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Kattelans, Thorsten. "The least squares spectral collocation method for incompressible flows." Berlin Köster, 2009. http://d-nb.info/997987812/04.
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Yuan, Huifang. "Spectral collocation methods for the fractional PDEs in unbounded domain." HKBU Institutional Repository, 2018. https://repository.hkbu.edu.hk/etd_oa/558.
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This thesis is concerned with a particular numerical approach for solving the fractional partial differential equations (PDEs). In the last two decades, it has been observed that many practical systems are more accurately described by fractional differential equations (FDEs) rather than the traditional differential equation approaches. Consequently, it has become an important research area to study the theoretical and numerical aspects of various types of FDEs. This thesis will explore high order numerical methods for solving FDEs numerically. More precisely, spectral methods which exhibits exponential order of accuracy will be investigated. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss quadrature points. In this work, Hermite and modified rational functions are employed to serve as basis functions for solutions that decay exponentially and algebraically, respectively. The main emphasis of this thesis is to propose the spectral collocation method for FDEs posed in unbounded domains. Components of the differentiation matrix involving fractional Laplacian are derived which can then be computed recursively using the properties of confluent hypergeometric function or hypergeometric function. The first part of the thesis introduces preliminaries useful for other parts of the thesis. Review of the relevant definitions and properties of special functions such as Hermite functions, Bessel functions, hypergeometric functions, Gegenbauer polynomials, mapped Jacobi polynomials, modified rational functions are presented. Fractional Sobolev space is introduced and some lemmas on interpolation approximation in the fractional Sobolev space are also included. In the second part of the thesis, we present the spectral collocation method based on Hermite functions. Two bases are used, namely, the over-scaled Hermite function and generalized Hermite function, which are orthogonal functions on the whole line with appropriate weight functions. We will show that the fractional Laplacian of these two kinds of Hermite functions can be represented by confluent hypergeometric function. Behaviors of the condition numbers for the resulting spectral differentiation matrices with respect to the number of expansion terms are investigated. Moreover, approximation in two-dimensional space using the tensorized bases, application to multi-term problems and use of scaling to match different decay rate are also considered. Convergence analysis for generalized Hermite function are derived and numerical errors for two bases are analyzed. The third part of the thesis deals with the spectral collocation method based on modified rational functions. We first give a brief introduction for computation of the fractional Laplacian using modified rational functions, which is represented by hypergeometric functions. Then the differentiation matrix involving the fractional Laplace operator is given. Convergence analysis for modified Chebyshev rational functions and modified Legendre rational functions are derived and numerical experiments are carried out.
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Cameron, James M. "Spectral collocation and path-following methods for reaction-diffusion equations in one and two space dimensions." Thesis, Heriot-Watt University, 1994. http://hdl.handle.net/10399/1346.
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Merle, Matthieu. "Approches numériques pour l'analyse globale d'écoulements pariétaux en régime subsonique." Thesis, Paris, ENSAM, 2015. http://www.theses.fr/2015ENAM0026/document.
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Dans le cadre de l'étude des écoulements ouverts, deux types de dynamiques coexistent. Les écoulements de type oscillateur qui présentent une fréquence propre d'oscillation indépendante des perturbations extérieures (dynamique intrinsèque), ainsi que les écoulements de type amplificateur sélectif de bruit comme les écoulements de jets ou de couches limites décollées, caractérisés par une plus large gamme de fréquences dépendantes essentiellement de bruit extérieur (dynamique extrinsèque). Les études de couches limites décollées en régime incompressible ont montré un lien entre le phénomène auto-entretenu de basse fréquence qui apparaît et l'interaction non normale des modes globaux instables existants pour ce type de configuration. L'objectif de ce travail consiste à étendre cette interprétation lorsque l'écoulement est en régime subsonique. Dans ce but, un travail d'adaptation des conditions aux limites non-réfléchissantes aux problèmes de stabilité globale a été réalisé. Une méthode de zone absorbante de type Perfectly Matched Layer a été implémentée dans un code de simulation numérique utilisant des méthodes de collocation spectrale. Une méthode de décomposition de domaine adaptée aux calculs des solutions stationnaires ainsi qu'aux problèmes de stabilité globale a également été utilisée pour permettre la validation des conditions aux limites implémentées sur un cas d'écoulement rayonnant de cavité ouverte. Enfin, les études de stabilité d'un écoulement de couche limite décollée derrière une géométrie de type bosse ont été réalisées. L'étude des instabilités bidimensionnelles, responsables du phénomène basse fréquence (flapping), et réalisées en régime subsonique montre que le mécanisme observé en régime incompressible est aussi observé en régime subsonique. La stabilité de cet écoulement vis-à-vis de perturbations tri-dimensionnelles, et plus particulièrement les instabilités centrifuges ont aussi été étudiées en fonction du nombre de MachIn open flows context, there are generally two types of dynamic : oscillators, such as cylinder flow, exhibit a well defined frequency insensitive to external perturbations (intrinsic dynamics) and noise amplifiers, such as boundary layers, jets or in some cases the separated flows, which are characterized by wider spectrum bands that depend essentially on the external noise (dynamic extrinsic). Previous studies have shown that separated flows are subject to self-induced oscillations of low frequency in incompressible regime. These studies have revealed links between the interaction of non-normal modes and low oscillations in an incompressible boundary-layer separation and it will be to establish the validity of this interpretation in a compressible regime. In this regard, non-reflecting boundary conditions have been developed to solve the eigenvalue problem formed by linearised Navier-Stokes equations. An absorbing region known as Perfectly Matched Layer has been implemented in order to damp acoustic perturbations which are generated when the compressibility of the flow is considered. A multi-domain approach using spectral collocation discretisation has also been developed in order to study the influence of this absorbing region on the stability analysis of an open cavity flow which is known to generate acoustic perturbations. Finally, we focused on separated boundary layer induced by a bump geometry in order to understand what are the effects of compressibility on the bidimensional low frequency phenomenon and also on transverse instabilities which are known to be unstable for a lots of separated flows
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Alici, Haydar. "Pseudospectral Methods For Differential Equations: Application To The Schrodingertype Eigenvalue Problems." Master's thesis, METU, 2003. http://etd.lib.metu.edu.tr/upload/1086198/index.pdf.
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In this thesis, a survey on pseudospectral methods for differential equations is presented. Properties of the classical orthogonal polynomials required in this context are reviewed. Differentiation matrices corresponding to Jacobi, Laguerre,and Hermite cases are constructed. A fairly detailed investigation is made for the Hermite spectral methods, which is applied to the Schrödinger eigenvalue equation with several potentials. A discussion of the numerical results and comparison with other methods are then introduced to deduce the effciency of the method.
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Karkar, Sami. "Méthodes numériques pour les systèmes dynamiques non linéaires : application aux instruments de musique auto-oscillants." Phd thesis, Aix-Marseille Université, 2012. http://tel.archives-ouvertes.fr/tel-00742651.
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Ces travaux s'articulent autour du calcul des solutions périodiques dans les systèmes dynamiques non linéaires, au moyen de méthodes numériques de continuation. La recherche de solutions périodiques se traduit par un problème avec conditions aux limites périodiques, pour lequel nous avons implémenté deux méthodes d'approximation : - Une méthode spectrale dans le domaine fréquentiel : l'équilibrage harmonique d'ordre élevé, qui repose sur une formulation quadratique des équations. Nous proposons en outre une formulation originale permettant d'étendre cette méthode aux cas de non-linéarités non rationnelles. - Une méthode pseudo-spectrale par éléments dans le domaine temporel : la collocation à l'aide fonctions polynômiales par morceaux. Ces méthodes transforment le problème continu en un système d'équations algébriques non linéaires, dont les solutions sont calculées par continuation à l'aide de la méthode asymptotique numérique. L'ensemble de ces outils, intégrés au code de calcul MANLAB et complétés d'une analyse linéaire de stabilité, sont alors utilisés pour l'étude des régimes périodiques d'une classe particulière de systèmes dynamiques non linéaires : les instruments de musique auto-oscillants. Un modèle physique non-régulier de clarinette est étudié en détail : à partir de la branche de solutions statiques et ses bifurcations, on calcule les différentes branches de solutions périodiques, ainsi que leur stabilité et leurs bifurcations. Ce modèle est ensuite adapté au cas du saxophone, pour lequel on intègre une caractérisation acoustique expérimentale, afin de mieux tenir compte de la géométrie complexe de l'instrument. Enfin, nous étudions un modèle physique simplifié de violon, avec une non-régularité liée frottement de Coulomb. Cette dernière application illustre ainsi la polyvalence des outils développés face aux différents types de non-régularité.
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Singh, Pranav. "High accuracy computational methods for the semiclassical Schrödinger equation." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/274913.
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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.
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Ehrenstein, Uwe. "Méthodes spectrales de résolution des équations de Stokes et de Navier-Stokes : application à des écoulements de convection double diffusive." Nice, 1986. http://www.theses.fr/1986NICE4056.
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Résolution des équations en formulation fonction de courant tourbillon. Pour le problème de Stokes périodique, les équations sont approchées par les séries de Fourier, polynômes de Tchebychev en espace et par des schémas aux différences finies en temps ; stabilité des systèmes résultants. Application des méthodes de Tau-Tchebychev et collocation-Tchebychev au problème de Stokes stationnaire et non périodique. Application de l'algorithme de collocation-Tchebychev aux équations de Navier-Stokes instationnaires puis aux équations des écoulements de convection double-diffusive
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Azaiez, Majdi. "Calcul de la pression dans le problème de stokes pour des fluides visqueux incompressibles par une méthode spectrale de collocation." Paris 11, 1991. http://www.theses.fr/1990PA112364.
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Dans ce memoire nous presentons une methode spectrale de collocation pour resoudre les equations de stokes pour des fluides visqueux incompressibles, et en particulier pour satisfaire, a la precision spectrale la contrainte de continuite. On presente d'abord un solveur de helmholtz et on donne ensuite une premiere discretisation des equations de stokes utilisant, pour la vitesse et la pression, des polynomes de meme degre. Enfin on presente une methode d'approximation ameliorant la premiere
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Zakaria, Abdellatif. "Étude de divers schémas pseudo-spectraux de type collocation pour la résolution des équations aux dérivées partielles : application aux équations de Navier-stokes." Nice, 1985. http://www.theses.fr/1985NICE4022.
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On présente un ensemble de schémas pseudo spectraux de type collocation, basés sur des approximations spatiales en polynômes de Tchebychev et s'appliquant a la résolution des problèmes instationnaires. On étudie la stabilité numérique de ces schémas dans le cas d'une équation de diffusion, d'advection-diffusion et d'advection. On considère une méthode mixte spectrale aux différences finies, bien adaptée aux problèmes non linéaires stationnaires. On l'applique a la résolution des équations de Navier-Stokes pour les mouvements stationnaires d'un fluide visqueux incompressible
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Robitaillié-Montané, Cécilia. "Une approche non locale pour l'étude des instabilités linéaires : application à l'écoulement de couche limite compressible le long d'une ligne de partage." Toulouse, ENSAE, 2005. http://www.theses.fr/2005ESAE0005.
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Ce travail propose une généralisation de l'approche locale classique de stabilité linéaire pouvant être appliquée à des écoulements complexes. L'approche locale est réservée dans le cas incompressible à des écoulements dits parallèles et conduit à un problème donné sous forme d’une équation différentielle ordinaire. L'approche non locale développée permet d’étudier n'importe quel écoulement stationnaire ayant une évolution constante dans une des trois directions d’espace. Le problème est alors donné sous forme d’équations aux dérivées partielles (EDP). L'imposition de conditions limites homogènes, parfois délicates à établir, conduit a l'écriture d’un problème aux valeurs propres. Les deux directions d’espace intervenant dans les EDP sont discrétisées à l’aide d’une méthode de collocation spectrale. Le problème est ainsi représenté par des matrices de grande taille dont la détermination des valeurs propres fait appel à la méthode d’Arnoldi. L’approche non locale est ensuite mise en œuvre pour étudier le développement d'instabilités naturelles au sein de l'écoulement de couche limite le long du bord d’attaque d’une voilure d’avion. L'écoulement est modélisé par l'écoulement de Hiemenz en flèche. Les études sont réalisées d’abord en incompressible puis en compressible. La perturbation la plus instable correspond toujours au mode de Görtler-Hämmerlin donné par l'approche locale en incompressible mais d’autres modes d’instabilité de fréquences voisines sont également mis en évidence. De façon semblable au comportement d’une couche limite qui se développe sur une plaque plane on note les effets stabilisants d’une augmentation de nombre de Mach jusqu’à un certain seuil et d’un refroidissement de paroi. Enfin, la présence de modes de type acoustique se propageant dans la direction normale à la paroi est également discutée dans ce mémoire.
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Shen, Jie. "Résolution numérique des équations de Stokes et Navier-Stokes par les méthodes spectrales." Paris 11, 1987. http://www.theses.fr/1987PA112221.
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Nous présentons des méthodes spectrales pour la résolution des équations de Stokes et de Navier-Stokes. D'abord, nous construisons un solveur rapide de Helmholtz, basé sur la méthode de Tchebychev-Tau, qui sera fréquemment utilisé par la suite. Ensuite, nous considérons deux méthodes pour l'approximation du problème de Stokes: l'une est de type Uzawa; l'autre est appelée la méthode de la matrice d'influence. Divers résultats théoriques et numériques sont présentés. Enfin, nous proposons deux schémas pour l'approximation des équations de NavierStokes d'évolution. Nous montrons que ces deux schémas sont tous deux inconditionnellement stables et convergentsWe present spectral methods for solving Stokes et Navier-Stokes equations. First of all, we construct a fast Helmholtz solver, based on Chebychev-Tau method, which we will use frequently later. Then, we consider two methods for the approximation of Stokes problem : one is based on the Uzawa algorithm, the other is called the influence matrix method. Several theoritical and numerical results are presented. Finally, we propose two schemes for the approximation of Navier-Stokes equations. We prove that both schemes are unconditionary stable and convergent
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Boyer, Germain. "Étude de stabilité et simulation numérique de l’écoulement interne des moteurs à propergol solide simplifiés." Thesis, Toulouse, ISAE, 2012. http://www.theses.fr/2012ESAE0029/document.
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Cette thèse vise à modéliser les instabilités hydrodynamiques générant des détachements tourbillonnaires pariétaux (ou VSP) responsables des Oscillations De Pression dans les moteurs à propergol solide longs et segmentés par interaction avec l’acoustique du moteur. Ces instabilités sont modélisées en tant que modes de stabilité linéaire globaux de l’écoulement d’un conduit à parois débitantes. En supposant que les structures pariétales émergent d’une perturbation de l’écoulement de base, des modes discrets et indépendants du maillage utilisé sont calculés. Dans ce but, une discrétisation par collocation spectrale multi-domaine est implémentée dans un solveur parallèle afin de s’affranchir de la croissance polynomiale des fonctions propres et de la présence de couches limites. Les valeurs propres ainsi calculées dépendent explicitement des frontières du domaine, à savoir la position de la perturbation et celle de la sortie, et sont ensuite validées par simulation numérique directe. On montre alors qu’elles permettent bien de décrire la réponse à une perturbation initiale de l’écoulement modifié par une rupture de débit pariétale. Ensuite, la simulation d’une réponse forcée par l’acoustique se fait sous forme de structures tourbillonnaires dont les fréquences discrètes sont en accord avec celles des modes de stabilité. Ces structures sont réfléchies en ondes de pression de même fréquences remontant l’écoulement. Finalement, la simulation numérique et la théorie de la stabilité permettent de montrer que le VSP, dont la réponse est linéaire vis-à-vis d’un forçage compressible comme l’acoustique, est le phénomène moteur des Oscillations De PressionThe current work deals with the modeling of the hydrodynamic instabilities that play a major role in the triggering of the Pressure Oscillations occurring in large segmented solid rocket motors. These instabilities are responsible for the emergence of Parietal Vortex Shedding (PVS) and they interact with the boosters acoustics. They are first modeled as eigenmodes of the internal steady flowfield of a cylindrical duct with sidewall injection within the global linear stability theory framework. Assuming that the related parietal structures emerge from a baseflow disturbance, discrete meshindependant eigenmodes are computed. In this purpose, a multi-domain spectral collocation technique is implemented in a parallel solver to tackle numerical issues such as the eigenfunctions polynomial axial amplification and the existence of boundary layers. The resulting eigenvalues explicitly depend on the location of the boundaries, namely those of the baseflow disturbance and the duct exit, and are then validated by performing Direct Numerical Simulations. First, they successfully describe flow response to an initial disturbance with sidewall velocity injection break. Then, the simulated forced response to acoustics consists in vortical structures wihich discrete frequencies that are in good agreement with those of the eigenmodes. These structures are reflected into upstream pressure waves with identical frequencies. Finally, the PVS, which response to a compressible forcing such as the acoustic one is linear, is understood as the driving phenomenon of the Pressure Oscillations thanks to both numerical simulation and stability theory
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Sendrowski, Janek. "Feigenbaum Scaling." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-96635.
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In this thesis I hope to provide a clear and concise introduction to Feigenbaum scaling accessible to undergraduate students. This is accompanied by a description of how to obtain numerical results by various means. A more intricate approach drawing from renormalization theory as well as a short consideration of some of the topological properties will also be presented. I was furthermore trying to put great emphasis on diagrams throughout the text to make the contents more comprehensible and intuitive.
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Hu, Shih-Cong, and 胡世聰. "Spectral Collocation Methods for Semilinear Problems." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/d6ks4p.
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碩士國立中山大學
應用數學系研究所
96
In this thesis, we extend the spectral collocation methods(SCM) (i.e., pseudo-spectral method) in Quarteroni and Valli [27] for the semilinear, parameter-dependentproblems(PDP) in the square with the Dirichlet boundary condition. The optimal error bounds are derived in this thesis for both H1 and L2 norms. For the solutions sufficiently smooth, the very high convergence rates can be obtained. The algorithms of the SCM are simple and easy to carry out. Only a few of basis functions are needed so that not only can the high accuracy of the PDP solutions be achieved, but also a great deal of CPU time may be saved. Moreover, for PDP the stability analysis of SCM is also made, to have the same growth rates of condition number as those for Poisson’s equation. Numerical experiments are carried out to verify the theoretical analysis made.
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Su, Yuhong. "Collocation spectral methods in the solution of poisson equation." Thesis, 1998. http://hdl.handle.net/2429/8275.
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The Poisson equation is a very important partial differential equation for many
branches of science and engineering. A fast, robust and accurate Poisson equation solver
can find immediate applications in many fields such as electrical engineering, plasma
physics, incompressible fluid mechanics and space science. In this thesis, the collocation
pseudospectral method is employed to solve Poisson equation.
Many papers propose how to apply spectral methods to solve partial differential
equations in Cartesian coordinates, but not much attention has been paid to the use
of spectral methods in polar coordinates and cylindrical coordinates. In this thesis, the
application of the collocation spectral methods to the solution of one and two-dimensional
Poisson equations on a rectangular domain in Cartesian coordinates; and on a disk, a
part of disk and an annulus in polar coordinates. Also considered is the three-dimensional
Poisson equation in a cube in Cartesian coordinates; and a cylinder, a cylindrical annulus
and part of a cylinder in cylindrical coordinates.
Two of the most important approaches in this thesis are: First we put forward a
new collocation spectral method that can avoid the coordinate singularities and solve
the Poisson equation in a disk directly by the eigenvalue technique. This can simplify
the use of the spectral method in polar coordinates and cylindrical coordinates, where
coordinate singularities cause problems for spectral methods. Second, we also give an
algorithm that can directly solve the discrete Poisson equation in cylindrical coordinates
after discretization by the collocation spectral method. Basically, the idea is that we
combine r and z directions together and transform the equation into a form that can be
solved by an eigenvalue technique. This method is very fast and efficient.
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陳駿逸. "On Spectral Collocation Method Applied in Solving Partial Differential Equation." Thesis, 2004. http://ndltd.ncl.edu.tw/handle/88313077179952084918.
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Chang, Shu-Hao, and 張書豪. "Spectral collocation methods for the numerical solutions of Gross-Pitaevskii equation." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/80755376529457953218.
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碩士國立中興大學
應用數學系所
98
We describe an efficient spectral collocation method (SCM) for the numerical solutions of the Gross-Pitaevskii equation (GPE), where the Legendre polynomials are used as the basis functions for the trial function space. We show that the SCM obtained in this way is equivalent to the spectral- Galerkin method (SGM) involving the Gauss-Legendre integration. The SCM is incorporated in the context of continuaton methods for computing energy levels and wave functions of the stationary state nonlinear Schr¨odinger equation. Numerical results on the Bose-Einstein condensates (BEC) in a periodic potential are reported.
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Guo, Deng-Yao, and 郭登堯. "Adaptive Chebyshev Spectral Collocation on Solving One- Dimensional Shock Wave Problem." Thesis, 1996. http://ndltd.ncl.edu.tw/handle/63894535034737516639.
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Hu, Shin Yum, and 胡馨云. "Spectral Collocation Method on solving the linear stability of axisymmetric jet." Thesis, 1995. http://ndltd.ncl.edu.tw/handle/07541619141893067838.
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Huang, Chia-Chien, and 黃家健. "A Full-Vectorial Multidomain Spectral Collocation Method for Modal Analysis of Optical Waveguides." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/23362687003555126385.
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博士國立臺灣大學
應用力學研究所
91
Abstract An accurate and efficient solution method using full-vertorial multidomain spectral collocation method is proposed for computing optical waveguides with discontinuous refractive index profiles. The method is formulated in terms of the transverse magnetic fields. The use of domain decomposition divides the usual single domain into a few subdomains at the interfaces of discontinuous refractive index profiles. Each subdomain can be expanded by a suitable set of orthogonal basis functions and patched at these interfaces by matching the physical boundary conditions. In addition, a new technique which incorporating the effective index method and the Wentzel-Kramers-Brillouin method for the a priori determination of the scaling factor in Hermite-Gauss or Laguerre-Gauss basis functions is introduced to considerably save the computational time without experimenting with it. The present method shares the same desirable property of spectral collocation method that can provide fast and accurate solution but avoids the oscillatory solutions and improves the poor convergence problem of simple spectral collocation method with single domain where regions of discontinuous refractive index profiles exist. Computations of several 2-D and 3-D waveguide structures, such as three-layer, planar diffused, metal-clad, planar directional coupler, diffused channel, rectangular dielectric, and semiconductor rib waveguides have been carried out to test the accuracy and efficiency of the present method. Detailed comparisons of the present results with exact solutions or previously published data based on other methods are included and all the results are found to be in excellent agreement.
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Subich, Christopher. "Simulation of the Navier-Stokes Equations in Three Dimensions with a Spectral Collocation Method." Thesis, 2011. http://hdl.handle.net/10012/5926.
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This work develops a nonlinear, three-dimensional spectral collocation method for the simulation of the incompressible Navier-Stokes equations for geophysical and environmental flows. These flows are often driven by the interaction of stratified fluid with topography, which is accurately accounted for in this model using a mapped coordinate system. The spectral collocation
method used here evaluates derivatives with a Fourier trigonometric or Chebyshev polynomial expansion as appropriate, and it evaluates the nonlinear terms directly on a collocated grid. The coordinate mapping renders ineffective fast solution methods that rely on separation of variables,
so to avoid prohibitively expensive matrix solves this work develops a low-order finite-difference preconditioner for the implicit solution steps. This finite-difference preconditioner is itself too expensive to apply directly, so it is solved pproximately with a geometric multigrid method, using semicoarsening and line relaxation to ensure convergence with locally anisotropic grids. The model is discretized in time with a third-order method developed to allow variable timesteps. This multi-step method explicitly evaluates advective terms and implicitly evaluates pressure and viscous terms. The model’s accuracy is demonstrated with several test cases: growth rates of Kelvin-Helmholtz billows, the interaction of a translating dipole with no-slip boundaries, and the generation of internal waves via topographic interaction. These test cases also illustrate the model’s use from a high-level programming perspective. Additionally, the results of several large-scale simulations are discussed: the three-dimensional dipole/wall interaction, the evolution of internal waves with shear instabilities, and the stability of the bottom boundary layer beneath internal waves. Finally, possible future developments are discussed to extend the model’s capabilities and optimize its performance within the limits of the underlying numerical algorithms.
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Chen, Po Chi, and 陳柏旗. "Spectral Collocation Method on solving the Linear Stablity of the Flow in a Nutation Damper." Thesis, 1995. http://ndltd.ncl.edu.tw/handle/44368521074117042964.
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Oguntala, George A., and Raed A. Abd-Alhameed. "Thermal Analysis of Convective-Radiative Fin with Temperature-Dependent Thermal Conductivity Using Chebychev Spectral Collocation Method." 2018. http://hdl.handle.net/10454/15344.
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yesIn this paper, the Chebychev spectral collocation method is applied for the thermal analysis of convective-radiative straight fins with the temperature-dependent thermal conductivity. The developed heat transfer model was used to analyse the thermal performance, establish the optimum thermal design parameters, and also, investigate the effects of thermo-geometric parameters and thermal conductivity (nonlinear) parameters on the thermal performance of the fin. The results of this study reveal that the rate of heat transfer from the fin increases as the convective, radioactive, and magnetic parameters increase. This study establishes good agreement between the obtained results using Chebychev spectral collocation method and the results obtained using Runge-Kutta method along with shooting, homotopy perturbation, and adomian decomposition methods.
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Makibelo, Vuyelwa. "Application of the bivariate chebyshev spectral collocation quasi-linearisation method for non-similar boundary layer equations." Thesis, 2017. https://hdl.handle.net/10539/25824.
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