Métodos de resolução numérica de equações diferenciais parciais que utilizam ondaletas como base vêm sendo desenvolvidos nas últimas décadas, mas existe uma carência de estudos mais profundos das características computacionais dos mesmos. Neste estudo analisou-se detalhadamente um método espectral de Galerkin com base de ondaletas harmônicas. Revisou-se a teoria matemática referente às ondaletas harmônicas, que mostrou ter grande similaridade com a teoria referente à base trigonométrica de Fourier. Diversos testes numéricos foram realizados. Ao analisarmos a resolução da equação do transporte linear, e também de transporte não linear (equação de Burgers), obtivemos boas aproximações da solução esperada. O custo computacional obtido foi similar ao método com base de Fourier, mas com ondaletas harmônicas foi possível usar a localidade das ondaletas para detectar características de localidade do sinal. Analisamos ainda uma abordagem pseudo-espectral para os casos não lineares, que resultaram em um expressivo aumento de eficiência. Tendo em vista o uso das propriedades de localidade das ondaletas, usamos o método de Galerkin com base de ondaletas harmônicas para resolver um sistema de equações referente a um modelo de propagação de frentes de precipitação. O método mostrou boas aproximações das soluções esperadas, custo computacional ótimo e ainda a possibilidade de se obter espectralmente informações sobre a localização da frente de precipitação. / Numerical methods to solve partial differential equations based on wavelets have been developed in the last two decades, but there is a lack of studies on their computational characteristics. In this study a Galerkin spectral method using harmonic wavelets base has been thoroughly analyzed. We performed a review on the mathematics of harmonic wavelets, that showed a great similarity with Fourier basis. Several numerical experiments were made. Analyzing the use of the Galerkin method, with harmonic wavelets, on linear and non linear transport equations, we achieved good approximations in respect to the expected solution. The computational cost resulted to be similar to the same method with Fourier basis. On the other hand, employing harmonic wavelets we were able to obtain local information of the solution by simple inspection of the spectral coeffcients. We also analyzed a pseudo-spectral method based on harmonic wavelets for the non linear equations, resulting in a great improvement in efficiency. Looking towards using the locality propriety of harmonic wavelets, we tested the Galerkin method on a precipitation front propagation model. The method resulted in good approximations to the expected solution, optimal computational cost and the possibility of obtaining information on the locality of the precipitation fronts spectrally.

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.

<p> The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, u_x]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0.</p><p>The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense.</p><p>The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.</p>

The inverse heat conduction problem also frequently referred as the sideways heat equation, in short SHE, is considered as a mathematical model for a real application, where it is desirable for someone to determine the temperature on the surface of a body. Since the surface itself is inaccessible for measurements, one is restricted to use temperature data from the interior measurements. From a  mathematical point of view, the entire situation leads to a non-characteristic Cauchy problem, where by using recorded temperature one can solve a well-posed nonlinear problem in the finite region for computing heat flux, and consequently obtain the Cauchy data [u, ux]. Further by using these data and by performing an appropriate method, e.g. a space marching method, one can eventually achieve the desired temperature at x = 0. The problem is severely ill-posed in the sense that the solution does not depend continuously on the data. The problem solved by two different methods, and for both cases we stabilize the computations by replacing the time derivative in the heat equation by a bounded operator. The first one, a spectral method based on finite Fourier space is illustrated to supply an analytical approach for approximating the time derivative. In order to get a better accuracy in the numerical computation, we use cubic spline function for approximating the time derivative in the least squares sense. The inverse problem we want to solve, by using Cauchy data, is a nonlinear heat conduction problem in one space dimension. Since the temperature data u = g(t) is recorded, e.g. by a thermocouple, it usually contains some perturbation in the data. Thus the solution can be severely ill-posed if the Cauchy data become very noisy. Two experiments are presented to test the proposed approach.

In spectral methods for numerical PDEs, when the solutions are periodical, the Fourier functions may be used. However, when the solutions are non-periodical, the Legendre and Chebyshev polynomials are recommended, reported in many papers and books. There seems to exist few reports for the study of non-periodical solutions by spectral Fourier methods under the Dirichlet conditions and other boundary conditions. In this paper, we will explore the spectral Fourier methods(SFM) and collocation Fourier methods(CFM) for elliptic and eigenvalue problems. The CFM is simple and easy for computation, thus for saving a great deal of the CPU time. The collocation Fourier methods (CFM) can be regarded as the spectral Fourier methods (SFM) partly with the trapezoidal rule. Furthermore, the error bounds are derived for both the CFM and the SFM. When there exist no errors for the trapezoidal rule, the accuracy of the solutions from the CFM is as accurate as the spectral method using Legendre and Chebyshev polynomials. However, once there exists the truncation errors of the trapezoidal rule, the errors of the elliptic solutions and the leading eigenvalues the CFM are reduced to O(h^2), where h is the mesh length of uniform collocation grids, which are just equivalent to those by the linear elements and the finite difference method (FDM). The O(h^2) and even the superconvergence O(h4) are found numerically. The traditional condition number of the CFM is O(N^2), which is smaller than O(N^3) and O(N^4) of the collocation spectral methods using the Legendre and Chebyshev polynomials. Also the effective condition number is only O(1). Numerical experiments are reported for 1D elliptic and eigenvalue problems, to support the analysis made. The simplicity of algorithms and the promising numerical computation with O(h^4) may grant the CFM to be competent in application in numerical physics, chemistry, engineering, etc., see [7].

Nous considérons dans cette thèse la discrétisation par la méthode spectrale et la simulation numérique de l'écoulement d'un fluide visqueux incompressible occupant le domaine ? modélisé par les équations de Navier-Stokes. Nous avons choisi de les coupler avec l'équation de la chaleur dans le cas ou la viscosité dépend de la température avec des conditions aux limites portant sur la vitesse et la température.La méthode s'avère optimale en ce sens que l'erreur obtenue n'est limitée que par la régularité de la solution. Elle est de type spectrale. Nous donnons des estimations d'erreur a priori optimales et nous confirmons l'étude théorique par des résultats numériques. Nous considérons aussi les équations de Navier-Stokes/chaleur instationnaires dont nous proposons une discrétisation en temps et en espace en utilisant le schéma d'Euler implicite et les méthodes spectrale. Quelques expériences numériques confirment l'intérêt de la discrétisation. / In this thesis we consider the discretization by spectral method and the numerical simulation of a viscous incompressible fluid in the domain ?, the model being the Navier-Stokes equations. We have chosen to couple them with the heat equation where the viscosity of the fluid depends on the temperature, with boundary conditions which involve the velocity and the temperature. The method is proved to be optimal in the sense that the order of convergence is only limited by the regularity of the solution. The numerical analysis of the discrete problem is performed and numerical experiments are presented, they turn out to be in good coherence with the theoretical results. Finally, we consider the unsteady Navier-Stokes/heat equations which models the time-dependent flow. We propose a discretization of this problem that relies on a backward Euler's scheme in time and spectral methods in space and present some numerical experiments which confirm the interest of the discretization.

This dissertation consists of three research projects of kinetic models: a structure preserving scheme for Poisson-Nernst-Planck equations and two efficient spectral methods for multi-dimensional Boltzmann equation.<br><br>The Poisson-Nernst-Planck (PNP) equations is widely used to describe the dynamics of ion transport in ion channels. We introduce a structure-preserving semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is shown to satisfy the following properties: mass conservation, unconditional positivity, and energy dissipation (hence preserving the steady-state). <br><br>Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. In this dissertation, we studied the state of the art in the fast Fourier method and discussed its limitation. Next, we proposed a new approach to implement the Fourier method, which can resolve those issues. <br><br>However, the Fourier method requires a domain truncation which is unphysical since the collision operator is defined in whole space R^d . In the last part of this dissertation, we introduce a Petrov-Galerkin spectral method for the Boltzmann equation in the unbounded domain. The basis functions (both test and trial functions) are carefully chosen mapped Chebyshev functions to obtain desired convergence and conservation properties. Furthermore, thanks to the close relationship of the Chebyshev functions and the Fourier cosine series, we can construct a fast algorithm with the help of the non-uniform fast Fourier transform (NUFFT).<br>

<p>This investigation is motivated by the problem of optimal design of cooling elements in modern battery systems used in hybrid/electric vehicles. We consider a simple model of two-dimensional steady-state heat conduction generated by a prescribed distribution of heat sources and involving a one-dimensional cooling element represented by a closed contour. The problem consists in finding an optimal shape of the cooling element which will ensure that the temperature in a given region is close (in the least squares sense) to some prescribed distribution. We formulate this problem as PDE-constrained optimization and use methods of the shape-differential calculus to obtain the first-order optimality conditions characterizing the locally optimal shapes of the contour. These optimal shapes are then found numerically using the conjugate gradient method where the shape gradients are conveniently computed based on adjoint equations. A number of computational aspects of the proposed approach is discussed and optimization results obtained in several test problems are presented.</p> / Master of Science (MSc)

In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent pattern, we show how to find the equation determining the spectrum. We give two length 4 applications, and a weighted pattern of length 3 where the associated operator only has one non-zero eigenvalue. Using generating functions we show that the error term in the asymptotic expression is the smallest possible.

Etude théorique de l'interaction entre un atome à deux ou trois électrons actifs et un champ laser de fort éclairement (10^13 à 10^15 W.cm-2) et de durée d'impulsion ultra-brève (femto à attoseconde) dans le domaine spectral XUV. Notre approche est basée sur la résolution de l'équation de Schrödinger dépendante du temps. L'impulsion laser est définie par un modèle semi-classique. Les fonctions d'onde sont construites en utilisant des B-splines. Nous étudions particulièrement la double ionisation à deux photons de l'hélium dans l'état fondamental ainsi que dans l'état excité 1s2s. Nous testons une méthode d'approximation pour traiter certains ions héliumoÏdes. Enfin nous abordons le Lithium, système à trois électrons actifs. Nous comparons la double ionisation à deux photons par voie séquentielle et directe avec He(1s2s) dont la structure asymétrique est proche de celle du lithium. / Theoretical study of the interaction between an atom and a two or three electron system with an ultra short (10^-15 to 10^-18 s) high intensity (10^13 à 10^15 W.cm-2) pulse in the XUV domain. Our approach is based on the solving of th time dependent Schrödinger equation. Laser pulse is defined by a semi classical model. Wave functions are built with B-splines.We study particularly helium two photons double ionization in fundamental and excited state 1s2s. Then we test an approximative method to treat helium-like ions.At last, we investigate lithium, a three active electrons system. We compare TPDI in sequential and direct channel with He(1s2s) which asymmetric structure looks like Li's one.