In the last decade, high-order methods have gained increased attention. These combine the convergence properties of spectral methods with the geometrical flexibility of low-order methods. However, the time step is restrictive, necessitating the implicit treatment of diffusion terms in addition to the pressure. Therefore, efficient solution of elliptic equations is of central importance for fast flow solvers. As the operators scale with O(p · N), where N is the number of degrees of freedom and p the polynomial degree, the runtime of the best available multigrid algorithms scales with O(p · N) as well. This super-linear scaling limits the applicability of high-order methods to mid-range polynomial orders and constitutes a major road block on the way to faster flow solvers. This work reduces the super-linear scaling of elliptic solvers to a linear one. First, the static condensation method improves the condition of the system, then the associated operator is cast into matrix-free tensor-product form and factorized to linear complexity. The low increase in the condition and the linear runtime of the operator lead to linearly scaling solvers when increasing the polynomial degree, albeit with low robustness against the number of elements. A p-multigrid with overlapping Schwarz smoothers regains the robustness, but requires inverse operators on the subdomains and in the condensed case these are neither linearly scaling nor matrix-free. Embedding the condensed system into the full one leads to a matrix-free operator and factorization thereof to a linearly scaling inverse. In combination with the previously gained operator a multigrid method with a constant runtime per degree of freedom results, regardless of whether the polynomial degree or the number of elements is increased. Computing on heterogeneous hardware is investigated as a means to attain a higher performance and future-proof the algorithms. A two-level parallelization extends the traditional hybrid programming model by using a coarse-grain layer implementing domain decomposition and a fine-grain parallelization which is hardware-specific. Thereafter, load balancing is investigated on a preconditioned conjugate gradient solver and functional performance models adapted to account for the communication barriers in the algorithm. With the new model, runtime prediction and measurement fit closely with an error margin near 5 %. The devised methods are combined into a flow solver which attains the same throughput when computing with p = 16 as with p = 8, preserving the linear scaling. Furthermore, the multigrid method reduces the cost of implicit treatment of the pressure to the one for explicit treatment of the convection terms. Lastly, benchmarks confirm that the solver outperforms established high-order codes.

This thesis develops the methodology for solving initial boundary value problems with the use of summation-by-parts discretizations. The combination of high orders of accuracy and a systematic approach to construct provably stable boundary and interface procedures makes this methodology especially suitable for scientific computations with high demands on efficiency and robustness. Most classes of high order methods can be applied in a way that satisfies a summation-by-parts rule. These include, but are not limited to, finite difference, spectral and nodal discontinuous Galerkin methods. In the first part of this thesis, the summation-by-parts methodology is extended to the time domain, enabling fully discrete formulations with superior stability properties. The resulting time discretization technique is closely related to fully implicit Runge-Kutta methods, and may alternatively be formulated as either a global method or as a family of multi-stage methods. Both first and second order derivatives in time are considered. In the latter case also including mixed initial and boundary conditions (i.e. conditions involving derivatives in both space and time). The second part of the thesis deals with summation-by-parts discretizations on multi-block and hybrid meshes. A new formulation of general multi-block couplings in several dimensions is presented and analyzed. It collects all multi-block, multi-element and  hybrid summation-by-parts schemes into a single compact framework. The new framework includes a generalized description of non-conforming interfaces based on so called summation-by-parts preserving interpolation operators, for which a new theoretical accuracy result is presented.

Uma grande limitação dos métodos de diferenças finitas é que eles estão restritos a malhas e domínios retangulares. Para descrever escoamentos em domínios complexos, como, por exemplo, problemas com superfícies livres, faz-se necessário o uso de técnicas acessórias. O método de interfaces imersas é uma dessas técnicas. Nesse trabalho, primeiramente foi desenvolvido um método de projeção, totalmente livre de pressão, para as equações de Navier-Stokes com variáveis primitivas em malha deslocada. Esse método é baseado em diferenças finitas compactas, possuindo segunda ordem temporal e quarta ordem espacial. Esse método foi combinado com o método de interface imersa de Linnick e Fasel [2] para resolver numericamente as equações de Stokes com quarta ordem de precisão. A verificação do código foi feita por meio do método das soluções manufaturadas e da comparação com resultados de outros autores em problemas clássicos da literatura. / A great limitation of finite differences methods is that they are restricted to retangular meshes and domains. In order to describe flows in complex domains, e.g. free surface problems, it is necessary to use accessory techniques. The immersed interface method is one of such techniques. In the present work, firstly, a projection method was developed, which is completely pressure-free, for the Navier-Stokes equations with primitive variables in a staggered mesh. This method is based on compact finite differences, with temporal second-order precision and spatial foruth-order precision. This method was combined with the immersed interface method from Linnick e Fasel [2] in order to numerically solve the Stokes equations with fourth-order precision. The verification of the code was performed with the manufactured solutions method and by comparing results with other authors for some classical problems in the literature.

Em diversos escoamentos sobre superfícies há a presença de protuberâncias, como por exemplo rebites, parafusos e juntas. Estas protuberâncias podem influenciar a camada limite, acelerando a transição do escoamento do estado laminar para o estado turbulento. Em alguns casos isto pode ser indesejável, já que o escoamento turbulento implica necessariamente em uma força de atrito maior do que aquela referente ao escoamento laminar. Existem alguns aspectos neste tipo de escoamento que ainda não estão bem compreendidos. O objetivo deste trabalho é estudar a influência de uma rugosidade isolada no escoamento sobre uma superfície. Este estudo contribui para se entender o que ocorre em casos de maior complexidade. O estudo é de natureza computacional, em que se utiliza simulação numérica direta das equações de Navier-Stokes. A técnica de fronteiras imersas é utilizada para representar a rugosidade no escoamento sobre a superfície. O código numérico é verificado por meio do método de soluções manufaturadas. Comparações entre resultados experimentais, da teoria de estabilidade linear e numéricos também são utilizados para a validação do código. Resultados obtidos com diferentes alturas de rugosidade e variações no gradiente de pressão permitiram analisar a influência de elemento rugoso tridimensional em escoamentos de camada limite. / The presence of protuberances on surfaces, for example, rivets, screws and gaskets, can influence the boundary layer by accelerating the transition from laminar flow to turbulent flow. In some cases this may be undesirable, since the turbulent flow involves frictional forces greater than the ones at the laminar regime. There are some aspects of the flow in the boundary layer perturbed by a single roughness element that are not well understood. The aim of this work is to study the influence of an isolated roughness on the boundary layer. This study is a step towards to the understanding of what can happen in more complex cases. The nature of this study is computational, therefore a Direct Numerical Simulation code is used. The immersed boundary method is used to represent the roughness in the flow on the surface. The numerical code is verified via theMethod ofManufactured Solutions. Comparisons between experimental data, Linear Stability Theory and numerical results are also used for the validation of the code. Results obtained with different roughness heights and variations in the pressure gradient allowed the analysis of the influence of a three-dimensional roughness element in boundary layer flows.

The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.

Em diversos escoamentos sobre superfícies há a presença de protuberâncias, como por exemplo rebites, parafusos e juntas. Estas protuberâncias podem influenciar a camada limite, acelerando a transição do escoamento do estado laminar para o estado turbulento. Em alguns casos isto pode ser indesejável, já que o escoamento turbulento implica necessariamente em uma força de atrito maior do que aquela referente ao escoamento laminar. Existem alguns aspectos neste tipo de escoamento que ainda não estão bem compreendidos. O objetivo deste trabalho é estudar a influência de uma rugosidade isolada no escoamento sobre uma superfície. Este estudo contribui para se entender o que ocorre em casos de maior complexidade. O estudo é de natureza computacional, em que se utiliza simulação numérica direta das equações de Navier-Stokes. A técnica de fronteiras imersas é utilizada para representar a rugosidade no escoamento sobre a superfície. O código numérico é verificado por meio do método de soluções manufaturadas. Comparações entre resultados experimentais, da teoria de estabilidade linear e numéricos também são utilizados para a validação do código. Resultados obtidos com diferentes alturas de rugosidade e variações no gradiente de pressão permitiram analisar a influência de elemento rugoso tridimensional em escoamentos de camada limite. / The presence of protuberances on surfaces, for example, rivets, screws and gaskets, can influence the boundary layer by accelerating the transition from laminar flow to turbulent flow. In some cases this may be undesirable, since the turbulent flow involves frictional forces greater than the ones at the laminar regime. There are some aspects of the flow in the boundary layer perturbed by a single roughness element that are not well understood. The aim of this work is to study the influence of an isolated roughness on the boundary layer. This study is a step towards to the understanding of what can happen in more complex cases. The nature of this study is computational, therefore a Direct Numerical Simulation code is used. The immersed boundary method is used to represent the roughness in the flow on the surface. The numerical code is verified via theMethod ofManufactured Solutions. Comparisons between experimental data, Linear Stability Theory and numerical results are also used for the validation of the code. Results obtained with different roughness heights and variations in the pressure gradient allowed the analysis of the influence of a three-dimensional roughness element in boundary layer flows.

Uma grande limitação dos métodos de diferenças finitas é que eles estão restritos a malhas e domínios retangulares. Para descrever escoamentos em domínios complexos, como, por exemplo, problemas com superfícies livres, faz-se necessário o uso de técnicas acessórias. O método de interfaces imersas é uma dessas técnicas. Nesse trabalho, primeiramente foi desenvolvido um método de projeção, totalmente livre de pressão, para as equações de Navier-Stokes com variáveis primitivas em malha deslocada. Esse método é baseado em diferenças finitas compactas, possuindo segunda ordem temporal e quarta ordem espacial. Esse método foi combinado com o método de interface imersa de Linnick e Fasel [2] para resolver numericamente as equações de Stokes com quarta ordem de precisão. A verificação do código foi feita por meio do método das soluções manufaturadas e da comparação com resultados de outros autores em problemas clássicos da literatura. / A great limitation of finite differences methods is that they are restricted to retangular meshes and domains. In order to describe flows in complex domains, e.g. free surface problems, it is necessary to use accessory techniques. The immersed interface method is one of such techniques. In the present work, firstly, a projection method was developed, which is completely pressure-free, for the Navier-Stokes equations with primitive variables in a staggered mesh. This method is based on compact finite differences, with temporal second-order precision and spatial foruth-order precision. This method was combined with the immersed interface method from Linnick e Fasel [2] in order to numerically solve the Stokes equations with fourth-order precision. The verification of the code was performed with the manufactured solutions method and by comparing results with other authors for some classical problems in the literature.

L’imagerie du sous-sol par équations d’onde est une application de l’ingénierie pétrolière qui mobilise des ressources de calcul très importantes. On dispose aujourd’hui de calculateurs puissants qui rendent accessible l’imagerie de régions complexes mais des progrès sont encore nécessaires pour réduire les coûts de calcul et améliorer la qualité des simulations. Les méthodes utilisées aujourd’hui ne permettent toujours pas d’imager correctement des régions très hétérogènes 3D parce qu’elles sont trop coûteuses et /ou pas assez précises. Les méthodes d’éléments finis sont reconnues pour leur efficacité à produire des simulations de qualité dans des milieux hétérogènes. Dans cette thèse, on a fait le choix d’utiliser une méthode de Galerkine discontinue (DG) d’ordre élevé à flux centrés pour résoudre l’équation des ondes acoustiques et on développe un schéma d’ordre élevé pour l’intégration en temps qui peut se coupler avec la technique de discrétisation en espace, sans générer des coûts de calcul plus élevés qu’avec le schéma d’ordre deux Leap-Frog qui est le plus couramment employé. Le nouveau schéma est comparé au schéma d’ordre élevé ADER qui s’avère plus coûteux car il requiert un plus grand nombre d’opérations pour un niveau de précision fixé. De plus, le schéma ADER utilise plus de mémoire, ce qui joue aussi en faveur du nouveau schéma car la production d’images du sous-sol consomme beaucoup de mémoire et justifie de développer des méthodes numériques qui utilisent la mémoire au minimum. On analyse également la précision des deux schémas intégrés dans un code industriel et appliqués à des cas test réalistes. On met en évidence des phénomènes de pollution numériques liés à la mise en oeuvre d'une source ponctuelle dans le schéma DG et on montre qu'on peut éliminer ces ondes parasites en introduisant un terme de pénalisation non dissipatif dans la formulation DG. On finit cette thèse en discutant les difficultés engendrées par l'utilisation de schémas numériques dans un contexte industriel, et en particulier l'effet des calculs en simple précision. / Oil engineering uses a wide variety of technologies including imaging wave equation which involves very large computing resources. Very powerful computers are now available that make imaging of complex areas possible, but further progress is needed both to reduce the computational cost and improve the simulation accuracy. The current methods still do not allow to image properly heterogeneous 3D regions because they are too expensive and / or not accurate enough. Finite element methods turn out to be efficient for producing good simulations in heterogeneous media. In this thesis, we thus chose to use a high order Discontinuous Galerkin (DG) method based upon centered fluxes to solve the acoustic wave equation and developed a high-order scheme for time integration which can be coupled with the space discretization technique, without generating higher computational cost than the second-order Leap Frog scheme which is the most widely used . The new scheme is compared to the high order ADER scheme which is more expensive because it requires a larger number of computations for a fixed level of accuracy. In addition, the ADER scheme uses more memory, which also works in favor of the new scheme since producing subsurface images consumes lots of memory and justifies the development of low-memory numerical methods. The accuracy of both schemes is then analyzed when they are included in an industrial code and applied to realistic problems. The comparison highlights the phenomena of numerical pollution that occur when injecting a point source in the DG scheme and shows that spurious waves can be eliminated by introducing a non-dissipative penalty term in the DG formulation. This work ends by discussing the difficulties induced by using numerical methods in an industrial framework, and in particular the effect of single precision calculations.

Le sujet de ce travail de thèse est l’application de la méthode de caractéristiques longues (MOC) pour résoudre l’équation du transport des neutrons pour des géométries à trois dimensions extrudées. Les avantages du MOC sont sa précision et son adaptabilité, le point faible était la quantité de ressources de calcul requises. Ce problème est même plus important pour des géométries à trois dimensions ou le nombre d’inconnues du problème est de l’ordre de la centaine de millions pour des calculs d’assemblage.La première partie de la recherche a été dédiée au développement des techniques optimisées pour le traçage et la reconstruction à-la-volé des trajectoires. Ces méthodes profitent des régularités des géométries extrudées et ont permis une forte réduction de l’empreinte mémoire et une réduction des temps de calcul. La convergence du schéma itératif a été accélérée par un opérateur de transport dégradé (DPN) qui est utilisé pour initialiser les inconnues de l’algorithme itératif and pour la solution du problème synthétique au cours des itérations MOC. Les algorithmes pour la construction et la solution des opérateurs MOC et DPN ont été accélérés en utilisant des méthodes de parallélisation à mémoire partagée qui sont le plus adaptés pour des machines de bureau et pour des clusters de calcul. Une partie importante de cette recherche a été dédiée à l’implémentation des méthodes d’équilibrage la charge pour améliorer l’efficacité du parallélisme. La convergence des formules de quadrature pour des cas 3D extrudé a aussi été explorée. Certaines formules profitent de couts négligeables du traitement des directions azimutales et de la direction verticale pour accélérer l’algorithme. La validation de l’algorithme du MOC a été faite par des comparaisons avec une solution de référence calculée par un solveur Monte Carlo avec traitement continu de l’énergie. Pour cette comparaison on propose un couplage entre le MOC et la méthode des Sous-Groupes pour prendre en compte les effets des résonances des sections efficaces. Le calcul complet d’un assemblage de réacteur rapide avec interface fertile/fissile nécessite 2 heures d’exécution avec des erreurs de quelque pcm par rapport à la solution de référence.On propose aussi une approximation d’ordre supérieur du MOC basée sur une expansion axiale polynomiale du flux dans chaque maille. Cette méthode permet une réduction du nombre de mailles (et d’inconnues) tout en gardant la même précision.Toutes les méthodes développées dans ce travail de thèse ont été implémentées dans la version APOLLO3 du solveur de transport TDT. / The topic of our research is the application of the Method of Long Characteristics (MOC) to solve the Neutron Transport Equation in three-dimensional axial geometries. The strength of the MOC is in its precision and versatility. As a drawback, it requires a large amount of computational resources. This problem is even more severe in three-dimensional geometries, for which unknowns reach the order of tens of billions for assembly-level calculations.The first part of the research has dealt with the development of optimized tracking and reconstruction techniques which take advantage of the regularities of three-dimensional axial geometries. These methods have allowed a strong reduction of the memory requirements and a reduction of the execution time of the MOC calculation.The convergence of the iterative scheme has been accelerated with a lower-order transport operator (DPN) which is used for the initialization of the solution and for solving the synthetic problem during MOC iterations.The algorithms for the construction and solution of the MOC and DPN operators have been accelerated by using shared-memory parallel paradigms which are more suitable for standard desktop working stations. An important part of this research has been devoted to the implementation of scheduling techniques to improve the parallel efficiency.The convergence of the angular quadrature formula for three-dimensional cases is also studied. Some of these formulas take advantage of the reduced computational costs of the treatment of planar directions and the vertical direction to speed up the algorithm.The verification of the MOC solver has been done by comparing results with continuous-in-energy Monte Carlo calculations. For this purpose a coupling of the 3D MOC solver with the Subgroup method is proposed to take into account the effects of cross sections resonances. The full calculation of a FBR assembly requires about 2 hours of execution time with differences of few PCM with respect to the reference results.We also propose a higher order scheme of the MOC solver based on an axial polynomial expansion of the unknown within each mesh. This method allows the reduction of the meshes (and unknowns) by keeping the same precision.All the methods developed in this thesis have been implemented in the APOLLO3 version of the neutron transport solver TDT.

Cette thèse présente la construction de schémas distribuant le résidu (RD) d'ordre très élevés, pour la discrétisation d'équations d'advection-diffusion multidimensionnelles et stationnaires sur maillages non structurés. Des schémas linéaires ainsi que des schémas non linéaires sont considérés. Une approximation de la solution polynomiale par morceaux et continue sur chaque élément est adoptée, de plus une procédure de reconstruction du gradient que celle de la solution numérique est utilisée afin d'avoir une représentation continue de la solution numérique et de son gradient. Il est montré que le gradient doit être reconstruit avec la même précision de la solution, sans quoi la précision formel du schéma numérique est perdue dans les cas où les effets de diffusion prévalent sur les effets d'advection, et aussi quand l'advection et la diffusion sont également importants. Ensuite, la méthode est étendue à des systèmes d'équations, en particulier aux équations de Navier-Stokes et aux équations RANS. La précision, l'efficacité et la robustesse du solveur RD implicite sont démontrées sur plusieurs cas tests. / The construction of compact high-order Residual Distribution schemes for the discretizationof steady multidimensional advection-diffusion problems on unstructuredgrids is presented. Linear and non-linear scheme are considered. A piecewise continuouspolynomial approximation of the solution is adopted and a gradient reconstructionprocedure is used in order to have a continuous representation of both thenumerical solution and its gradient. It is shown that the gradient must be reconstructedwith the same accuracy of the solution, otherwise the formal accuracy ofthe numerical scheme is lost in applications in which diffusive effects prevail overthe advective ones, and when advection and diffusion are equally important. Thenthe method is extended to systems of equations, with particular emphasis on theNavier-Stokes and RANS equations. The accuracy, efficiency, and robustness of theimplicit RD solver is demonstrated using a variety of challenging aerodynamic testproblems.