespaço de Krylov do gradiente conjugado e do resíduo mínimo generalizado. No caso da solução dos sistemas 3-diagonal, utiliza-se o algoritmo de Thomas e o algoritmo de Cholesky. A solução paralela foi obtida sob duas abordagens. A decomposição ou particionamento de dados, onde as operações e os dados são distribuídos entre os processos disponíveis e são resolvidos em paralelo. E, a decomposição de domínio, onde obtém-se a solução do problema global combinando as soluções de subproblemas locais. Em particular, emprega-se neste trabalho, o método de decomposição de domínio aditivo de Schwarz, como método de solução, e como pré-condicionador. Para maximizar a relação computação/comunicação, visto que a eficiência computacional da solução paralela depende diretamente do balanceamento de carga e da minimização da comunicação entre os processos, empregou-se algoritmos de particionamento de grafos para obter localmente os subproblemas, ou as partes dos dados. O modelo computacional paralelo resultante mostrou-se computacionalmente eficiente e com alta qualidade numérica. / A multi-physics parallel computational model was developed and implemented for the simulation of substance transport and for the two-dimensional (2D) and threedimensional (3D) hydrodynamic flow in water bodies. The motivation for this work is focused in the fact that the margins and coastal zones of rivers, lakes, estuaries, seas and oceans are places of human agglomeration, because of their importance for economic, transport, and leisure activities causing ecosystem disequilibrium. This fact stimulates the researches related to this topic. Therefore, the goal of this work is to build a computational model of high numerical quality, that allows the simulation of hydrodynamics and of scalar transport of substances behavior in water bodies of complex configuration, aiming at their rational management. Since the focuses of this thesis are the numerical and computational aspects of the algorithms, the main numerical-computational characteristics and properties that the solutions need to fulfill were analyzed. That is: stability, monotonicity, positivity and mass conservation. Solution strategies focus on advective and diffusive terms, horizontal and vertical terms of the transport equation. In this way, horizontal advection is solved using Sweby’s flow limiting method; and the vertical transport (advection and diffusion) is solved with Gross and Crank-Nicolson’s beta methods. Meshes of different resolutions are employed in the solution of the multi-physics problem. The resulting numerical scheme is semi-implicit, computationally efficient, stable and provides second order accuracy in space and in time. The equation systems resulting of the discretization, in finite differences, of the flow and 3D transport are of large scale, linear, sparse and symmetric positive definite (SPD). In the 2D case, the systems are linear, but the equation systems for the transport equation are not symmetric. Therefore, for the solution of SPD equation systems and of the non-symmetric systems we employ, respectively, the methods of Krylov’s sub-space of the conjugate gradient and of the generalized minimum residue. In the case of the solution of 3-diagonal systems, Thomas algorithm and Cholesky algorithm are used. The parallel solution was obtained through two approaches. In data decomposition or partitioning, operation and data are distributed among the processes available and are solved in parallel. In domain decomposition the solution of the global problem is obtained combining the solutions of the local sub-problems. In particular, in this work, Schwarz additive domain decomposition method is used as solution method and as preconditioner. In order to maximize the computation/communication relation, since the computational efficiency of the parallel solution depends directly of the load balancing and of the minimization of the communication between processes, graph-partitioning algorithms were used to obtain the sub-problems or part of the data locally. The resulting parallel computational model is computationally efficient and of high numerical quality.

This Ph.D. thesis aims to develop numerical models for two-dimensional and three-dimensional shallow water systems over mobile beds. In order to accomplish the goal of this dissertation, the following sub-projects are defined and completed. 1: The first sub-project consists in developing a robust two-dimensional coupled numerical model based on an unstructured mesh, which can simulate rapidly varying flows over an erodible bed involving wet–dry fronts that is a complex yet practically important problem. In this task, the central-upwind scheme is extended to simulation of bed erosion and sediment transport, a modified shallow water system is adopted to improve the model, a wetting and drying scheme is proposed for tracking wet-dry interfaces and stably predict the bed erosion near wet-dry area. The shallow water, sediment transport and bed evolution equations are coupled in the governing system. The proposed model can efficiently track wetting and drying interfaces while preserving stability in simulating the bed erosion near the wet-dry fronts. The additional terms in shallow water equations can improve the accuracy of the simulation when intense sediment-exchange exists; the central-upwind method adopted in the current study shows great accuracy and efficiency compared with other popular solvers; the developed model is robust, efficient and accurate in dealing with various challenging cases. 2: The second sub-project consists in developing a novel numerical scheme for a coupled two-dimensional hyperbolic system consisting of the shallow water equations with friction terms coupled with the equations modeling the sediment transport and bed evolution. The resulting 5*5 hyperbolic system of balance laws is numerically solved using a Godunov-type central-upwind scheme on a triangular grid. A spatially second-order and temporally third-order central-upwind scheme has been derived to discretize the conservative hyperbolic sub-system. However, such schemes need a correct evaluation of local wave speeds to avoid instabilities. To address such an issue, a mathematical result by the Lagrange theorem is used in the proposed scheme. Consequently, a computationally expensive process of finding all of the eigenvalues of the Jacobian matrices is avoided: The upper/lower bounds on the largest/smallest local speeds of propagation are estimated using the Lagrange theorem. In addition, a special discretization of the bed-slope term is proposed to guarantee the well-balanced property of the designed scheme. 3: The third sub-project consists in designing a novel scheme to estimate bed-load fluxes which can produce more accurate results than the previously reported coupled model. Using a pair of local wave speeds different from those used for the flow, a novel wave estimator in conjunction with the central upwind method is proposed and successfully applied to the coupled water-sediment system involving a rapid bed-erosion process. It was demonstrated that, in comparison with the decoupled model, applying the proposed novel scheme to approximate the bed-load fluxes can successfully avoid the numerical oscillations caused by simple and less stable schemes, e.g. simple upwind methods; in comparison with the coupled model using same flux-estimator for both hydrodynamic and morphological systems, the proposed numerical scheme successfully prevents excessive numerical diffusion for prediction of bed evolution. Consequently, the proposed scheme has advantages in terms of accuracy which are shown in several numerical tests. In addition, analytical expressions have been provided for calculating the eigenvalues of the coupled shallow-water-Exner system, which greatly enhances the efficiency of the proposed method. 4: The fourth sub-project consists in developing a three-dimensional numerical model for the simulation of unsteady non-hydrostatic shallow water flows on unstructured grids using the finite volume method. The free surface variations are modeled by a characteristics-based scheme which simulates sub- and super-critical flows. Three-dimensional velocity components are considered in a collocated arrangement with a sigma coordinate system. A special treatment of the pressure term is developed to avoid the water surface oscillations. Convective and diffusive terms are approximated explicitly, and an implicit discretization is used for the pressure term. The unstructured grid in the horizontal direction and the sigma coordinate in the vertical direction facilitate the use of the model in complicated geometries. 5: The fifth sub-project consists in developing a well-balanced three-dimensional shallow water model which is able to simulate shock waves over dry bed. Due to the hydrostatic simplification of the vertical momentum equation, the governing system of equations is not hyperbolic and can not be solved using standard hyperbolic solvers. That is, one can not use a high-order Godunov-type scheme to compute all fluxes through cell-interfaces. This may cause the model to fail in simulations of some unsteady-flows with discontinuities, e.g., dam-break flows and floods. To overcome this difficulty, a novel numerical scheme for the three-dimensional shallow water equations is proposed using a relaxation approach in order to convert the system to a hyperbolic one. Thus, a high-order Godunov-type central-upwind scheme based on the finite volume method can be applied to approximate the numerical fluxes. The proposed model can also preserve the ``lake at rest'' state and positivity of water depth over irregular bottom topographies based on special reconstruction of the corresponding parameters. 6: The sixth sub-project consists in extending the result of the fifth sub-project to development of a three-dimensional numerical model for shallow water flows over mobile beds, which is able to simulate morphological evolutions under shock waves, e.g. dam-break flows. The hydrodynamic model solves the three-dimensional shallow water equations using a finite volume method on prismatic cells in sigma coordinates based on the scheme prposed in sub-project 5. The morphodynamic model solves an Exner equation consisting of bed-load sediment transportation. The performance of the proposed model has been demonstrated by several laboratory experiments of dam-break flows over mobile beds.

Cette thèse est consacrée à l’étude théorique ainsi qu’au traitement numérique de fluides incompressibles à surface libre. La première partie concerne un système d’équations appelé le système de Green–Naghdi. Comme le système de Saint-Venant, il s’agit d’une approximation d’eaux peu-profondes du problème de Zakharov. La différence est que le système de Green–Naghdi est d’un degré plus élevé en ordre d’approximation. C’est pourquoi il contient tous les termes du système de Saint-Venant plus de termes d’ordre trois non-linéairement dispersives. Autrement dit, le système de Green–Naghdi peut être vu comme une perturbation dispersive du système de Saint-Venant. Ce dernier système étant hyperbolique, il entre dans le cadre classique développé pour des systèmes hyperboliques. En particulier, il est entropique (au sense de Lax) et symétrique. On peut donc lui appliquer les résultats d’existence et d’unicité bien connus pour des systèmes hyperboliques. Dans la première partie de ce travail, on généralise la notion de symétrie à une classe plus générale de systèmes contenant le système de Green–Naghdi. Ceci nous permet de symétriser les équations de Green–Naghdi et d’utiliser la symétrie obtenue pour déduire un résultat d’existence globale après avoir ajouté un terme dissipative d’ordre 2 au système. Ceci est fait en adaptant l’approche utilisée dans la littérature pour des systèmes hyperboliques. La deuxième partie de ce travail concerne le traitement numérique des équations de Navier–Stokes à surface libre avec un terme de tension de surface. Ici, la surface libre est modélisée en utilisant la formulation des lignes de niveaux. C’est pourquoi la condition cinématique (condition de l’évolution de surface libre) s’écrit sous la forme d’une équation d’advection satisfaite par la fonction de ligne de niveaux. Cette équation est résolue sur une domaine de calcul contenant strictement le domaine de fluide, sur de petits sous-intervalles du temps. Chaque itération de l’algorithme global correspond donc à l’advection du domaine du fluide sur le sous-intervalle du temps associé et ensuite de résoudre le système de Navier–Stokes discrétisé en temps sur le domaine du fluide. Cette discrétisation en temps est faite par la méthode des caractéristiques. L’outil clé qui nous permet de résoudre ce système uniquement sur le domaine du fluide est l’adaptation de maillage anisotrope. Plus précisément, à chaque itération le maillage est adapté au domaine du fluide tel que l’erreur d’approximation et l’erreur géométrique soient raisonnablement petites au voisinage du domaine du fluide. La résolution du problème discrétisé en temps sur le domaine du fluide est faite par l’algorithme d’Uzawa utilisé dans la cadre de la méthode des éléments finis. Par ailleurs, la condition de glissement de Navier est traité ici en ajoutant un terme de pénalisation à la formulation variationnelle associée. / This thesis is about theoretical study and numerical treatment of some problems raised in incompressible free-surface fluid dynamics. The first part concerns a model called the Green–Naghdi (GN) equations. Similarly to the non linear shallow water system (called also Saint-Venant system), the Green–Naghdi equations is a shallow water approximation of water waves problem. Indeed, GN equation is one order higher in approximation compared to Saint-Venant system. For this reason, it contains all the terms of Saint-Venant system in addition to some non linear third order dispersive terms. In other words, the GN equations is a dispersive perturbation of the Saint-Venant system. The latter system is hyperbolic and fits the general framework developed in the literature for hyperbolic systems. Particularly, it is entropic (in the sense of Lax) and symmertizable. Therefore, we can apply the well-posedness results known for symmetric hyperbolic system. During the first part of this work, we generalize the notion of symmetry to a more general type of equations including the GN system. This lets us to symmetrize the GN equation. Then, we use the suggested symmetric structure to obtain a global existence result for the system with a second order dissipative term by adapting the approach classically used for hyperbolic systems. The second part of this thesis concerns the numerical treatment of the free surface incompressible Navier–Stokes equation with surface tension. We use the level set formulation to represent the fluid free-surface. Thanks to this formulation, the kinematic boundary condition is treated by solving an advection equation satisfied by the level set function. This equation is solved on a computational domain containing the fluid domain over small time subintervals. Each iteration of the algorithm corresponds to the adevction of the fluid domain on a small time subinterval and to solve the time-discretized Navier–Stokes equations only on the fluid domain. The time discretization of the Navier–Stokes equation is done by the characteristic method. Then, the key tool which lets us solve this equation on the fluid domain is the anisotropic mesh adaptation. Indeed, at each iteration the mesh is adapted to the fluid domain such that we get convenient approximation and geometric errors in the vicinity of the fluid domain. This resolution is done using the Uzawa algorithm for a convenient finite element method. The slip boundary conditions are considered by adding a penalization term to the variational formulation associated to the problem.

In this thesis we study the properties of exoplanetary atmospheres. The first part describes methods and instruments for searching exoplanets, statistics of discovered exoplanets and the sampling factor. The second part describes the properties of chosen planets and moons in the Solar system (Venus, Mars and Titan) and also possible properties of the exoplanetary atmospheres that are only briefly understood. The third part describes the atmospheric models which incorporate full 3D model of the atmosphere, dynamical core, shallow-water model and 1D spherically-symmetric model. We also show the results of exoplanetary atmospheric models published in the scientific journals. This part also describes the icosahedral geodetic grid that is advantageous for the global climatic models, and also discretisation on sphere and the application of the operators (gradient, divergence, vorticity) on geodetic grid. The fourth part discusses results of the numerical solution of the atmospheric circulation with the forcing on geodetic grid. In this part we also show global maps of the variables after a particular time of the numerical integration and also the evolution of the variables at chosen points in time. In the discussion part we examine the results of our program. The results of the numerical integrations (chosen...

Dans ce manuscrit, nous parlons des matériaux composites à matrice céramiques (CMCs) qui sont envisagés pour intégrer les chambres de combustion de futurs moteurs aéronautiques civils. Pour faire face des conditions extrêmes, ces matériaux possèdent la particularité de s’auto-protéger vis-à-vis de l’oxydation par la formation d’un oxyde passivant qui limite la diffusion des espèces oxydantes au sein des fissures matricielles. Nous modélisons l’écoulement d’un oxyde dans une fissure par l’équation de Navier-Stokes, puis les mettons sous forme non dimensionnelles, et les dérivations de deux types de modèles sont intéressantes : les modèles de Saint-Venant et les modèles de lubrification. Ensuit nous nous engageons à chercher l’existence de solution faible de l’approximation de lubrification d’ordre 4 obtenue précédent dans le cas uni-dimensionnel. Enfin nous précisons la limite entre les équations de Saint-Venant et l’équation de lubrification. / In this work, we are interested in the ceramic matrix composite materials(CMCs) who will be used to integrate the combustion chambers of future civil aeronautical engines. To face extreme conditions, these materials possess the peculiarity to auto-protect itself towards the oxidation by the formation of an oxide passivate which limits the distribution of the oxidizing species within the matrix cracks. We model the flow of an oxide in a crack by the Navier-Stokes equation, then put them under an asymptotic analysis in order to get two types of asymptotic models : models of Saint-Venant (Shallow water model) and lubrication models. Next we are interested in looking for the existence of weak solution to the one-dimensional approximated lubrication equation of order 4 obtained before. Finally we talk about the limit between the Saint-Venant equations and the lubrication equation.