第1页 / 共487页
第2页 / 共487页
第3页 / 共487页
第4页 / 共487页
第5页 / 共487页
第6页 / 共487页
第7页 / 共487页
第8页 / 共487页
谱方法 算法 分析与应用.pdf
Cover
Springer Series in
Computational
Mathematics 41
Spectral Methods
ISBN 9783540710400
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 Series
2.1.2 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 Polynomials
3.1.4 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 Space
3.2.5.1 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 Space
3.4.3 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
4.2 Galerkin Method with Numerical Integration
4.3 Collocation Methods
4.3.1 Galerkin Reformulation
4.3.2 Petrov-Galerkin Reformulation
4.4 Preconditioned Iterative Methods
4.4.1 Preconditioning in the Modal Basis
4.4.1.1 Legendre Case
4.4.1.2 Chebyshev Case
4.4.2 Preconditioning in the Nodal Basis
4.4.2.1 Finite Difference Preconditioning
4.4.2.2 Finite Element Preconditioning
4.5 Error Estimates
4.5.1 Legendre-Galerkin Method
4.5.2 Chebyshev-Collocation Method
4.5.3 Galerkin Method with Numerical Integration
4.5.4 Helmholtz Equation
4.5.4.1 A Priori Estimates
4.5.4.2 Convergence Analysis
Problems
Chapter 5 Volterra Integral Equations
5.1 Legendre-Collocation Method for VIEs
5.1.1 Numerical Algorithm
5.1.2 Convergence Analysis
5.1.3 Numerical Results and Discussions
5.2 Jacobi-Galerkin Method for VIEs
5.3 Jacobi-Collocation Method for VIEs with WeaklySingular Kernels
5.4 Application to Delay Differential Equations
Problems
Chapter 6 Higher-Order Differential Equations
6.1 Generalized Jacobi Polynomials
6.2 Galerkin Methods for Even-Order Equations
6.2.1 Fourth-Order Equations
6.2.2 General Even-Order Equations
6.3 Dual-Petrov-Galerkin Methods for Odd-Order Equations
6.3.1 Third-Order Equations
6.3.2 General Odd-Order Equations
6.3.3 Higher Odd-Order Equations with Variable Coefficients
6.4 Collocation Methods
6.5 Error Estimates
6.5.1 Even-Order Equations
6.5.2 Odd-Order Equations
6.6 Applications
6.6.1 Cahn–Hilliard Equation
6.6.2 Korteweg–de Vries (KdV) Equation
6.6.3 Fifth-Order KdV Type Equations
Problems
Chapter 7 Unbounded Domains
7.1 Laguerre Polynomials/Functions
7.1.1 Basic Properties
7.1.1.1 Generalized Laguerre Polynomials
7.1.1.2 Generalized Laguerre Functions
7.1.2 Laguerre-Gauss-Type Quadratures
7.1.3 Computation of Nodes and Weights
7.1.4 Interpolation and Discrete Laguerre Transforms
7.1.5 Differentiation in the Physical Space
7.1.6 Differentiation in the Frequency Space
7.2 Hermite Polynomials/Functions
7.2.1 Basic Properties
7.2.1.1 Hermite Polynomials
7.2.1.2 Hermite Functions
7.2.2 Hermite-Gauss Quadrature
7.2.3 Computation of Nodes and Weights
7.2.4 Interpolation and Discrete Hermite Transforms
7.2.5 Differentiation in the Physical Space
7.2.6 Differentiation in the Frequency Space
7.3 Approximation by Laguerre and Hermite Polynomials/Functions
7.3.1 Inverse Inequalities
7.3.2 Orthogonal Projections
7.3.3 Interpolations
7.4 Spectral Methods Using Laguerre and Hermite Functions
7.4.1 Laguerre-Galerkin Method
7.4.2 Hermite-Galerkin Method
7.4.3 Numerical Results and Discussions
7.4.4 Scaling Factor
7.5 Mapped Spectral Methods and Rational Approximations
7.5.1 Mappings
7.5.2 Approximation by Mapped Jacobi Polynomials
7.5.3 Spectral Methods Using Mapped Jacobi Polynomials
7.5.3.1 A Generic Example
7.5.3.2 Error Estimates for a Model Problem
7.5.3.3 Implementations and A Comparison Study
7.5.4 Modified Legendre-Rational Approximations
7.5.5 Irrational Mappings
7.5.6 Miscellaneous Issues and Extensions
7.5.6.1 Other One-Dimensional Applications
7.5.6.2 Multidimensional Problems
Problems
Chapter 8 Separable Multi-Dimensional Domains
8.1 Two- and Three-Dimensional Rectangular Domains
8.1.1 Two-Dimensional Case
8.1.1.1 Matrix Diagonalization Method
8.1.1.2 Partial Diagonalization
8.1.1.3 Full Diagonalization
8.1.1.4 An Equivalent Approach Based on Separation of Variables
8.1.2 Three-Dimensional Case
8.2 Circular and Cylindrical Domains
8.2.1 Dimension Reduction and Pole Conditions
8.2.2 Spectral-Galerkin Method for a Bessel-Type Equation
8.2.2.1 Legendre-Galerkin Approximation
8.2.2.2 Chebyshev-Galerkin Approximation
8.2.3 Another Fourier-Chebyshev Galerkin Approximation
8.2.3.1 A Fourier-Chebyshev Interpolation Operator on the Unit Disk
8.2.3.2 Description of the Algorithm
8.2.4 Numerical Results and Discussions
8.2.5 Three-Dimensional Cylindrical Domains
8.3 Spherical Domains
8.3.1 Spectral Methods on the Surface of a Sphere
8.3.2 Spectral Methods in a Spherical Shell
8.4 Multivariate Jacobi Approximations
8.4.1 Notation and Preliminary Properties
8.4.2 Orthogonal Projections
8.4.3 Interpolations
8.4.4 Applications of Multivariate Jacobi Approximations
8.4.4.1 Rectangular Domains
8.4.4.2 Circular and Spherical Domains
8.5 Sparse Spectral-Galerkin Methods for High-Dimensional Problems
8.5.1 Hyperbolic Cross Jacobi Approximations
8.5.2 Optimized Hyperbolic Cross Jacobi Approximations
8.5.3 Extensions to Generalized Jacobi Polynomials
8.5.4 Sparse Spectral-Galerkin Methods
8.5.4.1 One-Dimensional Hierarchical Basis
8.5.4.2 Multi-Dimensional Hierarchical Basis and Sparse Grid
8.5.4.3 Sparse Spectral-Galerkin Method
Problems
Chapter 9 Applications in Multi-Dimensional Domains
9.1 Helmholtz Equation for Acoustic Scattering
9.1.1 Time-Harmonic Wave Equations
9.1.2 Dirichlet-to-Neumann (DtN) Map
9.1.3 Spectral-Galerkin Method
9.2 Stokes Equations
9.2.1 Stokes Equations and Uzawa Operator
9.2.2 Galerkin Method for the Stokes Problem
9.2.3 Error Analysis
9.3 Allen–Cahn and Cahn–Hilliard Equations
9.3.1 Simple Semi-Implicit Schemes
9.3.2 Convex Splitting Schemes
9.3.3 Stabilized Semi-Implicit Schemes
9.3.4 Spectral-Galerkin Discretizations in Space
9.3.5 Error Analysis
9.3.6 Effect of Spatial Accuracy
9.4 Unsteady Navier–Stokes Equations
9.4.1 Second-Order Rotational Pressure-Correction Scheme
9.4.2 Second-Order Consistent Splitting Scheme
9.4.3 Full Discretization
9.5 Axisymmetric Flows in a Cylinder
9.5.1 Governing Equations and the Time Discretization
9.5.1.1 Spatial Discretization
9.5.2 Treatment for the Singular Boundary Condition
9.6 Gross-Pitaevskii Equation
9.6.1 GPE and Its Time Discretization
9.6.2 Hermite-Collocation Method for the 1-D GPE
9.6.3 Laguerre Method for the 2-D GPE with RadialSymmetry
9.6.4 Laguerre-Hermite Method for the 3-D GPE with Cylindrical Symmetry
9.6.5 Numerical Results
Problems
Appendix A Properties of the Gamma Functions
Appendix B Essential Mathematical Concepts
B.1 Banach Space
B.2 Hilbert Space
B.3 Lax-Milgram Lemma
B.4 Lp-Space
B.5 Distributions and Weak Derivatives
B.6 Sobolev Spaces
B.7 Integral Identities: Divergence Theorem and Green's Formula
B.8 Some Useful Inequalities
B.8.1 Sobolev-Type Inequalities
B.8.2 Hardy-Type Inequalities
B.8.3 Gronwall Inequalities
Appendix C Basic Iterative Methods and Preconditioning
C.1 Krylov Subspace Methods
C.1.1 Conjugate Gradient (CG) Method
C.1.2 BiConjugate Gradient (BiCG) Method
C.1.3 Conjugate Gradient Squared (CGS) Method
C.1.4 BiConjugate Gradient Stabilized (BiCGStab) Method
C.1.5 Generalized Minimal Residual (GMRES) Method
C.2 Preconditioning
C.2.1 Preconditioned Conjugate Gradient (PCG) Method
C.2.2 Preconditioned GMRES Method
Appendix D Basic Time Discretization Schemes
D.1 Standard Methods for Initial-Valued ODEs
D.1.1 Runge–Kutta Methods
D.1.2 Multi-Step Methods
D.1.3 Backward Difference Methods (BDF)
D.2 Operator Splitting Methods
References
Index
41
Springer Series in
Computational
Mathematics
Editorial Board
R. Bank
R.L. Graham
J. Stoer
R. Varga
H. Yserentant
For further volumes:
http://www.springer.com/series/797
Jie Shen • Tao Tang • Li-Lian Wang
Spectral Methods
Algorithms, Analysis and Applications
ABC
Jie Shen
Department of Mathematics
Purdue University
N. University St. 150
West Lafayette, IN 47907-2067
USA
shen@math.purdue.edu
Tao Tang
Department of Mathematics
Hong Kong Baptist University
Waterloo Road 224
Kowloon
Hong Kong SAR
ttang@hkbu.edu.hk
Li-Lian Wang
Division of Mathematical Sciences
School of Physical & Mathematical Sciences
Nanyang Technological University
21 Nanyang Link
637371
Singapore
lilian@ntu.edu.sg
ISSN 0179-3632
ISBN 978-3-540-71040-0
DOI 10.1007/978-3-540-71041-7
Springer Heidelberg Dordrecht London New York
e-ISBN 978-3-540-71041-7
Library of Congress Control Number: 2011934044
Mathematics Subject Classification (2010): 65M70, 65M12, 65N15, 65N35, 65N22, 65F05, 35J25,
35J40, 35K15, 42C05
c Springer-Verlag Berlin Heidelberg 2011
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is
concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,
reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication
or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,
1965, in its current version, and permission for use must always be obtained from Springer. Violations
are liable to prosecution under the German Copyright Law.
The use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant protective
laws and regulations and therefore free for general use.
Cover design: deblik, Berlin
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
Preface
This book is developed from lecture notes of graduate courses taught over the years
by the authors at the Pennsylvania State University, Purdue University, Hong Kong
Baptist University and Nanyang Technological University of Singapore.
The aim of the book is to provide
spectral methods
• A detailed presentation of basic spectral algorithms
• A systematical presentation of basic convergence theory and error analysis for
• Some illustrative applications of spectral methods
For many basic algorithms presented in the book, we provide Matlab codes (which
will be made available online) which contain additional programming details be-
yond the mathematical formulas, so that the readers can easily use or modify these
codes to suite their need. We believe that these Matlab codes will help the read-
ers to have a better understanding of these spectral algorithms and provide a useful
starting point for developing their own application codes.
There are already quite a few monographs/books on spectral methods. The classi-
cal books by Gottlieb and Orszag (1977) and by Canuto et al. (1987)1 were intended
for researchers and advanced graduate students, and they are excellent references
for the historical aspects of spectral methods as well as in depth presentations of
various techniques and applications in computational fluid dynamics. The book by
Boyd (2001) focused on the Fourier and Chebyshev methods with emphasis on im-
plementations and applications. The book by Trefethen (2000) gave an excellent
exposition on the spectral-collocation methods through a set of elegant Matlab rou-
tines. The books by Deville et al. (2002) and by Karniadakis and Sherwin (2005)
concentrated on the spectral-element methods with details on parallel implementa-
tions and applications in fluid dynamics, while the more recent book by Hesthaven
and Warburton (2008) focused on the discontinuous Galerkin methods with a nodal
spectral-element approach. On the other hand, Hesthaven et al. (2007) focused on
1 An updated and expanded version of Canuto et al. (1987) is recently published. This new version
Canuto et al. (2006, 2007) incorporated many new developments made in the last 20 years and
provided a more systematical treatment for spectral methods.
v
vi
Preface
the spectral methods for time-dependent problems with a particular emphasis on
hyperbolic equations and problems with non-smooth solutions. The book length ar-
ticle by Bernardi and Maday (1997) and their monograph in French Bernardi and
Maday (1992a) provided an excellent exposition on the basic approximation theory
of spectral methods with a particular emphasis on Stokes equations, while the mono-
graph (Shen and Tang 2006) presented a basic introduction in a lecture note style
to the implementation and analysis of spectral methods. The emphasis of the book
by Guo (1998b), on the other hand, was on numerical analysis of spectral meth-
ods for nonlinear evolution problems. Finally, spectral methods have been playing a
very significant role in dealing with stochastic differential equations and uncertainty
quantifications, and we refer to the recent books by Le Maˆıtre and Knio (2010) and
by Xiu (2010) on these emerging topics.
The current book attempts to provide a self-contained presentation for the con-
struction, implementation and analysis of efficient spectral algorithms for some
model equations, of elliptic, dispersive and parabolic type, which have wide ap-
plications in science and engineering. It strives to provide a systematical approach
based on variational formulations for both algorithm development and numerical
analysis. Some of the unique features of the current book are
• Our analysis is based on the non-uniformly weighted Sobolev spaces which lead
to simplified analysis and more precise estimates, particularly for problems with
corner singularities. We also advocate the use of the generalized Jacobi polyno-
mials which are particularly useful for dealing with boundary value problems.
• We develop efficient spectral algorithms and present their error analysis for
Volterra integral equations, higher-order differential equations, problems in un-
bounded domains and in high-dimensional domains. These topics have rarely
been covered in detail in the existing books on spectral methods.
• We provide online a set of well structured Matlab codes which can be easily
modified and expanded or rewritten in other programming languages.
The Matlab codes as well as corrections/updates to the book will be available
at http://www.math.purdue.edu/∼shen/STWbook. In case this site becomes unavail-
able due to unforeseen circumstances in the future, the readers are advised to check
the Springer Web site for the updated Web link on the book.
We do not attempt to provide in this book an exhaustive account on the wide
range of topics that spectral methods have had impact on. In particular, we do not
include some important topics such as spectral methods for hyperbolic equations
and spectral-element methods, partly because these topics do not fit well in our
uniform framework, and mostly because there are already some excellent books
mentioned above on these topics. As such, no attempt is made to provide a compre-
hensive list of references on the spectral methods. The cited references reflect the
topics covered in the book, but inevitably, the authors’ bias. While we strive for cor-
rectness, it is most likely that errors still exist. We welcome comments, suggestions
and corrections.
The book can be used as a textbook for graduate students in both mathematics and
other science/engineering. Mathematical analysis and applications are organized
Preface
vii
mostly at the end of each chapter and presented in such a way that they can be
skipped without affecting the understanding of algorithms in the following chapters.
The first four chapters and Sects. 8.1–8.4 provide the basic ingredients on Fourier
and polynomial approximations and essential strategies for developing efficient
spectral-Galerkin and spectral-collocation algorithms. Section 8.5 deals with sparse
spectral methods for high-dimensional problems. The topics in Chaps. 5, 6 and 7
are independent of each other so the readers can choose according to their need.
Applications covered in Chap. 9, except for a slight dependence on Sects. 9.4–9.5,
are also independent of each other. For the readers’ convenience, we provide in the
Appendices some essential mathematical concepts, basic iterative algorithms and
commonly used time discretization schemes.
The book is also intended as a reference for active practitioners and researchers of
spectral methods. The prerequisite for the book includes standard entry-level grad-
uate courses in Numerical Analysis, Functional Analysis and Partial Differential
Equations (PDEs). Some knowledge on numerical approximations of PDEs will be
helpful in understanding the convergence theory and error analysis but hardly nec-
essary for understanding the numerical algorithms presented in this book.
The authors would like to thank all the people and organizations who have pro-
vided support for this endeavor. In particular, the authors acknowledge the general
support over the years by NSF and AFOSR of USA, Purdue University; Hong Kong
Research Grants Council, the National Natural Science Foundation of China, Hong
Kong Baptist University; Singapore Ministry of Education and Nanyang Technolog-
ical University. We are grateful to Mrs. Thanh-Ha Le Thi of Springer for her support
and for tolerating our multiple delays, and to Ms. Xiaodan Zhao of Nanyang Tech-
nological University for carefully checking the manuscript. Last but not the least,
we would like to thanks our wives and children for their love and support.
Indiana, USA
Hong Kong, China
Singapore
Jie Shen
Tao Tang
Li-Lian Wang
相关推荐
2023年江西萍乡中考道德与法治真题及答案.doc
2012年重庆南川中考生物真题及答案.doc
2013年江西师范大学地理学综合及文艺理论基础考研真题.doc
2020年四川甘孜小升初语文真题及答案I卷.doc
2020年注册岩土工程师专业基础考试真题及答案.doc
2023-2024学年福建省厦门市九年级上学期数学月考试题及答案.doc
2021-2022学年辽宁省沈阳市大东区九年级上学期语文期末试题及答案.doc
2022-2023学年北京东城区初三第一学期物理期末试卷及答案.doc
2018上半年江西教师资格初中地理学科知识与教学能力真题及答案.doc
2012年河北国家公务员申论考试真题及答案-省级.doc
2020-2021学年江苏省扬州市江都区邵樊片九年级上学期数学第一次质量检测试题及答案.doc
2022下半年黑龙江教师资格证中学综合素质真题及答案.doc
