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## ‣ Resolução numérica de EDPs utilizando ondaletas harmônicas; Numerical resolution of partial differential equations using harmonic wavelets

Peixoto, Pedro da Silva
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
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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...

## ‣ A mixed spectral method for incompressible viscous fluid flow in an infinite strip

Zhong-Qing,Wang; Ben-yu,Guo
Tipo: Artigo de Revista Científica Formato: text/html
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This paper considers the numerical simulation of incompressible viscous fluid flow in an infinite strip. A mixed spectral method is proposed using the Legendre approximation in one direction and the Legendre rational approximation in another direction. Numerical results demonstrate the efficiency of this approach. Some results on the mixed Legendre-Legendre rational approximation are established, from which the stability and convergence of proposed method follow.

## ‣ On a linearisation method for Reiner-Rivlin swirling flow

Makukula,Zodwa G.; Sibanda,Precious; Motsa,Sandile S.
Tipo: Artigo de Revista Científica Formato: text/html
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The steady flow of a Reiner-Rivlin fluid with Joule heating and viscous dissipationis studied. We present a novel technique for accelerating the convergence of the spectral-homotopy analysis method. Solutions of the nonlinear momentum and energy equations are obtained using the improved spectral homotopy analysis method. Solutions were also generated using the spectral-homotopy analysis method and benchmarked against results in the literature. Mathematical subject classification: Primary: 76A05, 76N05; Secondary: 76M25.

## ‣ A chebyshev collocation spectral method for numerical simulation of incompressible flow problems

Martinez,Johnny de Jesús; Esperança,Paulo de Tarso T.
Fonte: Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM Publicador: Associação Brasileira de Engenharia e Ciências Mecânicas - ABCM
Tipo: Artigo de Revista Científica Formato: text/html
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This paper concerns the numerical simulation of internal recirculating flows encompassing a two-dimensional viscous incompressible flow generated inside a regularized square driven cavity and over a backward-facing step. For this purpose, the simulation is performed by using the projection method combined with a Chebyshev collocation spectral method. The incompressible Navier-Stokes equations are formulated in terms of the primitive variables, velocity and pressure. The time integration of the spectrally discretized, incompressible Navier-Stokes equations is performed by a second-order mixed explicit/implicit time integration scheme. This scheme is a combination of the Crank-Nicolson scheme operating on the diffusive terms and a second-order Adams-Bashforth scheme acting on the advective terms. The projection method is used to split the solution of the incompressible Navier-Stokes equations in two decoupled problems: the Burgers equation to predict an intermediate velocity field and the Poisson equation for the pressure, which is used to correct the intermediate velocity field and satisfy the continuity equation. Numerical simulations for flows inside a two-dimensional regularized square driven cavity for Reynolds numbers up to 10000 and over a backward-facing step for Reynolds numbers up to 875 are presented and compared with numerical results previously published...

## ‣ A Legendre tau-Spectral Method for Solving Time-Fractional Heat Equation with Nonlocal Conditions

Bhrawy, A. H.; Alghamdi, M. A.
Fonte: Hindawi Publishing Corporation Publicador: Hindawi Publishing Corporation
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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We develop the tau-spectral method to solve the time-fractional heat equation (T-FHE) with nonlocal condition. In order to achieve highly accurate solution of this problem, the operational matrix of fractional integration (described in the Riemann-Liouville sense) for shifted Legendre polynomials is investigated in conjunction with tau-spectral scheme and the Legendre operational polynomials are used as the base function. The main advantage in using the presented scheme is that it converts the T-FHE with nonlocal condition to a system of algebraic equations that simplifies the problem. For demonstrating the validity and applicability of the developed spectral scheme, two numerical examples are presented. The logarithmic graphs of the maximum absolute errors is presented to achieve the exponential convergence of the proposed method. Comparing between our spectral method and other methods ensures that our method is more accurate than those solved similar problem.

## ‣ Simulação de escoamentos não-periódicos utilizando as metodologias pseudo-espectral e da fronteira imersa acopladas; Simulation of non-periodics flows using the fourier pseudo-spectral and immersed boundary methods

Mariano, Felipe Pamplona
Tipo: Dissertação
Português
Relevância na Pesquisa
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Para compreender fenômenos relacionados à combustão, aeroacústica, transição a turbulência entre outros, a Dinâmica de Fluídos Computacional (CFD) utiliza os métodos de alta ordem. Um dos mais conhecidos é o método pseudo-espectral de Fourier, o qual alia: alta ordem de precisão na resolução das equações, com um baixo custo computacional. Este está ligado à utilização da FFT e do método da projeção do termo da pressão, o qual desvincula os cálculos da pressão da resolução das equações de Navier-Stokes. O procedimento de calcular o campo de pressão, normalmente é o mais oneroso nas metodologias convencionais. Apesar destas vantagens, o método pseudo-espectral de Fourier só pode ser utilizado para resolver problemas com condições de contorno periódicas, limitando o seu uso no campo da dinâmica de fluídos. Visando resolver essa restrição uma nova metodologia é proposta no presente trabalho, que tem como objetivo simular escoamentos não-periódicos utilizando o método pseudo-espectral de Fourier. Para isso, é utilizada a metodologia da Fronteira Imersa, a qual representa as condições de contorno de um escoamento através de um campo de força imposto nas equações de Navier-Stokes. Como teste...

## ‣ Simulação de grandes escalas de jatos periódicos temporais utilizando a metodologia psedo-espectral de Fourier; Large eddy simulation of periodic temporal jets using the Fourier pseudo-spectral method

Moreira, Leonardo de Queiroz
Tipo: Dissertação
Português
Relevância na Pesquisa
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A turbulência nos fluídos é um dos problemas mais desafiadores da atualidade, em especial no que se refere às aplicações industriais que envolvem processos de mistura de componentes, transferência de calor, lubrificação e degelo, injeção de combustível em câmaras de combustão, sistemas de propulsão de aviões e aeronaves. Diante de considerável interesse, no presente trabalho objetivou-se a análise da transição a turbulência de jatos em desenvolvimento temporal a números de Reynolds moderados utilizando a metodologia LES. Primeiramente desenvolveu-se um código computacional ESPC3D, com alta ordem de resolução para simulação de escoamentos do tipo jatos em desenvolvimento temporal em transição e/ou turbulentos. O código foi desenvolvido no Laboratório de Transferência de Calor e Massa e Dinâmica dos Fluidos (LTCM). Resultados consistentes foram obtidos do ponto de vista da análise física utilizando o código ESPC3D, com o qual realizou-se simulações de grandes escalas empregando o método pseudo-espectral de Fourier. Os resultados das simulações permitem verificar a transição a turbulência bem como suas estruturas típicas. Foi possível também verificar a influência da modelagem da turbulência utilizando a metodologia LES...

## ‣ Modelagem matemática de escoamentos bifásicos usando o Meto- Do Espectral de Fourier; Mathematics modeling of two-phase ows using spectral method of Fourier

Villela, Mariana Fernandes dos Santos
Tipo: Dissertação
Português
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A simulação numérica de escoamentos bifásicos requer alta acurácia para se obter maiores detalhes do escoamento. Além disso, busca-se baixo custo computacional, pois de modo geral, as metodologias necessitam de um elevado refinamento da malha ou possuem um grande estêncil de discretização, o que as torna onerosas. Portanto, o presente trabalho propõe a utilização do método pseudo-espectral de Fourier para resolver problemas de escoamentos multifásicos, o qual tem alta ordem de convergência numérica e um baixo custo computacional, devido ao algoritmo denominado FFT (Fast Fourier Transform). Além destas vantagens, este método, ao resolver as equações de Navier-Stokes, desacopla a pressão da velocidade, através do método da projeção, sem a necessidade de resolver a equação de Poisson. Para tratar escoamentos bifásicos com geometria móvel e deformável, utiliza-se o método pseudo-espectral de Fourier acoplado com o método híbrido Front-Tracking/Front- Capturing. Este método híbrido trabalha com dois domínios, sendo um euleriano, onde se resolvem as equações para o uido (equação de conservação de massa e as equações de Navier- Stokes) e o outro, móvel, lagrangiano, utilizado para as interfaces. Para este método...

## ‣ Subdomain Chebyshev spectral method for 2D and 3D numerical differentiations in a curved coordinate system

Zhou, B.; Heinson, G.; Rivera-Rios, A.
Fonte: Scientific Research Publishing Publicador: Scientific Research Publishing
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A new numerical approach, called the “subdomain Chebyshev spectral method” is presented for calculation of the spatial derivatives in a curved coordinate system, which may be employed for numerical solutions of partial differential equations defined in a 2D or 3D geological model. The new approach refers to a “strong version” against the “weak version” of the subspace spectral method based on the variational principle or Galerkin’s weighting scheme. We incorporate local nonlinear transformations and global spline interpolations in a curved coordinate system and make the discrete grid exactly matches geometry of the model so that it is achieved to convert the global domain into subdomains and apply Chebyshev points to locally sampling physical quantities and globally computing the spatial derivatives. This new approach not only remains exponential convergence of the standard spectral method in subdomains, but also yields a sparse assembled matrix when applied for the global domain simulations. We conducted 2D and 3D synthetic experiments and compared accuracies of the numerical differentiations with traditional finite difference approaches. The results show that as the points of differentiation vector are larger than five...

## ‣ Chebyshev spectral method for incompressible viscous flow with boundary layer control via suction or blowing

Alescio, Giuseppe
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 157 p.; 5640255 bytes; 5648926 bytes; application/pdf; application/pdf
Português
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The MISES quasi 3-D design/analysis code implements a two-equation integral method with empirical closure relations to solve the boundary layer flow problem with or without suction, but lacks the option of flow control via blowing. The integral method is parameterized with the shape parameter H _ 6*/0 which cannot be applied to the blowing problem since 0 - 0 downstream of the injection slot causing H -, co - a computational disaster. In this thesis, two alternate approaches are proposed to solve the blowing problem. First, a two-equation integral method parameterized with the profile parameters of a multi-deck representation of a turbulent jet based on Coles' law of the wake was formulated. The appearance of spurious singularities in the Jacobian matrices associated with the system of equations and the vector of unknowns prevented this method from being implemented. Second, a Chebyshev spectral method using the wall function technique was applied to the defect form of the incompressible viscous momentum equation. A turbulent jet profile was computed with N = 40 modes, a number low enough to allow the method's implementation into the MISES framework.; (cont.) For the spectral approach, a stand-alone code was developed to solve laminar and turbulent flow over a flat plate with the following configurations: solid wall...

## ‣ The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are i

## ‣ Compressive spectral method for the simulation of the water waves

Bayindir, Cihan
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper an approach for decreasing the computational effort required for the spectral simulations of the water waves is introduced. Signals with majority of the components zero, are known as the sparse signals. Like majority of the signals in the nature it can be realized that water waves are sparse either in time or in the frequency domain. Using the sparsity property of the water waves in the time or in the frequency domain, the compressive sampling algorithm can be used as a tool for improving the performance of the spectral simulation of the water waves. The methodology offered in this paper depends on the idea of using a smaller number of spectral components compared to the classical spectral method with a high number of components. After performing the time integration with a smaller number of spectral components and using the compressive sampling technique, it is shown that the water wave field can be reconstructed with a significantly better efficiency compared to the classical spectral method with a high number of spectral components, especially for long time evolutions. For the sparse water wave model in the time domain the well-known solitary wave solutions of the Korteweg-deVries (KdV) equation is considered. For the sparse water wave model in the frequency domain the well-known Airy (linear) ocean waves with Jonswap spectrum is considered. Utilizing a spectral method...

## ‣ Time-dependent Hermite-Galerkin spectral method and its applications

Luo, Xue; Yau, Shing-Tung; Yau, Stephen S. -T.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theorethical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Korteweg-de Vries-Burgers (KdVB) equation and its special cases, including the heat equation and the Burgers' equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.; Comment: 16 pages, 7 tables

## ‣ A divisive spectral method for network community detection

Cheng, Jianjun; Li, Longjie; Leng, Mingwei; Lu, Weiguo; Yao, Yukai; Chen, Xiaoyun
Tipo: Artigo de Revista Científica
Português
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Community detection is a fundamental problem in the domain of complex-network analysis. It has received great attention, and many community detection methods have been proposed in the last decade. In this paper, we propose a divisive spectral method for identifying community structures from networks, which utilizes a sparsification operation to pre-process the networks first, and then uses a repeated bisection spectral algorithm to partition the networks into communities. The sparsification operation makes the community boundaries more clearer and more sharper, so that the repeated spectral bisection algorithm extract high-quality community structures accurately from the sparsified networks. Experiments show that the combination of network sparsification and spectral bisection algorithm is highly successful, the proposed method is more effective in detecting community structures from networks than the others.; Comment: 23pages, 10 figures, and 2 tables

## ‣ The automatic solution of partial differential equations using a global spectral method

Townsend, Alex; Olver, Sheehan
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank $2$, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree $(n_x,n_y)$ is computed in $\mathcal{O}((n_x n_y)^{3/2})$ operations. Partial differential operators of splitting rank $\geq 3$ are solved via a linear system involving a block-banded matrix in $\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3}))$ operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under $60$ seconds. An experimental implementation in the Julia language can currently perform the same solve in $10$ seconds.; Comment: 22 pages

## ‣ FMM-based vortex method for simulation of isotropic turbulence on GPUs, compared with a spectral method

Yokota, Rio; Barba, L. A.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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The Lagrangian vortex method offers an alternative numerical approach for direct numerical simulation of turbulence. The fact that it uses the fast multipole method (FMM)--a hierarchical algorithm for N-body problems with highly scalable parallel implementations--as numerical engine makes it a potentially good candidate for exascale systems. However, there have been few validation studies of Lagrangian vortex simulations and the insufficient comparisons against standard DNS codes has left ample room for skepticism. This paper presents a comparison between a Lagrangian vortex method and a pseudo-spectral method for the simulation of decaying homogeneous isotropic turbulence. This flow field is chosen despite the fact that it is not the most favorable flow problem for particle methods (which shine in wake flows or where vorticity is compact), due to the fact that it is ideal for the quantitative validation of DNS codes. We use a 256^3 grid with Re_lambda=50 and 100 and look at the turbulence statistics, including high-order moments. The focus is on the effect of the various parameters in the vortex method, e.g., order of FMM series expansion, frequency of reinitialization, overlap ratio and time step. The vortex method uses an FMM code (exaFMM) that runs on GPU hardware using CUDA...

## ‣ Refined Spectral Method as an extremely accurate technique for solving 2D time-independent Schrodinger equation

Pedram, P.; Mirzaei, M.; Gousheh, S. S.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We present a refinement of the Spectral Method by incorporating an optimization method into it and generalize it to two space dimensions. We then apply this Refined Spectral Method as an extremely accurate technique for finding the bound states of the two dimensional time-independent Schrodinger equation. We first illustrate the use of this method on an exactly solvable case and then use it on a case which is not so. This method is very simple to program, fast, extremely accurate (e.g. a relative error of 10^(-15) is easily obtainable in two dimensions), very robust and stable. Most importantly, one can obtain the energies and the wave functions of as many of the bound states as desired with a single run of the algorithm.; Comment: 18 pages, 9 figures

## ‣ Limitations in the spectral method for graph partitioning: detectability threshold and localization of eigenvectors

Kawamoto, Tatsuro; Kabashima, Yoshiyuki
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both unnormalized and normalized Laplacians in sparse graphs. The detectability threshold is the critical point at which the result of the spectral method is completely uncorrelated to the planted partition. We also analyze whether the localization of eigenvectors affects the partitioning performance in the detectable region. We use the replica method, which is often used in the field of spin-glass theory, and focus on the case of bisection. We show that the gap between the estimated threshold for the spectral method and the threshold obtained from Bayesian inference is considerable in sparse graphs, even without eigenvector localization. This gap closes in a dense limit.; Comment: 26 pages, 13 figures

## ‣ Spectral Element Method for Pricing European Options and Their Greeks

Yue, Tianyao
Tipo: Dissertação
Relevância na Pesquisa
47.15342%

Numerical methods such as Monte Carlo method (MCM), finite difference method (FDM) and finite element method (FEM) have been successfully implemented to solve financial partial differential equations (PDEs). Sophisticated computational algorithms are strongly desired to further improve accuracy and efficiency.

The relatively new spectral element method (SEM) combines the exponential convergence of spectral method and the geometric flexibility of FEM. This dissertation carefully investigates SEM on the pricing of European options and their Greeks (Delta, Gamma and Theta). The essential techniques, Gauss quadrature rules, are thoroughly discussed and developed. The spectral element method and its error analysis are briefly introduced first and expanded in details afterwards.

Multi-element spectral element method (ME-SEM) for the Black-Scholes PDE is derived on European put options with and without dividend and on a condor option with a more complicated payoff. Under the same Crank-Nicolson approach for the time integration, the SEM shows significant accuracy increase and time cost reduction over the FDM. A novel discontinuous payoff spectral element method (DP-SEM) is invented and numerically validated on a European binary put option. The SEM is also applied to the constant elasticity of variance (CEV) model and verified with the MCM and the valuation formula. The Stochastic Alpha Beta Rho (SABR) model is solved with multi-dimensional spectral element method (MD-SEM) on a European put option. Error convergence for option prices and Greeks with respect to the number of grid points and the time step is analyzed and illustrated.

; Dissertation

## ‣ A graph-spectral method for surface height recovery

Robles-Kelly, Antonio; Hancock, Edwin R
Fonte: Pergamon-Elsevier Ltd Publicador: Pergamon-Elsevier Ltd
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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This paper describes a graph-spectral method for 3D surface integration. The algorithm takes as its input a 2D field of surface normal estimates, delivered, for instance, by a shape-from-shading or shape-from-texture procedure. We commence by using the surface normals to obtain an affinity weight matrix whose elements are related to the surface curvature. The weight matrix is used to compute a row-normalized transition probability matrix, and we pose the recovery of the integration path as that of finding the steady-state random walk for the Markov chain defined by this matrix. The steady-state random walk is given by the leading eigenvector of the original affinity weight matrix. By threading the surface normals together along the path specified by the magnitude order of the components of the leading eigenvector we perform surface integration. The height increments along the path are simply related to the traversed path length and the slope of the local tangent plane. The method is evaluated on needle-maps delivered by a shape-from-shading algorithm applied to real-world data and also on synthetic data. The method is compared with the local geometric height reconstruction method of Bors, Hancock and Wilson, and the global methods of Horn and Brooks and Frankot and Chellappa.