A programação paralela tem sido freqüentemente adotada para o desenvolvimento de aplicações que demandam alto desempenho computacional. Com o advento das arquiteturas multi-cores e a existência de diversos níveis de paralelismo é importante definir estratégias de programação paralela que tirem proveito desse poder de processamento nessas arquiteturas. Neste contexto, este trabalho busca avaliar o desempenho da utilização das arquiteturas multi-cores, principalmente o oferecido pelas unidades de processamento gráfico (GPUs) e CPUs multi-cores na resolução de um problema térmico inverso. Algoritmos paralelos para a GPU e CPU foram desenvolvidos utilizando respectivamente as ferramentas de programação em arquiteturas de memória compartilhada NVIDIA CUDA (Compute Unified Device Architecture) e a API POSIX Threads. O algoritmo do método do gradiente conjugado pré-condicionado para resolução de sistemas lineares esparsos foi implementado totalmente no espaço da memória global da GPU em CUDA. O algoritmo desenvolvido foi avaliado em dois modelos de GPU, os quais se mostraram mais eficientes, apresentando um speedup de quatro vezes que a versão serial do algoritmo. A aplicação paralela em POSIX Threads foi avaliada em diferentes CPUs multi-cores com distintas microarquiteturas. Buscando um maior desempenho do código paralelizado foram utilizados flags de otimização as quais se mostraram muito eficientes na aplicação desenvolvida. Desta forma o código paralelizado com o auxílio das flags de otimização chegou a apresentar tempos de processamento cerca de doze vezes mais rápido que a versão serial no mesmo processador sem nenhum tipo de otimização. Assim tanto a abordagem utilizando a GPU como um co-processador genérico a CPU como a aplicação paralela empregando as CPUs multi-cores mostraram-se ferramentas eficientes para a resolução do problema térmico inverso. / Parallel programming has been frequently adopted for the development of applications that demand high-performance computing. With the advent of multi-cores architectures and the existence of several levels of parallelism are important to define programming strategies that take advantage of parallel processing power in these architectures. In this context, this study aims to evaluate the performance of architectures using multi-cores, mainly those offered by the graphics processing units (GPUs) and CPU multi-cores in the resolution of an inverse thermal problem. Parallel algorithms for the GPU and CPU were developed respectively, using the programming tools in shared memory architectures, NVIDIA CUDA (Compute Unified Device Architecture) and the POSIX Threads API. The algorithm of the preconditioned conjugate gradient method for solving sparse linear systems entirely within the global memory of the GPU was implemented by CUDA. It evaluated the two models of GPU, which proved more efficient by having a speedup was four times faster than the serial version of the algorithm. The parallel application in POSIX Threads was evaluated in different multi-core CPU with different microarchitectures. Optimization flags were used to achieve a higher performance of the parallelized code. As those were efficient in the developed application, the parallelized code presented processing times about twelve times faster than the serial version on the same processor without any optimization. Thus both the approach using GPU as a coprocessor to the CPU as a generic parallel application using the multi-core CPU proved to be more efficient tools for solving the inverse thermal problem.

La résolution de problèmes d’onde par une méthode d’éléments finis de frontière (BEM) conduit à des systèmes d’équations linéaires pleins dont la taille augmente très vite pour les applications pratiques. Il est alors impératif d’employer des méthodes de résolution dites rapides. La méthode des multipôles rapides (FMM) accélère la résolution de ces systèmes par des algorithmes itératifs. La méthode des H-matrices permet d’accélérer les solveurs directs nécessaires aux cas d’application massivement multi-seconds membres. Elle a été introduite et théoriquement justifiée dans le cas de l’équation de Laplace.Néanmoins elle s’avère performante au-delà de ce qui est attendu pour des problèmes d’onde relativement haute fréquence. L’objectif de cette thèse est de comprendre pourquoi la méthode fonctionne et proposer des améliorations pour des fréquences plus élevées.Une H-matrice est une représentation hiérarchique par arbre permettant un stockage compressé des données grâce à une séparation des interactions proches (ou singulières)et lointaines (dites admissibles). Un bloc admissible a une représentation de rang faible de type UVT tandis que les interactions singulières sont représentées par des blocs pleins de petites tailles. Cette méthode permet une approximation rapide d’une matrice BEM par une H-matrice ainsi qu’une méthode de factorisation rapide de type Cholesky dont les facteurs sont également de type H-matrice.Nous montrons la nécessité d’un critère d’admissibilité dépendant de la fréquence et introduisons un critère dit de Fresnel basé sur la zone de diffraction de Fresnel. Ceci permet de contrôler la croissance du rang d’un bloc et nous proposons une estimation précise de celui-ci à haute fréquence à partir de résultats sur les fonctions d’onde sphéroïdales.Nous en déduisons une méthode de type HCA-II, robuste et fiable, d’assemblage rapide compressé à la précision voulue.Nous étudions les propriétés de cet algorithme en fonction de divers paramètres et leur influence sur le contrôle et la croissance du rang en fonction de la fréquence.Nous introduisons la notion de section efficace d’interaction entre deux clusters vérifiant le critère de Fresnel. Si celle-ci n’est pas dégénérée, le rang du bloc croît au plus linéairement avec la fréquence ; pour une interaction entre deux clusters coplanaires nous montrons une croissance comme la racine carrée de la fréquence. Ces développements sont illustrés sur des maillages représentatifs des interactions à haute fréquence. / The boundary elements method (BEM) leads to dense linear systemswhose size growsrapidly in pratice ; hence the use of so-called fast methods. The fast multipole method(FMM) accelerates the resolution of BEM systems within an iterative scheme. The H-matrix method speeds up a direct resolution which is needed in massively multiple righthandsides problems. It has been provably introduced in the context of the Laplace equation.However, the use ofH-matrices for relatively high-frequency wave problems leadsto results above expectations. This thesis main goal is to provide an explanation of thesegood results and thus improve the method for higher frequencies.A H-matrix is a compressed tree-based hierarchical representation of the data associated with an admissibility criterion to separate the near (or singular) and far (or compres-sed) fields. An admissible block reads as a UVT rank deficient matrix while the singularblocks are dense with small dimensions. BEM matrices are efficiently represented byH-matrices and this method also allows for a fast Cholesky factorization whose factors arealsoH-matrices.Our work on the admissibility condition emphasizes the necessity of a frequency dependantadmissibility criterion. This new criterion is based on the Fresnel diffraction areathus labelled Fresnel admissibility condition. In that case a precise estimation of the rankof a high-frequency block is proposed thanks to the spheroidal wave functions theory.Consequently, a robust and reliable HCA-II type algorithm has been developed to ensurea compressed precision-controlled assembly. The influence of various parameters on thisnew algorithm behaviour is discussed ; in particular their influence on the control andthe growth of the rank according to the frequency.We define the interaction cross sectionfor two Fresnel-admissible clusters and show in the non-degenerate case that the rankgrowth is linear according to the frequency in the high-frequency regime ; interaction ofcoplanar clusters results in growth like the square root of the frequency. All these resultsare presented on meshes adapted to high-frequency interactions.

Les travaux présentés dans cette thèse visent à produire automatiquement des prototypes de circuits de communication et de contrôle à partir de spécifications temporelles déclaratives. Partant d'un ensemble de propriétés écrites en langage PSL, nous produisons un modèle RTL synthétisable automatiquement. La méthode proposée est modulaire, contrairement aux méthodes publiées antérieurement qui étaient fondées sur la théorie des automates. Pour chaque propriété, nous produisons un composant qui observe certains opérandes et génère des chronogrammes pour les autres opérandes : le module réactif. Tout d'abord, une bibliothèque des modules réactifs primitifs a été développée pour les opérateurs FL et SERE. Pour ce faire, une relation de dépendance a été définie pour chaque opérateur : fondée sur la sémantique de l'opérateur, elle exprime la dépendance entre ses opérandes. Ensuite, la relation de dépendance de chaque opérateur est interprétée comme un composant matériel qui met en œuvre l'opérateur : c'est le module réactif primitif de l'opérateur. À l'aide de cette formalisation, nous proposons une méthode pour déterminer automatiquement quels signaux d'une propriété sont observés et lesquels sont générés. Dans le cas où il n'est pas possible de déterminer le sens du signal, un solveur est ajouté pour identifier la valeur du signal. Le solveur sert aussi à déterminer la valeur d'un signal généré par plusieurs propriétés. Le circuit final est l'interconnexion des modules réactifs et des solveurs pour l'ensemble des propriétés. Un outil prototype, SyntHorus2, qui est une extension d'HORUS, a été mis développé. Il prend les propriétés PSL comme entrées et génère le code VHDL synthétisable du circuit. En outre, il génère des propriétés complémentaires pour vérifier si l'ensemble des spécifications est cohérent et complet. La méthode est efficace et synthétise des circuits de commande en quelques secondes. Les résultats que nous avons obtenus sur des jeux d'essais classiques montrent que notre technique compile les propriétés plus efficacement que les outils prototypes qui l'ont précédée. / The work presented in this thesis aims at automatically prototype communication and control designs from declarative temporal specifications. From a set of PSL properties, we produce a synthesizable RTL design automatically. The proposed method is modular, in contrast to previously published methods that were based on automata theory. From each property, we produce a component that observes some operands and generates waveforms for the other operands: the reactant. First, a library of primitive reactants has been provided for FL and SERE operators. To this goal, a dependency relation is defined for each operator that expresses the dependency among its operands using the operator's semantics. Then, the dependency relation of each operator is interpreted as a hardware component that implements the operator: the operator's primitive reactant. Using this formalization, a method is proposed to automatically decide which signals of a property are observed and which are generated. In the cases when specifying the signal direction is not possible, a solver is implemented to identify the signal value. In addition, the way of identifying the value of the signal that is generated in several properties is addressed. The final circuit is the interconnection of the properties' reactants and solvers. A prototype tool SyntHorus2, which is an extension to HORUS, has been developed. It takes PSL properties as its inputs, and generates the synthesizable VHDL code of the circuit. In addition, it generates some complementary properties to verify if the set of specification is coherent and complete. The method is efficient, and synthesizes control circuits in a few seconds. Results obtained on classical benchmarks show that our technique compiles properties more efficiently than previous prototype tools.

L'analyse isométrique s'est révélée être un outil très prometteur pour la conception et l'analyse. Une tâche difficile consiste toujours à faire passer l'IGA de concept à un outil de conception pratique pour l'industrie et ce travail contribue à cet effort. Ce travail porte sur l'implémentation de l'IGA dans le solveur explicite Altair Radioss afin de répondre aux applications de simulation de crash et d'emboutissage. Pour cela, les ingrédients nécessaires à une intégration native de l'IGA dans un code éléments finis traditionnel ont été identifiés et adaptés à l'architecture de code existante. Un élément solide B-Spline et NURBS a été développé dans Altair Radioss. Les estimations heuristiques des pas de temps élémentaires ou nodaux sont explorées pour améliorer l'efficacité des simulations et garantir leur stabilité. Une interface de contact existante a été étendue afin de fonctionner de manière transparente avec les éléments finis NURBS et de Lagrange. Un raffinement local est souvent nécessaire pour la bonne représentation de champs non linéaires tels que les champs de déformations plastiques. Une analyse est faite en termes de compatibilité pour l'analyse et de mise en oeuvre pour plusieurs bases de fonctions Spline telles que les Hierarchical B-Splines, les Truncated Hierarchical B-Splines, les T-Splines et les Locally Refined B-Splines (LR B-Splines). Les LR B-Splines sont implémentés. Un schéma de raffinement est proposé et définit un sous-ensemble de raffinements adapté à leur utilisation au sein de Radioss. Le processus de raffinement d’un maillage initialement grossier et régulier est développé au sein du solveur. Il permet à l’utilisateur d’établir du raffinement local par un ensemble d’instructions à fournir dans le jeu de donnée de la simulation. La solution globale est validée sur des cas tests industriels, pour des cas de validation classiquement utilisés pour les codes industriels comme l'emboutissage et les tests de chute. / IsoGeometric Analysis has shown to be a very promising tool for an integrated design and analysis process. A challenging task is still to move IGA from a proof of concept to a convenient design tool for industry and this work contributes to this endeavor. This work deals with the implementation of the IGA into Altair Radioss explicit finite element solver in order to address crash and stamping simulation applications. To this end, the necessary ingredients to a smooth integration of IGA in a traditional finite element code have been identified and adapted to the existing code architecture. A solid B-Spline element has been developed in Altair Radioss. The estimations of heuristic element and nodal stable time increment are explored to improve the accuracy of simulations and guarantee their stability. An existing contact interface has been extended in order to work seamlessly with both NURBS and Lagrange finite elements. As local refinement is needed for solution approximation, an analysis is made in terms of analysis suitability and implementation aspects for several Spline basis functions as Hierarchical B-Splines (HB-Splines), Truncated Hierarchical B-Splines (THB-Splines), T-Splines and Locally Refined B-Splines (LR B-Splines). The LR B-Spline basis is implemented. An improved refinement scheme is introduced and defines a set of analysis-suitable refinements to be used in Radioss. The refinement process of a regular coarse mesh is developed inside the solver. It allows the user to define a local refinement giving a set of instructions in the input file. The global solution is validated on industrial benchmarks, for validation cases conventionally used for industrial codes like stamping and drop test.

Nous présentons une nouvelle méthode pour la restauration d'images sismiques. Quand on l'observe, une image sismique est le résultat d'un système de dépôt initial qui a été transformé par un ensemble de déformations géologiques successives (flexions, glissement de la faille, etc) qui se sont produites sur une grande période de temps. L'objectif de la restauration sismique consiste à inverser les déformations pour fournir une image résultante qui représente le système de dépôt géologique tel qu'il était dans un état antérieur. Classiquement, ce procédé permet de tester la cohérence des hypothèses d'interprétations formulées par les géophysiciens sur les images initiales. Dans notre contribution, nous fournissons un outil qui permet de générer rapidement des images restaurées et qui aide donc les géophysiciens à reconnaître et identifier les caractéristiques géologiques qui peuvent être très fortement modifiées et donc difficilement identifiables dans l'image observée d'origine. Cette application permet alors d'assister ces géophysiciens pour la formulation d'hypothèses d'interprétation des images sismiques. L'approche que nous introduisons est basée sur un processus de minimisation qui exprime les déformations géologiques en termes de contraintes géométriques. Nous utilisons une approche itérative de Gauss-Newton qui converge rapidement pour résoudre le système. Dans une deuxième partie de notre travail nous montrons différents résultats obtenus dans des cas concrets afin d'illustrer le processus de restauration d'image sismique sur des données réelles et de montrer comment la version restaurée peut être utilisée dans un cadre d'interprétation géologique. / We present a new method for seismic image restoration. When observed, a seismic image is the result of an initial deposit system that has been transformed by a set of successive geological deformations (folding, fault slip, etc) that occurred over a large period of time. The goal of seismic restoration consists in inverting the deformations to provide a resulting image that depicts the geological deposit system as it was in a previous state. With our contribution, providing a tool that quickly generates restored images helps the geophysicists to recognize geological features that may be too strongly altered in the observed image. The proposed approach is based on a minimization process that expresses geological deformations in terms of geometrical constraints. We use a quickly-converging Gauss-Newton approach to solve the system. We provide results to illustrate the seismic image restoration process on real data and present how the restored version can be used in a geological interpretation framework.

No desenvolvimento de códigos paralelos, a biblioteca PETSc se destaca como uma ferramenta prática e útil. Com o uso desta ferramenta, este trabalho apresenta um estudo sobre resolvedores de sistemas lineares aplicados a escoamentos incompressíveis de fluidos em microescala, além de uma análise de seu comportamento em paralelo. Após um estudo dos diversos aspectos dos métodos de fronteira imersa, é apresentado um método de fronteira imersa paralelo de primeira ordem. Na sequência, é apresentada uma proposta de melhoria na precisão do método, baseada na minimização da distância entre a condição de contorno exata e aproximada, no sentido de mínimos quadrados. O desenvolvimento de uma ferramenta paralela eficiente é demonstrado na solução numérica de problemas envolvendo escoamentos incompressíveis de fluidos viscosos com fronteiras imersas / In the development of parallel codes, PETSc library has an important position as a practical and useful tool. With this tool, this work presents a study about linear system solvers applied to incompressible flow in microscale problems, furthermore an analysis of the parallel behavior of these methods is presented. After a study of several aspects of immersed boundary methods, and taking advantage of the flexibility of PETSc, a parallel first order immersed boundary method is presented. Thereafter, an improvement in the accuracy of the method is presented, based on the minimization of the distance between exact and approximated boundary conditions, in the least square sense. The development of a parallel and efficient tool is demonstrated in the numerical solution of incompressible viscous flow problems with immersed boundary

Este trabalho apresenta a aplicação de técnicas de processamento paralelo na resolução de equações algébricas oriundas do Método de Elementos Finitos aplicado ao Eletromagnetismo, nos regimes estático e harmônico. As técnicas de programação paralelas utilizadas foram OpenMP, CUDA e GPUDirect, sendo esta última para as plataformas do tipo Multi-GPU. Os métodos iterativos abordados incluem aqueles do subespaço Krylov: Gradientes Conjugados, Gradientes Biconjugados, Conjugado Residual, Gradientes Biconjugados Estabilizados, Gradientes Conjugados para equações normais (CGNE e CGNR) e Gradientes Conjugados ao Quadrado. Todas as implementações fizeram uso das bibliotecas CUSP, CUSPARSE e CUBLAS. Para problemas estáticos, os seguintes pré-condicionadores foram adotados, todos eles com implementações paralelizadas e executadas na GPU: Decomposições Incompletas LU e de Cholesky, Multigrid Algébrico, Diagonal e Inversa Aproximada. Para os problemas harmônicos, apenas os dois primeiros pré-condicionadores foram utilizados, porém na sua versão sequencial, com execução na CPU, resultando em uma implementação híbrida CPU-GPU. As ferramentas computacionais desenvolvidas foram testadas na simulação de problemas de aterramento elétrico. No caso do regime harmônico, em que o fenômeno é regido pela Equação de Onda completa com perdas e não homogênea, a formulação adotada foi aquela em dois potenciais, A-V aresta-nodal. Em todas as situações, os aplicativos desenvolvidos para GPU apresentaram speedups apreciáveis, demonstrando a potencialidade dessa tecnologia para a simulação de problemas de larga escala na Engenharia Elétrica, com excelente relação custo-benefício. / This work presents the use of parallel processing techniques in Graphics Processing Units (GPU) for the solution of algebraic equations arising from the Finite Element modeling of electromagnetic phenomena, both in steadystate and time-harmonic regime. The techniques used were parallel programming OpenMP, CUDA and GPUDirect, the latter for those platforms of type Multi-GPU. The iterative methods discussed include those of the Krylov subspace: Conjugate Gradients, Bi-conjugate Gradients, Conjugate Residual, Bi-conjugate Gradients Stabilized, Conjugate Gradients for Normal Equations (CGNE and CGNR) and Conjugate Gradients Squared. All implementations have made use of CUSP, CUSPARSE and CUBLAS libraries. For the static problems, the following pre-conditioners were adopted, all with parallelized implementations and executed on the GPU: Incomplete decompositions, both LU and Cholesky, Algebraic Multigrid, Diagonal and Approximate Inverse. For the time-harmonic varying problems, only the first two pre-conditioners were used, but in their sequential version and running in the CPU, which yielded a hybrid CPU-GPU implementation. The developed computational tools were tested in the simulation of electrical grounding systems. In the case of the harmonic regime, in which the phenomenon is governed by the driven, lossy wave equation, the formulation adopted was that in two potential, the ungauged edge A-V formulation. In all cases, the developed GPU-based tools showed considerable speedups, showing that this is a promising technology for the simulation of large-scale Electrical Engineering problems, with excellent cost-benefit.

Ce sujet concerne un travail pluridisciplinaire mathématique et astrophysique. Le but de cette thèse est l'étude des modèles d'hydrodynamique radiative dont l'application est bien évidemment très vaste en physique et astrophysique. Les modèles M1-multigroupes sont explorés pour décrire le transfert radiatif sans faire à priori d'hypothèse sur la profondeur optique du milieu. L'intérêt qui découle directement de ce travail est le développement du code d'hydrodynamique radiative HADES 2D permettant le calcul massivement parallèle. Il autorise des simulations dans des configurations astrophysiques réalistes en termes de nombre de Mach et de contraste de densité et de température entre les différents milieux. Nous nous sommes concentrés sur deux applications intéressantes : les jets d'étoiles jeunes et les chocs radiatifs dont les premières simulations seront présentées. / This topic is a multidisciplinary work between mathematics and astrophysics. The aim of this thesis is the study of radiation hydrodynamic models of which application is obviously very broad in physics and astrophysics. M1-multigroup models are explored to describe the radiative transfer without a priori assumption on the optical depth of the medium. The interest ensuing directly from this work is the development of a radiation hydrodynamic code, namely HADES 2D, for massively parallel computing. It allows simulations in realistic astrophysical configurations in terms of Mach number, density and temperature contrasts between different environments. We focused on two interesting applications: the jets from young stars and the radiative shocks of which first simulations will be presented.

A renewed commitment in the United States and abroad to electricity from renewable resources, such as wind, along with the recent deployment of very large turbines that rise to new heights, makes obtaining the most efficient and safe designs of the structures that support them ever more important. Towards this goal, the present research seeks to understand how optimization concepts and Microsoft Excel's optimization capabilities can be used in the design of wind turbine towers and foundations. Additionally, this research expands on the work of previous researchers to study how considering the tower and foundation as an integral system, where tower support conditions are not perfectly rigid, affects the optimal design. Specifically, optimization problems are formulated and solved with and without taking into account the effect of deflections, resulting from the foundation's rotational and horizontal stiffness, on natural frequency calculations. The general methodology used to transcribe the design of wind turbine towers and foundations into an optimization problem includes: 1) collecting information on design requirements and parameter values 2) deciding how to analyze the structure 3) formulating the optimization problem 4) implementation using Microsoft Excel. Key assumptions include: 1) use of an equivalent lumped mass method for estimating natural frequency 2) International Electrotechnical Commission (IEC) 61400-1 extreme loading condition controls design (i.e. fatigue loading condition is not considered) 3) extreme loads are obtained from manufacturer provided structural load document that satisfies loading cases outlined in IEC 61400-1 4) wind forces on the tower are calculated in accordance with IEC 61400-1 5) optimization variables are continuous. The sum of the tower material and fabrication cost and the total foundation cost is taken as the objective function. Important conclusions from this work include: 1) optimization concepts and Microsoft Excel's optimization capabilities can be used to obtain reasonable conceptual level designs and cost estimates 2) detailed designs and cost estimates could be achieved using a solver capable of handling discrete optimization problems 3) considering the tower and foundation as an integral system results in a more expensive, but safer, design 4) for the assumed parameter values, the constraint on the tower's natural frequency was found to control the tower design and the bearing capacity constraint was found to control the foundation design 5) relaxing or tightening the limit on the natural frequency will result in the greatest benefit or penalty, respectively, on the optimum solution.

Variable resolution method helps to study the local scale phenomenon of interest within the context of global scale atmosphere/ocean dynamics. Global spectral methods based on spherical harmonics as basis functions are known to resolve a given function defined on the sphere, in an uniform manner. Though known for its mathematical elegance and higher order accuracy, global spectral methods are considered to be restrictive for developing mesh-refinement strategies. The only mesh refinement strategy available until now is due to the pioneering work of F. Schmidt. Schmidt transformation can study only one region with higher resolution. The study of tropical dynamics is an interesting theme due to the presence of teleconnections between various phenomena, especially Indian Monsoon and the El-Nino. The Inter-Tropical Convergence Zone (ITCZ)is a continental scale phenamenon. It is in the ITCZ, many monsoon systems and tropical cyclones do occur. To study such phenomena under variable resolution method, high resolution is required in the entire tropical belt. Hitherto such a kind of mesh refinement strategies were not available in global spectral models. In this work, a new variable resolution method is developed that can help to study the tropical sub-scale phenomena with high resolution, in global spectral models. A new conformal coordinate transformation named ’High resolution Tropical Belt Transformation(HTBT)’ is developed to generate high resolution in the entire tropical belt. Mathematical demonstrations are given to show the existence of additional conformal transformations available on the sphere, indicating additional degrees of freedom available to create variable resolution global spectral method. Variable resolution global spectral method with high resolution over tropics is created through HTBT. The restriction imposed by Schmidt’s framework that the map-ping factor of the transformation need to have a finite-decomposition in the spectral space of the transformed domain is relaxed, by introduction of a new framework. The new framework uses transformed spherical harmonics Bnm as basis for spectral computations. With the use of FFT algorithm and Gaussian quadrature, the efficiency of the traditional spectral method is retained with the variable resolution global spectral method. The newly defined basis functions Bnm are the eigenvalues of the transformed Laplacian operator . This property of Bnm provide an elegant direct solver for the transformed Helmholtz operator on the sphere. The transformed Helmholtz equations are solved accurately with the variable resolution method. Advection experiments conducted with variable resolution spectral transport scheme on the HTBT variable grid produces near-dispersion free advection on the tropical belt. Transport across homogeneous resolution regions produce very less dispersion errors. Transport of a feature over the poles result in severe grid representation errors. It is shown that an increase in resolution around the poles greatly reduces this error. Transport of a feature from a point close to poles but not over it, does not produce such representation errors. Fourth-order Runge-Kutta scheme improves the accuracy of the transport scheme. The second order Magazenkov time-scheme proves to be better accurate than the leap-frog scheme with Asselin filter. The non-divergent barotropic vorticity equation is tested with two exact solutions namely Rochas solution and Rossby-Haurwitz wave solutions. Each of the solution tests certain unique and contrasting characteristic of the system. The numerical behaviour of the solutions show non-linear interactions in them. The singularity at the poles, arising due to the unbounded nature of the latitudinal derivative of the map factor of HTBT, triggers Gibbs phenomena for certain functions. However the recent advances in spectral methods, especially spectral viscosity method and Boyd-Vandeven filtering strategy provide ways to control the Gibbs oscillation and recover higher accuracy; make the variable resolution global spectral method viable for accurate meteorological computations.