Coarse grid projection (CGP) methodology is a novel multigrid method for systems involving decoupled nonlinear evolution equations and linear elliptic Poisson equations. The nonlinear equations are solved on a fine grid and the linear equations are solved on a corresponding coarsened grid. Mapping operators execute data transfer between the grids. The CGP framework is constructed upon spatial and temporal discretization schemes. This framework has been established for finite volume/difference discretizations as well as explicit time integration methods. In this article we present for the first time a version of CGP for finite element discretizations, which uses a semi-implicit time integration scheme. The mapping functions correspond to the finite-element shape functions. With the novel data structure introduced, the mapping computational cost becomes insignificant. We apply CGP to pressure correction schemes used for the incompressible Navier Stokes flow computations. This version is validated on standard test cases with realistic boundary conditions using unstructured triangular meshes. We also pioneer investigations of the effects of CGP on the accuracy of the pressure field. It is found that although CGP reduces the pressure field accuracy, it preserves the accuracy of the pressure gradient and thus the velocity field, while achieving speedup factors ranging from approximately 2 to 30. Exploring the influence of boundary conditions on CGP, the minimum speedup occurs for velocity Dirichlet boundary conditions, while the maximum speedup occurs for open boundary conditions. We discuss the CGP method as a guide for partial mesh refinement of incompressible flow computations and show its application for simulations of flow over a backward facing step and flow past a cylinder. / Master of Science

Thesis (MEng)--University of Stellenbosch, 2000. / ENGLISH ABSTRACT: This study investigates the implementation of the finite volume numerical method applicable to non-orthogonal control volumes and the application of the method to calculate the thermo-flow field within the collector area of a solar chimney power generating plant. The discretisation of the governing equations for the transient, Newtonian, incompressible and turbulent fluid flow, including heat transfer, is presented for a non-orthogonal coordinate frame. The standard k - E turbulence model, modified to include rough surfaces, is included and evaluated in the method. An implicit solution procedure (SIP-semi implicit procedure) as an alternative to a direct solution procedure for the calculation of the flow field on nonstaggered grids is investigated, presented and evaluated in this study. The Rhie and Chow interpolation practice was employed with the pressurecorrection equation to eliminate the presence of pressure oscillations on nonstaggered grids. The computer code for the solution of the three-dimensional thermo-flow fields is developed in FORTRAN 77. The code is evaluated against simple test cases for which analytical and experimental results exist. It is also applied to the analysis of the thermo-flow field of the air flow through a radial solar collector. KEYWORDS: NUMERICAL METHOD, FINITE VOLUME, NON-ORTHOGONAL, k+-e TURBULENCE MODEL, SIP / AFRIKAANSE OPSOMMING: Die studie ondersoek die implementering van 'n eindige volume numeriese metode van toepassing op nie-ortogonale kontrole volumes asook die toepassing van die metode om die termo-vloei veld binne die kollekteerder area van 'n sonskoorsteen krag aanleg te bereken. Die diskretisering van die behoudsvergelykings vir die tyd-afhanlike, Newtonse, onsamedrukbare en turbulente vloei, insluitende hitteoordrag, word beskryf vir 'n nie-ortogonale koordinaatstelsel. Die standaard k - E turbulensiemodel, aangepas om growwe oppervlakrandvoorwaardes te hanteer, is ingesluit en geevalueer in die studie. 'n Implisiete oplossings metode (SIP-semi implisiete prosedure) as alternatief vir 'n direkte oplossingsmetode is ondersoek en geimplimenteer vir die berekening van die vloeiveld met nie-verspringde roosters. 'n Rhie en Chow interpolasie metode is gebruik tesame met die drukkorreksie-vergelyking ten einde ossilasies in die drukveld in die nie-verspringde roosters te vermy. Die rekenaarkode vir die oplossing van die drie dimensionele termo-vloeiveld is ontwikkel in FORTRAN 77. Die kode is geevalueer teen eenvoudige toetsprobleme waarvoor analitiese en eksperimentele resultate bestaan. Die kode IS ook gebruik om die termo-vloeiveld binne 'n radiale son kollekteerder te analiseer. SLEUTELWOORDE: NUMERIESE METODE, EINDIGE VOLUME, NIE-ORTOGONAAL, k - E TURBULENSIE MODEL, SIP

This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equations for compressible flow with a right-hand side correction. This model is based on the work Fornasier et al. (2010). Due to the complexity of the model, we focus only on the one-dimensional case. For the numerical solution we use a semi-implicit discontinuous Galerkin method. Discretization of the right-hand side is chosen so that we preserve the structure of the semi-implicit scheme for the Euler equations presented in the work Feistauer, Kučera (2007). The proposed numerical scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. Powered by TCPDF (www.tcpdf.org)

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.

Dans ce travail, nous nous sommes intéressés à la mise en place de schémas semi-implicites pour l’amélioration des simulations numériques du déplacement d’un globule rouge dans le sang. Nous considérons la méthode levelset, où l’interface fluide-structure est représentée comme la ligne de niveau 0 d’une fonction auxiliaire et le couplage est effectué en ajoutant un terme source dans l’équation fluide.Le principe de ces schémas semi-implicites est de prédire la position et la forme de la structure par une équation de la chaleur et d’utiliser cette prédiction pour obtenir un terme de force dans l’équation fluide plus précis. Ce type de schémas semi-implicites a d’abord été mis en place dans le cadre d’un système diphasique ou d’une membrane élastique immergée afin d’utiliser un plus grand pas de temps que pour un couplage explicite. Cela a permis d’améliorer les conditions sur le pas de temps et ainsi augmenter l’efficacité globale de l’algorithme complet par rapport à un schéma explicite classique.Pour étendre ce raisonnement au cas d’un globule rouge, nous proposons un algorithme pour simuler le flot de Willmore en dimension 2 et 3. Notre méthode s’inspire des méthodes de mouvements d’interface générés par diffusion et nous arrivons à obtenir un flot non linéaire d’ordre 4 uniquement avec des résolutions d’équations de la chaleur. Pour assurer la conservation du volume et de l’aire d’un globule rouge, nous proposons ensuite une méthode de correction qui déplace légèrement l’interface afin de recoller aux contraintes.La combinaison des deux étapes précédentes décrit le comportement d’un globule rouge laissé au repos. Nous validons cette méthode en obtenant une formed’équilibre d’un globule rouge. Nous proposons enfin un schéma semi-implicite dans le cas d’un globule rouge qui ouvre la voie vers l’utilisation de cette méthode comme prédicteur de l’algorithme de couplage complet. / In this work, we propose new semi-implicit schemes to improve the numerical simulations of the motion of an immersed red blood cell. We consider the levelset method where the interface is described as the 0 isoline of an auxiliary function and the fluid-structure coupling is done by adding a source term in the fluid equation.The idea of these semi-implicit scheme is to predict the position and the shape of the structure through a heat equation and to use this prediction to improve the accuracy of the source term in the fluid equation. This type of semi-implicit scheme has firstly been implemented in the case of a multiphase flow and a immersed elastic membrane and has shown better temporal stability than an explicit scheme, resulting in an improved global efficiency.In order extend this method to the case of a red blood cell, we propose an algorithm to compute the Willmore flow in dimenson 2 and 3. In the spirit of the diffusion generated motion methods, our method simulate a non linear four order flow by only solving heat equations. To ensure the conservation of the volume and area of the the vesicle, we add to the method a correction step that slightly moves the interface so that we recover the constraints.Combnation of these two steps allows to compute the behavior of a red blood cell left at rest. We validate this method by obtaining the convergence to an equilibrium shape in both 2D and 3D. Finaly we introduce a semi-implicit scheme in the case of a red blood cell that shows how we can use this method as a prediction in the complete coupling model.

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.

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.

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.

Partial differential equations (PDEs) and stochastic partial differential equations (SPDEs) are powerful tools in modeling real-world phenomena in many fields such as geo-engineering. For instance processes such as oil or gas recovery from hydrocarbon reservoirs and mining heat from geothermal reservoirs can be modeled by PDEs or SPDEs. An important task is to understand the behavior of such phenomena. This can be achieved through explicit solutions of equations. Since explicit solutions of many PDEs and SPDEs are rarely known, developing numerical schemes is a good alternative to provide approximations of these explicit solutions. The study of numerical solutions of PDEs and SPDEs is therefore an active research area and has attracted a lot of attentions since at least two decades. The aims of this dissertation is to develop numerical schemes to approximate semilinear parabolic PDEs and SPDEs in space and in time. The approximation in space is done via the standard Galerkin finite element method and the approximation in time, which is the core of our work is done via various numerical integrators. This dissertation consists of two general parts. The first part of this thesis deals with autonomous PDEs and SPDEs. Here, our main interest is on semilinear PDEs and SPDEs where the nonlinear part is stronger than the linear part also called (stochastic) reactive dominated transport equations, or stiff problems. For such problems, many numerical techniques in the current scientific literature lose their good stability properties. We develop a new explicit exponential integrator (called exponential Rosenbrock-type method) and a new semi-implicit method (called linear implicit Rosenbrock-type method), appropriate for such PDEs and SPDEs. We analyze the strong convergence of our novel fully discrete schemes towards the mild solution of the (S)PDE and obtain convergence orders similar to existing ones in the literature. The second part of this thesis focuses on numerical approximations of semilinear non-autonomous parabolic PDEs and SPDEs. Such equations are more realistic than the autonomous ones and find applications in many fields such as fluid mechanics, quantum field theory, electromagnetism, etc. Numerics of non-autonomous semilinear parabolic PDEs and SPDEs are far from being well understood in the literature. We fill that gap in this thesis by developing a new exponential integrator (called Magnus-type method) and the semi-implicit method for such problems and provide their strong convergence towards the mild solution. Moreover, for both autonomous and non-autonomous SPDEs driven by additive noise, we achieve optimal convergence order in time 1 or approximately 1. Numerical simulations are provided to illustrate our theoretical findings in both autonomous and non-autonomous cases.

The heart is one the most important organs in the human body and many other live creatures. The electrical activity in the heart controls the heart function, and many heart diseases are linked to the abnormalities in the electrical activity in the heart. Mathematical equations and computer simulation can be used to model the electrical activity in the heart. The heart models are challenging to solve because of the complexity of the models and the huge size of the problems. Several cell models have been proposed to model the electrical activity in a single heart cell. These models must be coupled with a heart model to model the electrical activity in the entire heart. The bidomain model is a popular model to simulate the propagation of electricity in myocardial tissue. It is a continuum-based model consisting of non-linear ordinary differential equations (ODEs) describing the electrical activity at the cellular scale and a system of partial differential equations (PDEs) describing propagation of electricity at the tissue scale. Because of this multi-scale, ODE/PDE structure of the model, splitting methods that treat the ODEs and PDEs in separate steps are natural candidates as numerical methods. First, we need to solve the problem at the cellular scale using ODE solvers. One of the most popular methods to solve the ODEs is known as the Rush-Larsen (RL) method. Its popularity stems from its improved stability over integrators such as the forward Euler (FE) method along with its easy implementation. The RL method partitions the ODEs into two sets: one for the gating variables, which are treated by an exponential integrator, and another for the remaining equations, which are treated by the FE method. The success of the RL method can be understood in terms of its relatively good stability when treating the gating variables. However, this feature would not be expected to be of benefit on cell models for which the stiffness is not captured by the gating equations. We demonstrate that this is indeed the case on a number of stiff cell models. We further propose a new partitioned method based on the combination of a first-order generalization of the RL method with the FE method. This new method leads to simulations of stiff cell models that are often one or two orders of magnitude faster than the original RL method. After solving the ODEs, we need to use bidomain solvers to solve the bidomain model. Two well-known, first-order time-integration methods for solving the bidomain model are the semi-implicit method and the Godunov operator-splitting method. Both methods decouple the numerical procedure at the cellular scale from that at the tissue scale but in slightly different ways. The methods are analyzed in terms of their accuracy, and their relative performance is compared on one-, two-, and three-dimensional test cases. As suggested by the analysis, the test cases show that the Godunov method is significantly faster than the semi-implicit method for the same level of accuracy, specifically, between 5 and 15 times in the cases presented. Second-order bidomain solvers can generally be expected to be more effective than first-order bidomain solvers under normal accuracy requirements. However, the simplest and the most commonly applied second-order method for the PDE step, the Crank-Nicolson (CN) method, may generate unphysical oscillations. We investigate the performance of a two-stage, L-stable singly diagonally implicit Runge-Kutta method for solving the PDEs of the bidomain model and present a stability analysis. Numerical experiments show that the enhanced stability property of this method leads to more physically realistic numerical simulations compared to both the CN and Backward Euler (BE) methods.