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Spectral Methods : Algorithms, Analysis and Applications / by Jie Shen, Tao Tang, Li-Lian Wang
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Springer Series in Computational Mathematics ; 41 | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer-Verlag Berlin Heidelberg, 2011Description: Online-Ressource (XVI, 470p. 55 illus., 8 illus. in color, digital)ISBN:- 9783540710417
- 1283369508
- 9781283369503
- 515.7222 22
- 518
- QA71-90
- QA371
Contents:
Summary: Introduction -- Fourier Spectral Methods for Periodic Problems -- Orthogonol Polynomials and Related Approximation Results -- Second-Order Two-Point Boundary Value Problems -- Integral Equations -- High-Order Differential Equations -- Problems in Unbounded Domains -- Multi-Dimensional Domains -- Mathematical Preliminaries -- Basic iterative methods -- Basic time discretization schemes -- Instructions for routines in Matlab. .Summary: Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for spectral methods. 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. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.PPN: PPN: 1651024448Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Spectral Methods; Preface; Contents; Symbol List; Chapter 1 Introduction; 1.1 Weighted Residual Methods; 1.2 Spectral-Collocation Method; 1.3 Spectral Methods of Galerkin Type; 1.3.1 Galerkin Method; 1.3.2 Petrov-Galerkin Method; 1.3.3 Galerkin Method with Numerical Integration; 1.4 Fundamental Tools for Error Analysis; 1.5 Comparative Numerical Examples; 1.5.1 Finite-Difference Versus Spectral-Collocation; 1.5.2 Spectral-Galerkin Versus Spectral-Collocation; Problems; Chapter 2 Fourier Spectral Methods for Periodic Problems; 2.1 Continuous and Discrete Fourier Transforms
2.1.1 Continuous Fourier Series2.1.2 Discrete Fourier Series; 2.1.3 Differentiation in the Physical Space; 2.1.4 Differentiation in the Frequency Space; 2.2 Fourier Approximation; 2.2.1 Inverse Inequalities; 2.2.2 Orthogonal Projection; 2.2.3 Interpolation; 2.3 Applications of Fourier Spectral Methods; 2.3.1 Korteweg-de Vries (KdV) Equation; 2.3.2 Kuramoto-Sivashinsky (KS) Equation; 2.3.3 Allen-Cahn Equation; Problems; Chapter 3 Orthogonal Polynomials and Related Approximation Results; 3.1 Orthogonal Polynomials; 3.1.1 Existence and Uniqueness; 3.1.2 Zeros of Orthogonal Polynomials
3.1.3 Computation of Zeros of Orthogonal Polynomials3.1.4 Gauss-Type Quadratures; 3.1.5 Interpolation and Discrete Transforms; 3.1.6 Differentiation in the Physical Space; 3.1.7 Differentiation in the Frequency Space; 3.1.8 Approximability of Orthogonal Polynomials; 3.1.8.1 A Short Summary of this Section; 3.2 Jacobi Polynomials; 3.2.1 Basic Properties; 3.2.1.1 Sturm-Liouville Equation; 3.2.1.2 Rodrigues' Formula; 3.2.1.3 Recurrence Formulas; 3.2.1.4 Maximum Value; 3.2.2 Jacobi-Gauss-Type Quadratures; 3.2.3 Computation of Nodes and Weights; 3.2.4 Interpolation and Discrete Jacobi Transforms
3.2.5 Differentiation in the Physical Space3.2.5.1 Jacobi-Gauss-Lobatto Differentiation Matrix; 3.2.5.2 Jacobi-Gauss-Radau Differentiation Matrix; 3.2.5.3 Jacobi-Gauss Differentiation Matrix; 3.2.6 Differentiation in the Frequency Space; 3.3 Legendre Polynomials; 3.3.1 Legendre-Gauss-Type Quadratures; 3.3.2 Computation of Nodes and Weights; 3.3.3 Interpolation and Discrete Legendre Transforms; 3.3.4 Differentiation in the Physical Space; 3.3.5 Differentiation in the Frequency Space; 3.4 Chebyshev Polynomials; 3.4.1 Interpolation and Discrete Chebyshev Transforms
3.4.2 Differentiation in the Physical Space3.4.3 Differentiation in the Frequency Space; 3.5 Error Estimates for Polynomial Approximations; 3.5.1 Inverse Inequalities for Jacobi Polynomials; 3.5.2 Orthogonal Projections; 3.5.3 Interpolations; 3.5.3.1 Jacobi-Gauss Interpolation; 3.5.3.2 Jacobi-Gauss-Radau Interpolation; 3.5.3.3 Jacobi-Gauss-Lobatto Interpolation; Problems; Chapter 4 Spectral Methods for Second-Order Two-Point Boundary Value Problems; 4.1 Galerkin Methods; 4.1.1 Weighted Galerkin Formulation; 4.1.2 Legendre-Galerkin Method; 4.1.3 Chebyshev-Galerkin Method
4.1.4 Chebyshev-Legendre Galerkin Method
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