Currently, existing devices for extracting electrical power from ocean waves all suffer from problems with low efficiency due to a lack of information about the incoming waves in advance. The complex dynamic nonlinear characteristics of the ocean make the prediction of these incoming waves a great challenge. This paper aims to investigate a deterministic short-term wave-by-wave prediction system, that can accurately predict the wave height and timing of the incoming waves, based on measurements from a submerged pressure sensor. The complete prediction process contains three steps, namely reconstruction, assimilation, and prediction. A nonlinear Weakly Dispersive Reconstruction method (WDM) is firstly employed to accurately calculate the surface elevation from the measured pressures. Afterwards, a variational assimilation method is used to convert the time series surface elevation to a spatial wavefield, to obtain initial conditions for the prediction. Lastly, a High Order Spectral Method (HOSM) deterministically predicts the evolution of the 2D irregular wavefield based on the acquired initial conditions. To verify the performance of this proposed prediction system, tests were conducted with data from irregular sea states with varying wave parameters, generated by simulations as well as model experiments in the controlled environment of a wave tank. The results show that the surface elevation can be predicted within 5% from the reference, for a future period of about 10 seconds for wavefields commonly experienced by a wave energy converter. Based on the results, a prediction is possible, but the accuracy heavily depends on the current sea state and the chosen prediction distance.The results have been compared against similar tests made using radar data and proven to be a viable alternative for certain sea states. In conclusion, pressure measurements, as a mean to sample an ocean wavefield, is shown to be a good option when combined with nonlinear reconstruction and prediction methods to assist the power harvesting capabilities of a wave energy converter. / Nuvarande enheter för att extrahera elektrisk energi från havsvågor lider av stora problem med låg effektivitet på grund av brist på information om de inkommande vågorna. Det komplexa ickelinjära dynamiska beteendet hos havsvågor gör förutsägelsen av dem till en stor utmaning. Det här arbetet syftar till att undersöka ett deterministiskt kortsiktigt system för att förutspå våg för våg, som noggrant kan förutspå våghöjd och tidpunkt för de inkommande vågorna, baserat på mätdata från en dränkbar trycksensor. Den kompletta förutsägelseprocessen innehåller tre steg, rekonstruktion, assimilering och förutsägelse. En ickelinjär weakly dispersive reconstruction method används först för att med hög noggrannhet beräkna ythöjningen från det uppmätta trycket. Därefter, används en variational assimilation method för att konvertera en tidsserie av ythöjningen till ett rumsligt vågfält, för att erhålla initialvillkor för förutsägelsen. Slutligen används en High Order Spectral Method för att deterministiskt förutspå utvecklingen av det tvådimensionella oregelbundna vågfältet baserat på de förvärvade initialvillkoren. För att verifiera prestandan av det föreslagna förutsägelsesystemet, så genomfördes tester med data från olika oregelbundna havstillstånd med varierande parametrar, genererade av simuleringar, såväl som modellexperiment utförda i en kontrollerad miljö i form av en vågtank. Resultaten från testerna visar att ythöjningen förutspås inom 5% från referensen, för en period på 10 sekunder framåt i tiden, för vågor som ett vågkraftverk vanligtvis utsätts för. Baserat på resultatet, så är det möjligt att förutspå inkommande vågor, men noggrannheten beror kraftigt på det aktuella havstillståndet och det valda avståndet för förutsägelsen. Resultaten har jämförts mot liknande tester gjorda med radardata och visat sig vara ett genomförbart alternativ för vissa havstillstånd. Sammanfattningsvis visas det att tryckmätningar, som ett medel för att mäta ett havsvågfält, är ett bra alternativ när de kombineras med ickelinjära rekonstruktions- och förutsägelsemetoder för att hjälpa till att öka ett vågkraftverks energigenerering.

Thesis (PhD (Electrical and Electronic Engineering))--University of Stellenbosch, 2007. / Most speech recognition and language identification engines are based on hidden Markov models (HMMs). Higher-order HMMs are known to be more powerful than first-order HMMs, but have not been widely used because of their complexity and computational demands. The main objective of this dissertation was to develop a more time-efficient method of decoding high-order HMMs than the standard Viterbi decoding algorithm currently in use. We proposed, implemented and evaluated two decoders based on the Forward-Backward Search (FBS) paradigm, which incorporate information obtained from low-order HMMs. The first decoder is based on time-synchronous Viterbi-beam decoding where we wish to base our state pruning on the complete observation sequence. The second decoder is based on time-asynchronous A* search. The choice of heuristic is critical to the A* search algorithms and a novel, task-independent heuristic function is presented. The experimental results show that both these proposed decoders result in more time-efficient decoding of the fully-connected, high-order HMMs that were investigated. Three significant facts have been uncovered. The first is that conventional forward Viterbi-beam decoding of high-order HMMs is not as computationally expensive as is commonly thought. The second (and somewhat surprising) fact is that backward decoding of conventional, high-order left-context HMMs is significantly more expensive than the conventional forward decoding. By developing the right-context HMM, we showed that the backward decoding of a mathematically equivalent right-context HMM is as expensive as the forward decoding of the left-context HMM. The third fact is that the use of information obtained from low-order HMMs significantly reduces the computational expense of decoding high-order HMMs. The comparison of the two new decoders indicate that the FBS-Viterbi-beam decoder is more time-efficient than the A* decoder. The FBS-Viterbi-beam decoder is not only simpler to implement, it also requires less memory than the A* decoder. We suspect that the broader research community regards the Viterbi-beam algorithm as the most efficient method of decoding HMMs. We hope that the research presented in this dissertation will result in renewed investigation into decoding algorithms that are applicable to high-order HMMs.

No contexto de Dinâmica de Fluidos Computacional, métodos de simulação de objetos imersos em Malhas Cartesianas têm se mostrado vantajosos tanto em termos de Custo Computacional quanto em termos de precisão numérica. Entretanto, a representação física de objetos imersos nesses domínios computacionais impõe a perda de validade dos esquemas de Diferenças Finitas empregados, na região das superfícies introduzidas. Este trabalho analisa um Método de Interfaces Imersas quanto ao desempenho em aplicações a esquemas de solução numérica de Alta Ordem de precisão. Através de Testes de Refinamento de Malha, é feita a apreciação da ordem de decaimento dos erros das soluções numéricas em comparação com as soluções analíticas para 2 problemas unidimensionais. O primeiro envolve a solução da Equação de Calor unidimensional sujeita a uma Condição Inicial Unitária, e o segundo relaciona-se ao cálculo das duas primeiras derivadas espaciais das funções analíticas Seno e Tangente Hiperbólica. Também é promovida uma análise de forma fragmentária do método, a fim de individualizar a contribuição dos elementos envolvidos no comportamento das soluções geradas. Os resultados obtidos indicam eventuais alterações na ordem de precisão dos esquemas de Diferenças Finitas originalmente aplicados. Esse comportamento e visto como uma dependência que o método escolhido apresenta em relação a função discretizada. Por fim, são elaboradas considerações sobre restrições de aplicabilidade do método escolhido. / In the Computational Fluid Dynamics context, methods for simulating immersed objects in Cartesian Grids have shown advantages regarding both Computational Cost and numerical precision. Nevertheless, the physical representation of immersed objects within these computational domains leads to the loss of validity of the emplyed Finite Dierence Schemes near the immersed surfaces. This work analizes a Immersed Interface Method regarding its performance in High Order Schemes applications. The error decay order for numerical solutions of two 1D problems is observed. The rst problem relates to the solution of the Heat Equation subjected to the unitary initial condition. The second relates to the computation of the rst two derivatives of analytical functions Sin and Hyperbolic Tangent. It\'s also conducted a fragmentary analysis, which is intended to identify the contribution of each element of this method to the character of the generated solution. The results indicate some eventual changes in the Order of the Finite Dierences Schemes employed. This behaviour is regarded as a dependency of this method to the nature of the discretized function. Finaly, some remarks regarding restrictions to this method\'s applicability are made.

Desde o importante trabalho de Barth e Frederickson (1990), um certo número de pesquisadores têm estudado o método de Volumes Finitos de alta-ordem k-exato, por exemplo o grupo do Prof. Ollivier-Gooch: Ollivier-Gooch e van Altena (2002), Nejat (2007), Michalak (2009), etc. Outras discretizações espaciais de alta-ordem bastante populares são o método Galerkin Descontínuo e o método de Diferença Espectral; processos iterativos que involucram estes esquemas tem sido acelerados, nos últimos anos, por métodos p-multigrid. Porém, esta aceleração não tem sido aplicada no contexto do método de Volumes Finitos de alta-ordem, pelo menos para conhecimento do autor desta tese. Por isso, o objetivo desta pesquisa é adaptar o p-multigrid desenvolvido por Liang et al. (2009b) no contexto da Diferença Espectral, para o ambiente dos Volumes Finitos estudado pelo Prof. Ollivier-Gooch. A pesquisa começa implementando o solver VF-RK, de Volumes Finitos com avanço Runge-Kutta, para resolver as equações de advecção-difusão e de Euler aplicados a problemas estacionários, por exemplo, o escoamento transônico ao redor do NACA 0012. Depois, estuda-se o método p-multigrid no contexto da Diferença Espectral; o p-multigrid acelera o processo iterativo comutando níveis polinomiais de alta e de baixa-ordem. Após esse estudo, a adaptação ao âmbito dos Volumes Finitos é realizada resultando num p-multigrid relativamente mais simples porque, em contraposição com o p-multigrid para Diferença Espectral, não precisa de operadores de restrição e prolongação para a comunicação entre diferentes níveis polinomiais. A pesquisa conclui com uma comparação com o método de Volumes Finitos de 4a ordem sem p-multigrid (solver VF-RK). Nesse sentido, implementa-se o solver pMG, baseado no p-multigrid proposto, para resolver os problemas estacionários considerados na primeira parte do trabalho; o smoother do p-multigrid é o esquema Runge-Kutta do código VF-RK, e cada problema estacionário é resolvido utilizando diferentes Vciclos procurando sempre soluções de 4a ordem. Os resultados indicam que o método p-multigrid proposto é mais eficiente que o método de Volumes Finitos de 4a ordem sem p-multigrid, isto é, os dois métodos oferecem a mesma precisão mas o primeiro pode levar menos de 50% do tempo de CPU do segundo. / Since Barth and Frederickson\'s important work (Barth e Frederickson, 1990), a number of researchers have studied high-order k-exact Finite Volume method, for example Prof. Ollivier-Gooch\'s group: Ollivier-Gooch e van Altena (2002), Nejat (2007), Michalak (2009), etc. Other quite popular high-order spatial discretizations are the Discontinuous Galerkin methods and the Spectral Difference methods; the iterative processes involving these schemes have been accelerated in recent years by p-multigrid methods. However, this acceleration has not been applied in the context of the high-order Finite Volume method, at least for the knowledge of the author of this thesis. Therefore, the objective of this research is to adapt the p-multigrid developed by Liang et al. (2009b) in the context of Spectral Difference methods, to the environment of Finite Volume studied by Prof. Ollivier-Gooch. This research begins by implementing the solver VF-RK, Finite Volume solver with Runge-Kutta advance, to compute the advection-diffusion equation and Euler equations applied to steady state problems, for example, the transonic flow around NACA 0012. Then, it is studied the p-multigrid method in the context of Spectral Difference schemes; p-multigrid accelerates the iterative process by switching polynomial levels of high- and low-order. After this study, the adaptation to the context of the Finite Volume scheme is performed resulting in a relatively simple p-multigrid because, in contrast to the p-multigrid for Spectral Difference schemes, it doesn\'t need restriction and prolongation operators for communication between different polynomial levels. The research concludes with a comparison with 4th order Finite Volume method without p-multigrid (solver VF-RK). Accordingly, the solver pMG, based on the proposed p-multigrid, is implemented to resolve the steady state problems considered in the first part of the work; the p-multigrid smoother is the Runge-Kutta scheme from VF-RK code, and each steady state problem is solved using different Vcycles, looking for 4th order solutions ever. The results indicate that the proposed p-multigrid method is more efficient than the 4th order Finite Volume method without p-multigrid: the two methods give the same accuracy but the first one can take less than 50% of second one\'s CPU time.

Nesta tese foram desenvolvidas metodologias sintéticas para a abertura de N-tosilaziridinas, provenientes de aminoácidos, via arilcianocupratos de ordem superior, os quais foram gerados pela reação de troca telúrio/cobre entre teluretos arílicos e cianocupratos; tais aberturas levam à formação de estruturas análogas a fenetilaminas em bons rendimentos, as quais possuem atividades farmacológicas interessantes. Adicionalmente, um conjunto de amino teluretos contendo diferentes grupos protetores foram preparados a partir dos respectivos aminoálcoois, em bons rendimentos (72 a 83%), e avaliados como potenciais espécies nucleofílicas. Os teluretos, que contêm como grupo protetor benzoíla, foram utilizados em reações de troca telúrio/lítio, gerando as espécies dilitiadas, as quais foram capturadas com diversos eletrófilos, levando à formação de aminoálcoois de cadeia maior e também à formação de análogos de fenetilaminas em bons rendimentos. Nesta tese também foram estudadas reações que envolvem química de arinos, os quais podem ser gerados de várias maneiras. Dentre essas formas para produzir os arinos, destacamos o uso de 2-(trimetilsilil)fenil triflato, o qual, na presença de fontes de íons fluoreto em solvente polar aprótico, gera a espécie reativa. Empregamos essa metodologia para geração de benzinos na reação de inserção em ligações do tipo σ de disselenetos de diarila. Tendo visto o sucesso da metodologia desenvolvida, decidimos sintetizar outros arinos derivados do 2-(trimetilsilil)fenil triflato e empregá-los em outro tipo de reação de inserção em ligações do tipo σ, desta vez em ligações entre selênio/estanho de tributil(fenilselanil)estanana, levando a uma difuncionalização nas posições 1 e 2 do anel aromático. / In this PhD thesis we developed a synthetical methodology for the ring opening reaction of N-tosyl aziridines by high order arylcyanocuprates that have been generated by tellurium/copper exchange reaction between aryl tellurides and cyanocuprates. These ring opening reactions lead to an interesting class of phenethylamines in good yields. Additionally, a set of tellurium amines containing different protecting groups were conveniently prepared in good yields (72 - 83%) from aminoalcohols and evaluated as potencial nucleophilic species. The corresponding tellurium amines have been evaluated in the tellurium/lithium exchange reaction. The nitrogen-containing organolithium compounds were efficiently prepared by using N-Bz tellurium amines and a mixture of n-butyllithium and lithium naphthalenide (LiNp) to performe the exchange reaction. The dianion intermediates were trapped with a wide range of electrophiles, resulting in this corresponding products in good to excellent yields. The reaction was also employed in the synthesis of phenethylamines. In this work we also studied reactions involving aryne chemistry, that are reactive species. A number of methods for benzyne formation have been reported in the literature. Among them we pay particular attention to the use of 2-(trimethylsilyl)aryl triflates in the presence of fluoride ion sources and aprotic polar solvents that generate this reactive species. We employed this methodology for generation of benzyne in the insertion reaction of diaryl diselenide σ bond. In view of the success of the developed methodology, we decided to synthesize other ayne derivatives from 2-(trimethylsilyl)phenyl triflate and use them on another type of σ bond insertion reaction. The insertion reaction of arynes into Se-Sn sigma bond of tributyl(phenylselanyl)stannane, leading a double functionality in the aromatic system.

Neste trabalho apresentamos uma análise de possíveis processos de tunelamento em sistemas unidimensionais através do estudo do potencial de barreira dupla com região intermediária confinante, conseguimos verificar a existência de um processo de tunelamento que ocorre através da ocupação virtual da região intermediária. Uma modelagem deste fenômeno é proposta baseando-se em uma teoria perturbativa realizada em termos de estados \"quase-localizados\" da partícula. Além da descrição qualitativa do fenômeno, determinamos as condições físicas para que este processo de tunelamento, também chamado de cotunelamento ou tunelamento de ordem superior, ocorra. Como resultado, recuperamos com boa aproximação o coeficiente de transmissão exato do sistema escolhido. Por fim, um outro resultado obtido durante o desenvolvimento deste trabalho foi uma prova conclusiva da convergência do método de diferenças finitas FDTD aplicada à equação de Schrödinger. Esta prova, diferente das demais encontradas na literatura, conseguiu demonstrar de maneira conclusiva a inclusão do limite superior para o passo temporal, de maneira a assegurar a convergência das soluções numéricas, algo até então testado numericamente, mas sem uma prova rigorosa da sua validade. / In this work, we present a study about tunnelling processes occurring in one-dimensional systems. Choosing a double well potential with a confining region as a case study, we verified that there is a tunnelling process which is due to virtual occupation of the confining region. Using perturbative theory of quasi-localized states to describe the particles dynamics, we provide a qualitative description of the phenomenon of tunnelling through virtual occupation, and we are capable of determining the conditions for which such a tunneling (also known as cotunnelling or high-order tunneling) should be present. With this analysis we could calculate with good approximation the particle transmission coefficient through the barrier. Finally, we also provide a rigorous proof of the convergence conditions for the numerical calculation of the Schrödinger equation using the finite difference method.

Cette thèse porte sur la généralisation de la méthode NXFEM proposée par A. et P. Hansbo pour le problème d’interface elliptique. La modélisation et simulation numérique d’écoulements dans des domaines fracturés sont au coeur de nombreuses applications, telles que le milieu pétrolier (modélisation de réservoirs, présence de failles, propagation d’un signal, repérage de couches), l’aérospatiale (problème de chocs, de rupture), en génie civil (fissuration du béton), mais également dans la biologie cellulaire (déformation des globules rouges). En outre, de nombreux projets de recherche nécessitent le développement des méthodes robustes pour la prise en compte de singularités, ce qui fait partie des motivations et des objectifs de l'équipe Concha, ainsi que de cette thèse. Une modification de cette méthode a tout d’abord été proposée afin d’obtenir la robustesse à la fois par rapport à la géométrie du maillage coupé par l’interface et par rapport aux paramètres de diffusion. Nous nous sommes ensuite intéressés à sa généralisation à tout type de maillages 2D-3D (triangles, quadrilatères, tétraèdres, hexaèdres), et pour tout type d’éléments finis (conformes, non conformes, Galerkin discontinus) pour des interfaces planes et courbes. Les applications ont été orientées vers des problèmes d’écoulements en milieux poreux fracturés : adaptation de la méthode NXFEM à la résolution d’un modèle asymptotique de failles, à des problèmes instationnaires, de transports, ou encore à des domaines multi-fracturés. / This thesis focuses on the generalization of the NXFEM method proposed by A. and P. Hansbo for elliptic interface problem. Numerical modeling and simulation of flow in fractured media are at the heart of many applications, such as petroleum and porous media (reservoir modeling, presence of faults, signal propagation, identification of layers ...), aerospace (problems of shock, rupture), civil engineering (concrete cracking), but also in cell biology (deformation of red blood cells). In addition, many research projects require the development of robust methods for the consideration of singularities, which is one of the motivations and objectives of the Concha team and of this thesis. First a modification of this method was proposed to obtain a robust method not only with respect to the mesh-interface geometry, but also with respect to the diffusion parameters. We then looked to its generalization to any type of 2D-3D meshes (triangles, quadrilaterals, tetrahedra, hexahedra), and for any type of finites elements (conforming, nonconforming, Galerkin discontinuous) for plane and curved interfaces. The applications have been referred to the flow problems in fractured porous media : adaptation of NXFEM method to solve an asymptotic model of faults, to unsteady problems, transport problems, or to multi-fractured domains.

Discontinuous Galerkin methods form a class of numerical methods to find a solution of partial differential equations by combining features of finite element and finite volume methods. Methods are defined using a weak form of a particular model problem, allowing for discontinuities in the discrete trial and test spaces. Using a discontinuous discrete space mesh provides proper flexibility and a compact discretisation pattern, allowing a multidomain and multiphysics simulation. Discontinuous Galerkin methods with a higher approximation polynomial order, the socalled p-version, performs better in terms of convergence rate, compared with the low order h-version with smaller element sizes and bigger mesh. However, the condition number of the Galerkin system grows subsequently. This causes surge in the amount of required storage, computational complexity and in the time required for computation. We use the following three approaches to keep the advantages and eliminate the disadvantages. The first approach will be a specific choice of basis functions which we call C1 polynomials. These ensure that the majority of integrals over the edge of the mesh elements disappears. This reduces the total number of non-zero elements in the resulting system. This decreases the computational complexity without loss in precision. This approach does not affect the number of iterations required by chosen Conjugate Gradients method when compared to the other choice of basis functions. It actually decreases the total number of algebraic operations performed. The second approach is the introduction of suitable preconditioners. In our case, the Additive two-layer Schwarz method, developed in [4], for the iterative Conjugate Gradients method is considered. This directly affects the spectral condition number of the system matrix and decreases the number of iterations required for the computation. This approach, however, increases the total number of algebraic operations and might require more operational time. To tackle the rise in the number of algebraic operations, we introduced a modified Additive two-layer non-overlapping Schwarz method with a Multigrid process. This using a fixed low-order approximation polynomial degree on a coarse grid. We show that this approach is spectrally equivalent to the first preconditioner, and requires less time for computation. The third approach is a development of an efficient mathematical framework for distributed data structure. This allows a high performance, massively parallel, implementation of the discontinuous Galerkin method. We demonstrate that it is possible to exploit properties of the system matrix and C1 polynomials as basis functions to optimize the parallel structures. The previously mentioned parallel data structure allows us to parallelize at the same time both the matrix-vector multiplication routines for the Conjugate Gradients method, as well as the preconditioner routines on the solver level. This minimizes the transfer ratio amongst the distributed system. Finally, we combined all three approaches and created a framework, which allowed us to successfully implement all of the above.

La maladie veineuse thromboembolique (MVTE) est un problème de santé publique (plus de 100000 cas par an en France). Elle regroupe deux entités cliniques : la thrombose veineuse profonde (TVP) des membres inférieurs et l’embolie pulmonaire (EP). La TVP correspond à la formation inadaptée d’un thrombus veineux (appelé aussi caillot sanguin) dans les veines profondes (poplitées, fémorales, iliaques). Un thrombus est principalement constitué de globules rouges et de plaquettes dans un réseau de fibrine. La complication majeure d’une TVP est la survenue d’une EP, c’est-à-dire que le thrombus s’est détaché de la paroi veineuse, ou s’est fragmenté, et est entraîné par la circulation sanguine jusqu’à une artère pulmonaire. Cette complication a un taux de mortalité assez élevée autour de 10000 à 20000 cas mortels par an en France. La survenue d’une TVP est multifactorielle associant des facteurs génétiques et acquis pouvant être répartis en trois catégories : la stase veineuse, l’altération de la paroi d’une veine et une hypercoagulabilité. En analysant la structure du thrombus, notre projet vise à identifier le facteur principal responsable de la TVP et à évaluer le risque d’EP. Pour caractériser sa structure, nous disposons de deux modes d’imagerie acoustique : l’échographie et l’élastographie (carte de dureté). Nous proposons d’extraire des descripteurs de ces images acoustiques par deux approches, l’une basée sur les ondelettes (le scattering operator) et l’autre sur les statistiques d’ordre supérieur (les multicorrélations). Ces descripteurs sont ensuite analysées par diverses techniques de classification (analyse en composantes principales, k-moyennes, classification spectrale) pour retrouver la cause principale des TVP ou la présence d’EP. / Venous thromboembolism (VTE) is an important public health issue (over 100000 individuals in France per year). VTE is a combination of a deep venous thrombosis (DVT) and a pulmonary embolism (PE). DVT is an inappropriate formation of a thrombus (also called blood clot) in one of the deep veins of the body, usually in the leg (popliteal, femoral, iliac). There are mainly three components in a thrombus: platelets, red blood cells and a mesh of fibrins. The main complication of a DVT is a pulmonary embolism (PE) which occurs when a thrombus breaks loose and travels to the lungs. PE affects an estimated 10000 - 20 000 individuals just in France per year. Three physiopathological mechanisms cancontribute, isolated or combined, to the development of a DVT: venous stasis, endothelial injury and hypercoagulability. Our project is aiming to relate the thrombus structure, its main triggering factor and the risk of a PE. To characterize the thrombus structure, we are collecting ultrasonography (echogenicity) and elastography (stiffness) of human thrombus. We propose to extract features from these to kind of ultrasound images using two approaches: one basedon wavelets (the scattering operator) and another based on high order statistics (multicorrelations). Then, the obtained features are analysed using several classification technics (principal component analysis, k-means, spectral clustering) to find the main cause of the DVT or the presence of PE.

A general finite element method is presented to solve the Euler equations in a Lagrangian reference frame. This FEM framework allows for separate arbitrarily high order representation of kinematic and thermodynamic variables. An accompanying hydrodynamics code written in Matlab is presented as a test-bed to experiment with various basis function choices. A wide range of basis function pairs are postulated and a few choices are developed further, including the bi-quadratic Q2-Q1d and Q2-Q2d elements. These are compared with a corresponding pair of low order bi-linear elements, traditional Q1-Q0 and sub-zonal pressure Q1-Q1d. Several test problems are considered including static convergence tests, the acoustic wave hourglass test, the Sod shocktube, the Noh implosion problem, the Saltzman piston, and the Sedov explosion problem. High order methods are found to offer faster convergence properties, the ability to represent curved zones, sharper shock capturing, and reduced shock-mesh interaction. They also allow for the straightforward calculation of thermodynamic gradients (for multi-physics calculations) and second derivatives of velocity (for monotonic slope limiters), and are more computationally efficient. The issue of shock ringing remains unresolved, but the method of hyperviscosity has been identified as a promising means of addressing this. Overall, the curvilinear finite elements presented in this thesis show promise for integration in a full hydrodynamics code and warrant further consideration.