The Chebyshev-Legendre method implementing Legendre methods on Chebyshev points

Cover of: The Chebyshev-Legendre method |

Published by National Aeronautics and Space Administration, Langley Research Center, National Technical Information Service, distributor in Hampton, Va, [Springfield, Va .

Written in English

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  • Legendre"s polynomials.,
  • Chebyshev polynomials.

Edition Notes

Book details

StatementWai Sun Don, David Gottlieb.
SeriesICASE report -- no. 93-40., NASA contractor report -- 191497., NASA contractor report -- NASA CR-191497.
ContributionsGottlieb, David., Langley Research Center.
The Physical Object
Pagination1 v.
ID Numbers
Open LibraryOL18062809M

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A new collocation method for the numerical solution of partial differential equations is presented. This method uses the Chebyshev collocation points, but, because of the way the boundary conditions are implemented, it has all the advantages of the Legendre The Chebyshev-Legendre method book The Chebyshev–Legendre Method: Implementing Legendre Methods on Chebyshev Points Article (PDF Available) in SIAM Journal on Numerical Analysis 31(6) August with Reads.

A Chebyshev–Legendre method which implements the Legendre method at Chebyshev nodes was proposed by Don and Gotlieb for the parabolic and hyperbolic equations. The approach enjoys advantages of both the Legendre and Chebyshev by: The new method is based on a Legendre-Galerkin formulation, but only the Chebyshev-Gauss-Lobatto points are used in the computation.

Hence, it enjoys advantages of both the LegendreGalerkin and. The Chebyshev-Legendre spectral method for the two-dimensional vorticity equations is considered.

The Legendre Galerkin Chebyshev collocation method is used with the Chebyshev-Gauss collocation. In this paper, a Chebyshev--Legendre spectral viscosity (CLSV) method is developed for nonlinear conservation laws with initial and boundary conditions.

The boundary conditions are dealt with by a penalty method. The viscosity is put only on the high modes, so accuracy may be recovered by postprocessing the CLSV approximation. In this paper, we propose and analyze a spectral Chebyshev–Legendre approximation for fractional order integro-differential equations of Fredholm type.

The fractional derivative is described in the Caputo sense. Our proposed method is illustrated by considering some examples whose exact solutions are. () Chebyshev-Legendre method for discretizing optimal control problems.

Journal of Shanghai University (English Edition)() The Chebyshev–Legendre collocation method for a class of optimal control problems. In this paper, we derive the so-called Chebyshev–Legendre method for a class of optimal control problems governed by ordinary differential equations.

We use Legendre expansions to approximate the control and state functions and we employ the Chebyshev–Gauss–Lobatto (CGL) points as the interpolating points.

Thus the unknown variables of the equivalent nonlinear programming problems. Some other interesting approaches to the construction of approximate inertial manifolds have focused on the application of wavelets [1,18] multi-grid methods [61] and Chebyshev Legendre.

The Chebyshev–Legendre spectral method was introduced in to take advantage of both the Legendre and Chebyshev polynomials. It’s main idea is to use the Legendre–Galerkin formulation which preserves the The Chebyshev-Legendre method book of the underlying problem and lead to a simple sparse linear system, while the physical values are evaluated at the Gauss.

We extend the Chebyshev-Legendre spectral method to multi-domain case for solving the two-dimensional vorticity equations. The schemes are formulated in Legendre-Galerkin method while the nonlinear term is collocated at Chebyshev-Gauss collocation points. We introduce proper basis functions in order that the matrix of algebraic system is sparse.

the Chebyshev–Legendre pseudospectral viscosity method and the Chebyshev–Le gendre pseudospectral super-viscosity method for nonlinear conservation laws. As in Shen (). A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions.

Kinetic & Related Models,13 (4): doi: /krm [4] Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Chebyshev-Legendre Galerkin method and describe the fast Chebyshev-Legendre transform between the values at CGL points and the coefficients of Legendre expansion.

In Section 4, we present some numerical results which demon- strate the efficiency of the new method. 2 Legendre-Galerkin and Cheby- shev-Galerkin methods. In this paper, we consider Chebyshev–Legendre Pseudo-Spectral (CLPS) method for solving coupled viscous Burgers (VB) equations.

A leapfrog scheme is used in time direction, while CLPS method is used for space direction. Chebyshev–Gauss–Lobatto (CGL) nodes are used for practical computation.

Around x = 0 the operational Tau method in the standard basis is suitable to approximate solutions, while in the Legendre and Chebyshev basics are suitable for outbye x = 0. (2) Comparing the Tau method, Makroglou method and the Trapezoidal rule presented by Jerri show that the Tau method has more accuracy in computations.

In this paper, we consider Chebyshev-Legendre Pseudo-Spectral (CLPS) method for solving coupled viscous Burgers (VB) equations. A leapfrog scheme is used in time direction, while CLPS method is used for space direction. Chebyshev-Gauss-Lobatto (CGL) nodes are used for practical computation.

We investigate the existence of solutions for a sum-type fractional integro-differential problem via the Caputo differentiation. By using the shifted Legendre and Chebyshev polynomials, we provide a numerical method for finding solutions for the problem.

In. Get this from a library. The Chebyshev-Legendre method: implementing Legendre methods on Chebyshev points. [Wai Sun Don; David Gottlieb; Langley Research Center.]. We analyze the asymptotic rates of convergence of Chebyshev, Legendre and Jacobi polynomials.

One complication is that there are many reasonable measures of optimality as enumerated here. Another is that there are at least three exceptions to the general principle that Chebyshev polynomials give the fastest rate of convergence from the larger family of Jacobi polynomials.

We introduce a novel coarse ridge orientation smoothing algorithm based on orthogonal polynomials, which can be used to estimate the orientation field (OF) for fingerprint areas of no ridge information. This method does not need any base information of singular points (SPs).

The algorithm uses a consecutive application of filtering- and model-based orientation smoothing methods. Comparison with the finite element method We may compare the spectral method (before actually describing it) to the finite element method. One difference is this: the trial functions τ k in the finite element method are usually 1 at the mesh-point, x k = khwith h =2π/N, and 0 at the other mesh-points, whereas eikx is nonzero everywhere.

The Chebyshev–Legendre collocation method for a class of optimal control problems International Journal of Computer Mathematics, Vol. 85, No.

2 On the convergence of nonlinear optimal control using pseudospectral methods for feedback linearizable systems. The Chebyshev-Legendre (C-L) quadrature set has been implemented in the neutron transport section of the TWODANT code. The C-L quadrature set has two advantages as compared to other quadrature sets.

First, it is possible to easily generate S/sub N/ orders up to S/sub / with weights that are positive. In this paper, a Chebyshev--Legendre spectral viscosity (CLSV) method is developed for nonlinear conservation laws with initial and boundary conditions.

The boundary conditions are dealt with by a penalty method. The viscosity is put only on the high. Two of these methods are based on cardinal Chebyshev basis function with Galerkin method. Gauss-quadrature formula and El-gendi method are used to convert the problem into system of ordinary differential equations.

In the third proposed method, the cardinal Legendre basis function with Galerkin method. The Museum of HP Calculators. MoHPC HPC Software Library This library contains copyrighted programs that are used here by permission. The Finite Element Method Elasticity and Solid Mechanics 4 Fourier Series and Integrals Fourier Series for Periodic Functions Chebyshev, Legendre, and Bessel The Discrete Fourier Transform and the FFT Convolution and Signal Processing Fourier Integrals Deconvolution and Integral Equations.

Boyd, John P., Computing the zeros, maxima and inflection points of Chebyshev, Legendre and Fourier series: Solving transcendental equations by spectral interpolation and polynomial rootfinding. Engrg. Math. v56 i3. Google Scholar [6]. Boyd, John P., Computing real roots of a polynomial in Chebyshev series form through subdivision.

The following Matlab project contains the source code and Matlab examples used for chebyshev to legendre conversion.

Given a Chebyshev polynomial expansion where the coefficients are stored in a column vector, this script computes the expansion in terms of Legendre polynomials. Chebyshev Legendre Galerkin(CLG) method coupled with higher-order shear and normal deformable plate theory is proposed to analyze free and forced vibrations of laminated composite plates.

The laminates of various boundary conditions, side-to-thickness ratios, and material properties are considered by present method and the numerical results agree well. The Chebyshev pseudospectral method for optimal control problems is based on Chebyshev polynomials of the first is part of the larger theory of pseudospectral optimal control, a term coined by Ross.

Unlike the Legendre pseudospectral method, the Chebyshev pseudospectral (PS) method does not immediately offer high-accuracy quadrature solutions. Chebyshev-Legendre spectral method and inverse problem analysis for the space fractional Benjamin-Bona-Mahony equation Hui Zhang, Xiaoyun Jiang.

Chebyshev-Legendre method for discretizing optimal control problems 18 April | Journal of Shanghai University (English Edition), Vol. 13, No. 2 Three-Dimensional Trajectory Optimization in Constrained Airspace.

The chapter contains first the general formulation of the spectral approximation as a weighted residual method, i.e., the projection and interpolation operators, test and trial (shape) functions etc.

Then, the functional framework of the tau and Galerkin methods based on Chebyshev polynomials is provided, mainly discussing the projection operators. 'Gil Strang has given the discipline of computational science and engineering its first testament in this new and comprehensive book.

It surely extends Gil's long tradition of practical, wide-ranging, and insightful books that are invaluable for students, teachers, and researchers alike. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method.

Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR. The Chebyshev-Legendre Method: Implementing Legendre methods on Chebyshev points.

(with W.S. Don) SIAM Journal on Numerical Analysis, 31 no. 6, () pp. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains.

The book contains a large number of figures which are designed to illustrate various concepts stressed in the book.

Preface This book expands lecture notes by the authors for a course on Introduction of Spec- tral Methods taught in the past few years at Penn State University, Simon Fraser University, the Chinese University of Hong Kong, Hong Kong Baptist University.Ross–Fahroo pseudospectral method — class of pseudospectral method including Chebyshev, Legendre and knotting Ross–Fahroo lemma — condition to make discretization and duality operations commute Ross' π lemma — there is fundamental time constant within which a control solution must be computed for controllability and stability.To cite Chebfun in a publication, we recommend the following: T.

A. Driscoll, N. Hale, and L. N. Trefethen, editors, Chebfun Guide, Pafnuty Publications, Oxford, Below is a list of some of the Chebfun-related papers written by members of the Chebfun Team.

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