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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 numerical study of adaptive space and time discretisations for Gross–Pitaevskii equations

Thalhammer, Mechthild; Abhau, Jochen
Tipo: Artigo de Revista Científica
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As a basic principle, benefits of adaptive discretisations are an improved balance between required accuracy and efficiency as well as an enhancement of the reliability of numerical computations. In this work, the capacity of locally adaptive space and time discretisations for the numerical solution of low-dimensional nonlinear Schrödinger equations is investigated. The considered model equation is related to the time-dependent Gross–Pitaevskii equation arising in the description of Bose–Einstein condensates in dilute gases. The performance of the Fourier-pseudo spectral method constrained to uniform meshes versus the locally adaptive finite element method and of higher-order exponential operator splitting methods with variable time stepsizes is studied. Numerical experiments confirm that a local time stepsize control based on a posteriori local error estimators or embedded splitting pairs, respectively, is effective in different situations with an enhancement either in efficiency or reliability. As expected, adaptive time-splitting schemes combined with fast Fourier transform techniques are favourable regarding accuracy and efficiency when applied to Gross–Pitaevskii equations with a defocusing nonlinearity and a mildly varying regular solution. However...

## ‣ Numerical Investigation of turbulent coupling boundary layer of air-water interaction flow

Liu, Song, S.M. Massachusetts Institute of Technology
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 164 p.
Português
Relevância na Pesquisa
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Air-water interaction flow between two parallel flat plates, known as Couette flow, is simulated by direct numerical simulation. The two flowing fluids are coupled through continuity of velocity and shear stress condition across the interface. Pseudo-spectral method is used in each flow subdomain with Fourier expansion in streamwise and spanwise directions and finite difference in vertical direction. Statistically quasi-steady flow properties, such as mean velocity profiles, turbulent intensities, Reynolds stress and turbulent kinetic energy (TKE) budget terms show significant differences between air-water interface turbulence near the water side (IntT-w) and wall-bounded turbulence(WT) while there are some similarities between IntT-w and free surface turbulence (FST). Due to the velocity fluctuation at the interface, water side near interface turbulence flow (IntT-w) is characterized with a thinner viscous sub-layer and decreased intercept parameter B in log-law layer, strengthened Reynolds stress and eddy viscosity, together with a stronger production term, decreasing-then-increasing dissipation term and negative turbulent diffusion term in TKE budget.; (cont.) Abundant physical phenomena exist on the water side turbulent flow with four major types of three-dimensional vortex structures identified near the interface by variable-interval spacing averaging (VISA) techniques. Each type of vortex structures is found to play an essential role in the turbulent energy balance and passive scalar transport.; by Song Liu.; Thesis (S.M. in Mechanical Engineering and S.M. in Ocean Engineering)--Massachusetts Institute of Technology...

## ‣ 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
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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...

## ‣ Solução numérica das equações de Navier-Stokes usando uma hibridação das metodologias Fronteira Imersa e Pseudo-Espectral de Fourier; Numerical solution of Navier-Stokes equations using a hybrid methodology of immersed boundary and Fourier pseudo-spectral

Mariano, Felipe Pamplona
Português
Relevância na Pesquisa
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## ‣ 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
Relevância na Pesquisa
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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...

## ‣ Modelagem matemática de jatos em desenvolvimento espacial usando a metodologia pseudoespectral de Fourier

Moreira, Leonardo de Queiroz
Português
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## ‣ Solution of the Schr\"odinger equation using exterior complex scaling and fast Fourier transform

Serov, Vladislav V.; Sergeeva, Tatiana A.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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The split-operator pseudo-spectral method based on the fast Fourier transform (SO-FFT) is a fast and accurate method for the numerical solution of the time-dependent Schr\"odinger-like equations (TDSE). As well as other grid-based approaches, SO-FFT encounters a problem of the unphysical reflection of the wave function from the grid boundaries. Exterior complex scaling (ECS) is an effective method widely applied for the suppression of the unphysical reflection. However, SO-FFT and ECS have not been used together heretofore because of the kinetic energy operator coordinate dependence that appears in ECS applying. We propose an approach for the combining the ECS with SO-FFT for the purpose of the solution of TDSE with outgoing-wave boundary conditions. Also, we propose an effective ECS-friendly FFT-based preconditioner for the solution of the stationary Schr\"odinger equation by means of the preconditioned conjugate gradients method.; Comment: 20 pages, 7 figures

## ‣ Numerical Study of Nearly Singular Solutions of the 3-D Incompressible Euler Equations

Hou, Thomas Y.; Li, Ruo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
48.713384%
In this paper, we perform a careful numerical study of nearly singular solutions of the 3D incompressible Euler equations with smooth initial data. We consider the interaction of two perturbed antiparallel vortex tubes which was previously investigated by Kerr in \cite{Kerr93,Kerr05}. In our numerical study, we use both the pseudo-spectral method with the 2/3 dealiasing rule and the pseudo-spectral method with a high order Fourier smoothing. Moreover, we perform a careful resolution study with grid points as large as $1536\times1024\times 3072$ to demonstrate the convergence of both numerical methods. Our computational results show that the maximum vorticity does not grow faster than doubly exponential in time while the velocity field remains bounded up to T=19, beyond the singularity time $T=18.7$ reported by Kerr in \cite{Kerr93,Kerr05}. The local geometric regularity of vortex lines near the region of maximum vorticity seems to play an important role in depleting the nonlinear vortex stretching dynamically.; Comment: 26 pages, 32 figures

## ‣ Time domain analysis of superradiant instability for the charged stringy black hole-mirror system

Li, Ran; Tian, Yu; Zhang, Hongbao; Zhao, Junkun
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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It has been proved that the charged stringy black holes are stable under the perturbations of massive charged scalar fields. However, superradiant instability can be generated by adding the mirror-like boundary condition to the composed system of charged stringy black hole and scalar field. The unstable boxed quasinormal modes have been calculated by using both analytical and numerical method. In this paper, we further provide a time domain analysis by performing a long time evolution of charged scalar field configuration in the background of the charged stringy black hole with the mirror-like boundary condition imposed. We have used the ingoing Eddington-Finkelstein coordinates to derive the evolution equation, and adopted Pseudo-spectral method and the forth-order Runge-Kutta method to evolve the scalar field with the initial Gaussian wave packet. It is shown by our numerical scheme that Fourier transforming the evolution data coincides well with the unstable modes computed from frequency domain analysis. The existence of the rapid growth mode makes the charged stringy black hole a good test ground to study the nonlinear development of superradiant instability.; Comment: 7 pages, 6 figures, and 5 tables. References added

## ‣ A pseudo-spectral method for a non-local KdV-Burgers equation posed on $\mathbb R$

de la Hoz, Francisco; Cuesta, Carlota Maria
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper, we present a new pseudo-spectral method to solve the initial value problem associated to a non-local KdV-Burgers equation involving a Caputo-type fractional derivative. The basic idea is, using an algebraic map, to transform the whole real line into a bounded interval where we can apply a Fourier expansion. Special attention is given to the correct computation of the fractional derivative in this setting.

## ‣ Transpose-free Fast Fourier Transform for Turbulence Simulation

Chatterjee, A. G.; Verma, M. K.; Chaudhuri, M.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Pseudo-spectral method is one of the most accurate techniques for simulating turbulent flows. Fast Fourier transform (FFT) is an integral part of this method. In this paper, we present a new procedure to compute FFT in which we save operations during interprocess communications by avoiding transpose of the array. As a result, our transpose-free FFT is 15\% to 20\% faster than FFTW.

## ‣ Analysis of Non-Linear Mode Coupling of Cosmological Density Fluctuations by the Pseudo-Spectral Method

Gouda, Naoteru
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The pseudo-spectral method is proposed for following the evolution of density and velocity fluctuations at the weakly non-linear stage in the expanding universe with a good accuracy. In this method, the evolution of density and velocity fluctuations is integrated in the Fourier Space with using FFT. This method is very useful to investigate accurately the non-linear dynamics in the weakly non-linear regime. Because the pseudo-spectral method works directly in the Fourier space, it should be especially useful for examining behavior in the Fourier domain, for an example, the effects of the non-linear coupling of different wave modes on the evolution of the power spectrum. I show the results of this analysis both in one and three dimensional systems.; Comment: 12 pages+3 figures, uuencoded, tar compressed Postscript. To appear in Prog.Theor.Phys.vol.94

## ‣ Stability and spectral convergence of Fourier method for nonlinear problems. On the shortcomings of the 2/3 de-aliasing method

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers' equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two main flavors. One is the spectral Fourier method. The other is the 2/3 pseudo-spectral Fourier method, where one removes the highest 1/3 portion of the spectrum; this is often the method of choice to maintain the balance of quadratic energy and avoid aliasing errors. Two main themes are discussed in this paper. First, we prove that as long as the underlying exact solution has a minimal C^{1+\alpha} spatial regularity, then both the spectral and the 2/3 pseudo-spectral Fourier methods are stable. Consequently, we prove their spectral convergence for smooth solutions of the inviscid Burgers equation and the incompressible Euler equations. On the other hand, we prove that after a critical time at which the underlying solution lacks sufficient smoothness, then both the spectral and the 2/3 pseudo-spectral Fourier methods exhibit nonlinear instabilities which are realized through spurious oscillations. In particular...

## ‣ An Immersed Boundary Fourier Pseudo-spectral Method for Simulation of Confined Two-dimensional Incompressible Flows

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The present paper is devoted to implementation of the immersed boundary technique into the Fourier pseudo-spectral solution of the vorticity-velocity formulation of the two-dimensional incompressible Navier--Stokes equations. The immersed boundary conditions are implemented via direct modification of the convection and diffusion terms, and therefore, in contrast to many other similar methods, there is not an explicit external forcing function in the present formulation. The desired immersed boundary conditions are approximated on some regular grid points, using different orders (up to second-order) polynomial extrapolations. At the beginning of each timestep, the solenoidal velocities (also satisfying the desired immersed boundary conditions), are obtained and fed into a conventional pseudo-spectral solver, together with a modified vorticity. The zero-mean pseudo-spectral solution is employed, and therefore, the method is applicable to the confined flows with zero mean velocity and vorticity, and without mean vorticity dynamics. In comparison to the classical Fourier pseudo-spectral solution, the method needs ${\cal O}(4(1+\log N)N)$ more operations for boundary condition settings. Therefore, the computational cost of the method, as a whole...

## ‣ Direct control of the small-scale energy balance in 2D fluid dynamics

Frank, Jason; Leimkuhler, Benedict; Myerscough, Keith
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We explore the direct modification of the pseudo-spectral truncation of 2D, incompressible fluid dynamics to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the purpose of matching simulation statistics to given information, arising from observations, theoretical prediction or high fidelity simulation. In the scheme outlined here, Nos\'e-Hoover thermostats, commonly used in molecular dynamics, are introduced as feedback controls applied to energy shells of the Fourier-discretized Navier-Stokes equations. As we demonstrate in numerical experiments, the dynamical properties (quantified using autocorrelation functions) are only modestly perturbed by our device, while ensemble dispersion is significantly enhanced in comparison with simulations of a corresponding truncation incorporating hyperviscosity.

## ‣ Computing Nearly Singular Solutions Using Pseudo-Spectral Methods

Hou, Thomas Y.; Li, Ruo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In this paper, we investigate the performance of pseudo-spectral methods in computing nearly singular solutions of fluid dynamics equations. We consider two different ways of removing the aliasing errors in a pseudo-spectral method. The first one is the traditional 2/3 dealiasing rule. The second one is a high (36th) order Fourier smoothing which keeps a significant portion of the Fourier modes beyond the 2/3 cut-off point in the Fourier spectrum for the 2/3 dealiasing method. Both the 1D Burgers equation and the 3D incompressible Euler equations are considered. We demonstrate that the pseudo-spectral method with the high order Fourier smoothing gives a much better performance than the pseudo-spectral method with the 2/3 dealiasing rule. Moreover, we show that the high order Fourier smoothing method captures about $12 \sim 15%$ more effective Fourier modes in each dimension than the 2/3 dealiasing method. For the 3D Euler equations, the gain in the effective Fourier codes for the high order Fourier smoothing method can be as large as 20% over the 2/3 dealiasing method. Another interesting observation is that the error produced by the high order Fourier smoothing method is highly localized near the region where the solution is most singular...

## ‣ Numerical solution for a general class of nonlocal nonlinear wave equations

Borluk, Handan; Muslu, Gulcin M.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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A class of nonlocal nonlinear wave equation arises from the modeling of a one dimensional motion in a nonlinearly, nonlocally elastic medium. The equation involves a kernel function with nonnegative Fourier transform. We discretize the equation by using Fourier spectral method in space and we prove the convergence of the semidiscrete scheme. We then use a fully-discrete scheme, that couples Fourier pseudo-spectral method in space and 4th order Runge-Kutta in time, to observe the effect of the kernel function on solutions. To generate solitary wave solutions numerically, we use the Petviashvili's iteration method.

## ‣ FluSI: A novel parallel simulation tool for flapping insect flight using a Fourier method with volume penalization

Engels, Thomas; Kolomenskiy, Dmitry; Schneider, Kai; Sesterhenn, Jörn