Numerical Analysis of Spectral Methods

Gottlieb, David; Orszag, Steven A.

doi:10.1137/1.9781611970425

Cited by 2,544 publications

(1,614 citation statements)

References 7 publications

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“…An introduction to the theory of spectral methods is given in the monograph by Gottlieb & Orszag (1977). Here we summarize the implementation of spectral methods for the present channel flow simulations.…”

Section: Methodsmentioning

confidence: 99%

Transition to turbulence in plane Poiseuille and plane Couette flow

1980

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Direct numerical solutions of the three-dimensional time-dependent Navier-Stokes equations are presented for the evolution of three-dimensional finite-amplitude disturbances of plane Poiseuille and plane Couette flows. Spectral methods using Fourier series and Chebyshev polynomial series are used. It is found that plane Poiseuille flow can sustain neutrally stable two-dimensional finite-amplitude disturbances at Reynolds numbers larger than about 2800. No neutrally stable two-dimensional finite-amplitude disturbances of plane Couette flow were found.Three-dimensional disturbances are shown to have a strongly destabilizing effect. It is shown that finite-amplitude disturbances can drive transition to turbulence in both plane Poiseuille flow and plane Couette flow at Reynolds numbers of order 1000. Details of the resulting flow fields are presented. It is also shown that plane Poiseuille flow cannot sustain turbulence at Reynolds numbers below about 500.

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Section: Methodsmentioning

confidence: 99%

Transition to turbulence in plane Poiseuille and plane Couette flow

1980

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“…We begin with a time-dependent periodic problem [159,123,93,19,103]. As an example, we consider the Schrödinger equation (2.6) over a 2π-torus, Ω = T d x , which is covered by a Cartesian grid of (2N + 1) d equispaced gridpoints:…”

Section: Spectral Methodsmentioning

confidence: 99%

“…Since the problem is nonperiodic, the discrete equations (3.27) are augmented with appropriate Dirichlet-or Neumann-type boundary conditions at x = ±1. We now seek a spectral approximant in terms of algebraic polynomials [93,214]. We proceed by expressing the spectral approximant in terms of the orthogonal family of Legendre polynomials, {p k (x)} k≥0 :…”

Section: Spectral Methodsmentioning

confidence: 99%

A review of numerical methods for nonlinear partial differential equations

Tadmor

2012

Bull. Amer. Math. Soc.

143

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Abstract. Numerical methods were first put into use as an effective tool for solving partial differential equations (PDEs) by John von Neumann in the mid1940s. In a 1949 letter von Neumann wrote "the entire computing machine is merely one component of a greater whole, namely, of the unity formed by the computing machine, the mathematical problems that go with it, and the type of planning which is called by both." The "greater whole" is viewed today as scientific computation: over the past sixty years, scientific computation has emerged as the most versatile tool to complement theory and experiments, and numerical methods for solving PDEs are at the heart of many of today's advanced scientific computations. Numerical solutions found their way from financial models on Wall Street to traffic models on Main Street. Here we provide a bird's eye view on the development of these numerical methods with a particular emphasis on nonlinear PDEs.

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“…Along with extensive applications of Legendre and Chebyshev spectral methods for bounded domains (cf. [1,2,6,9,11,12]), considerable progress has been made recently in spectral methods for unbounded domains. Among these methods, a direct and commonly used approach is based on certain orthogonal approximations on infinite intervals, i.e., the Hermite and Laguerre spectral methods ( see, e.g., [5,10,13,16,18,23,24,28,32]).…”

Section: Introductionmentioning

confidence: 99%

Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel

Guo

Wang

2009

Math. Comp.

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Abstract. In this paper, we propose a composite generalized LaguerreLegendre spectral method for partial differential equations on two-dimensional unbounded domains, which are not of standard types. Some approximation results are established, which are the mixed generalized Laguerre-Legendre approximations coupled with domain decomposition. These results play an important role in the related spectral methods. As an important application, the composite spectral scheme with domain decomposition is provided for the Fokker-Planck equation in an infinite channel. The convergence of the proposed scheme is proved. An efficient algorithm is described. Numerical results show the spectral accuracy in the space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are applicable to many other problems on unbounded domains. In particular, some quasi-orthogonal approximations are very appropriate for solving PDEs, which behave like parabolic equations in some directions, and behave like hyperbolic equations in other directions. They are also useful for various spectral methods with domain decompositions, and numerical simulations of exterior problems.

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Numerical Analysis of Spectral Methods

Cited by 2,544 publications

References 7 publications

Transition to turbulence in plane Poiseuille and plane Couette flow

Transition to turbulence in plane Poiseuille and plane Couette flow

A review of numerical methods for nonlinear partial differential equations

Composite generalized Laguerre-Legendre spectral method with domain decomposition and its application to Fokker-Planck equation in an infinite channel

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