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Chebyshev and Fourier Spectral Methods

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ISBN-10: 0486411834

ISBN-13: 9780486411835

Edition: 2nd 2001 (Revised)

Authors: John P. Boyd

List price: $37.95
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Book details

List price: $37.95
Edition: 2nd
Copyright year: 2001
Publisher: Dover Publications, Incorporated
Publication date: 12/3/2001
Binding: Hardcover
Pages: 688
Size: 6.25" wide x 9.00" long x 1.25" tall
Weight: 1.980
Language: English

Terence Ranger is Emeritus Rhodes Professor of Race Relations, University of Oxford and Visiting Professor of History at the University of Zimbabwe.

Preface
Acknowledgments
Errata and Extended-Bibliography
Introduction
Series expansions
First Example
Comparison with finite element methods
Comparisons with Finite Differences
Parallel Computers
Choice of basis functions
Boundary conditions
Non-Interpolating and Pseudospectral
Nonlinearity
Time-dependent problems
FAQ: Frequently Asked Questions
The Chrysalis
Chebyshev and Fourier Series
Introduction
Fourier series
Orders of Convergence
Convergence Order
Assumption of Equal Errors
Darboux's Principle
Why Taylor Series Fail
Location of Singularities
Corner Singularities and Compatibility Conditions
FACE: Integration-by-Parts Bound
Asymptotic Calculation of Fourier Coefficients
Convergence Theory: Chebyshev Polynomials
Last Coefficient Rule-of-Thumb
Convergence Theory for Legendre Polynomials
Quasi-Sinusoidal Rule of Thumb
Witch of Agnesi Rule-of-Thumb
Boundary Layer Rule-of-Thumb
Galerkin and Weighted Residual Methods
Mean Weighted Residual Methods
Completeness and Boundary Conditions
Inner Product and Orthogonality
Galerkin Method
Integration-by-Parts
Galerkin Method: Case Studies
Separation-of-Variables and the Galerkin Method
Heisenberg Matrix Mechanics
The Galerkin Method Today
Interpolation, Collocation and All That
Introduction
Polynomial interpolation
Gaussian Integration and Pseudospectral Grids
Pseudospectral Is Galerkin Method via Quadrature
Pseudospectral Errors
Cardinal Functions
Introduction
Whittaker Cardinal or "Sinc" Functions
Trigonometric Interpolation
Cardinal Functions for Orthogonal Polynomials
Transformations and Interpolation
Pseudospectral Methods for BVPs
Introduction
Choice of Basis Set
Boundary Conditions: Behavioral and Numerical
"Boundary-Bordering"
"Basis Recombination"
Transfinite Interpolation
The Cardinal Function Basis
The Interpolation Grid
Computing Basis Functions and Derivatives
Higher Dimensions: Indexing
Higher Dimensions
Corner Singularities
Matrix methods
Checking
Summary
Linear Eigenvalue Problems
The No-Brain Method
QR/QZ Algorithm
Eigenvalue Rule-of-Thumb
Four Kinds of Sturm-Liouville Problems
Criteria for Rejecting Eigenvalues
"Spurious" Eigenvalues
Reducing the Condition Number
The Power Method
Inverse Power Method
Combining Global and Local Methods
Detouring into the Complex Plane
Common Errors
Symmetry and Parity
Introduction
Parity
Modifying the Grid to Exploit Parity
Other Discrete Symmetries
Axisymmetric and Apple-Slicing Models
Explicit Time-Integration Methods
Introduction
Spatially-Varying Coefficients
The Shamrock Principle
Linear and Nonlinear
Example: KdV Equation
Implicitly-Implicit: RLW and QG
Partial Summation, the FFT and MMT
Introduction
Partial Summation
The Fast Fourier Transform: Theory
Matrix Multiplication Transform
Costs of the Fast Fourier Transform
Generalized FFTs and Multipole Methods
Off-Grid Interpolation
Fast Fourier Transform: Practical Matters
Summary
Aliasing, Spectral Blocking, and Blow-Up
Introduction
Aliasing and Equality-on-the-Grid
"2 h-Waves" and Spectral Blocking
Aliasing Instability: History and Remedies
Dealiasing and the Orszag Two-Thirds Rule
Energy-Conserving: Constrained Interpolation
Energy-Conserving Schemes: Discussion
Aliasing Instability: Theory
Summary
Implicit Schemes and the Slow Manifold
Introduction
Dispersion and Amplitude Errors
Errors and CFL Limit for Explicit Schemes
Implicit Time-Marching Algorithms
Semi-Implicit Methods
Speed-Reduction Rule-of-Thumb
Slow Manifold: Meteorology
Slow Manifold: Definition and Examples
Numerically-Induced Slow Manifolds
Initialization
The Method of Multiple Scales (Baer-Tribbia)
Nonlinear Galerkin Methods
Weaknesses of the Nonlinear Galerkin Method
Tracking the Slow Manifold
Three Parts to Multiple Scale Algorithms
Splitting and Its Cousins
Introduction
Fractional Steps for Diffusion
Pitfalls in Splitting, I: Boundary Conditions
Pitfalls in Splitting, II: Consistency
Operator Theory of Time-Stepping
High Order Splitting
Splitting and Fluid Mechanics
Semi-Lagrangian Advection
Concept of an Integrating Factor
Misuse of Integrating Factor Methods
Semi-Lagrangian Advection: Introduction
Advection and Method of Characteristics
Three-Level, 2D Order Semi-Implicit
Multiply-Upstream SL
Numerical Illustrations and Superconvergence
Two-Level SL/SI Algorithms
Noninterpolating SL and Numerical Diffusion
Off-Grid Interpolation
Off-Grid Interpolation: Generalities
Spectral Off-grid
Low-order Polynomial Interpolation
McGregor's Taylor Series Scheme
Higher Order SL Methods
History and Relationships to Other Methods
Summary
Matrix-Solving Methods
Introduction
Stationary One-Step Iterations
Preconditioning: Finite Difference
Computing Iterates: FFT/Matrix Multiplication
Alternative Preconditioners
Raising the Order Through Preconditioning
Multigrid: An Overview
MRR Method
Delves-Freeman Block-and-Diagonal Iteration
Recursions and Formal Integration: Constant Coefficient ODEs
Direct Methods for Separable PDE's
Fast Iterations for Almost Separable PDEs
Positive Definite and Indefinite Matrices
Preconditioned Newton Flow
Summary and Proverbs
Coordinate Transformations
Introduction
Programming Chebyshev Methods
Theory of 1-D Transformations
Infinite and Semi-Infinite Intervals
Maps for Endpoint and Corner Singularities
Two-Dimensional Maps and Corner Branch Points
Periodic Problems and the Arctan/Tan Map
Adaptive Methods
Almost-Equispaced Kosloff/Tal-Ezer Grid
Methods for Unbounded Intervals
Introduction
Domain Truncation
Domain Truncation for Rapidly-decaying Functions
Domain Truncation for Slowly-Decaying Functions
Domain Truncation for Time-Dependent Wave Propagation: Sponge Layers
Whittaker Cardinal or "Sinc" Functions
Hermite functions
Semi-Infinite Interval: Laguerre Functions
New Basis Sets via Change of Coordinate
Rational Chebyshev Functions: TB[subscript n]
Behavioral versus Numerical Boundary Conditions
Strategy for Slowly Decaying Functions
Numerical Examples: Rational Chebyshev Functions
Semi-Infinite Interval: Rational Chebyshev TL[subscript n]
Numerical Examples: Chebyshev for Semi-Infinite Interval
Strategy: Oscillatory, Non-Decaying Functions
Weideman-Cloot Sinh Mapping
Summary
Spherical and Cylindrical Geometry
Introduction
Polar, Cylindrical, Toroidal, Spherical
Apparent Singularity at the Pole
Polar Coordinates: Parity Theorem
Radial Basis Sets and Radial Grids
One-Sided Jacobi Basis for the Radial Coordinate
Boundary Value and Eigenvalue Problems on a Disk
Unbounded Domains Including the Origin in Cylindrical Coordinates
Annular Domains
Spherical Coordinates: An Overview
The Parity Factor for Scalars: Sphere versus Torus
Parity II: Horizontal Velocities and Other Vector Components
The Pole Problem: Spherical Coordinates
Spherical Harmonics: Introduction
Legendre Transforms and Other Sorrows
FFT in Longitude/MMT in Latitude
Substitutes and Accelerators for the MMT
Parity and Legendre Transforms
Hurrah for Matrix/Vector Multiplication
Reduced Grid and Other Tricks
Schuster-Dilts Triangular Matrix Acceleration
Generalized FFT: Multipoles and All That
Summary
Equiareal Resolution
Spherical Harmonics: Limited-Area Models
Spherical Harmonics and Physics
Asymptotic Approximations, I
Asymptotic Approximations, II
Software: Spherical Harmonics
Semi-Implicit: Shallow Water
Fronts and Topography: Smoothing/Filters
Fronts and Topography
Mechanics of Filtering
Spherical splines
Filter Order
Filtering with Spatially-Variable Order
Topographic Filtering in Meteorology
Resolution of Spectral Models
Vector Harmonics and Hough Functions
Radial/Vertical Coordinate: Spectral or Non-Spectral?
Basis for Axial Coordinate in Cylindrical Coordinates
Axial Basis in Toroidal Coordinates
Vertical/Radial Basis in Spherical Coordinates
Stellar Convection in a Spherical Annulus: Glatzmaier (1984)
Non-Tensor Grids: Icosahedral, etc.
Robert Basis for the Sphere
Parity-Modified Latitudinal Fourier Series
Projective Filtering for Latitudinal Fourier Series
Spectral Elements on the Sphere
Spherical Harmonics Besieged
Elliptic and Elliptic Cylinder Coordinates
Summary
Special Tricks
Introduction
Sideband Truncation
Special Basis Functions, I: Corner Singularities
Special Basis Functions, II: Wave Scattering
Weakly Nonlocal Solitary Waves
Root-Finding by Chebyshev Polynomials
Hilbert Transform
Spectrally-Accurate Quadrature Methods
Introduction: Gaussian and Clenshaw-Curtis Quadrature
Clenshaw-Curtis Adaptivity
Mechanics
Integration of Periodic Functions and the Trapezoidal Rule
Infinite Intervals and the Trapezoidal Rule
Singular Integrands
Sets and Solitaries
Symbolic Calculations
Introduction
Strategy
Examples
Summary and Open Problems
Teh Tau-Method
Introduction
[tau]-Approximation for a Rational Function
Differential Equations
Canonical Polynomials
Nomenclature
Domain Decomposition Methods
Introduction
Notation
Connecting the Subdomains: Patching
Weak Coupling of Elemental Solutions
Variational Principles
Choice of Basis and Grid
Patching versus Variational Formalism
Matrix Inversion
The Influence Matrix Method
Two-Dimensional Mappings and Sectorial Elements
Prospectus
Books and Reviews
A Bestiary of Basis Functions
Trigonometric Basis Functions: Fourier Series
Chebyshev Polynomials: T[subscript n](x)
Chebyshev Polynomials of the Second Kind: U[subscript n](x)
Legendre Polynomials: P[subscript n](x)
Gegenbauer Polynomials
Hermite Polynomials: H[subscript n](x)
Rational Chebyshev Functions: TB[subscript n](y)
Laguerre Polynomials: L[subscript n](x)
Rational Chebyshev Functions: TL[subscript n](y)
Graphs of Convergence Domains in the the Complex Plane
Direct Matrix-Solvers
Matrix Factorizations
Banded Matrix
Matrix-of-Matrices Theorem
Block-Banded Elimination: the "Lindzen-Kuo" Algorithm
Blck and "Bordered" Matrices
Cyclic Banded Matrices (Periodic Boundary Conditions)
Parting shots
Newton Iteration
Introduction
Examples
Eigevalue Problems
Summary
The Continuation Method
Introduction
Examples
Initialization Strategies
Limit Points
Bifurcation points
Pseudoarclength Continuation
Change-of-Coordinate Derivative Transformations
Cardinal Functions
Introduction
General Fourier Series: Endopint Grid
Fourier Cosine Series: Endpoint Grid
Fourier Sine Series: Endpoint Grid
Cosine Cardinal Functions: Interior Grid
Sine Cardinal Functions: Interior Grid
Sinc(x): Whittaker cardinal function
Chebyshev Gauss-Lobatto ("Endpoints")
Chebyshev Polynomials: Interior or "Roots" Grid
Legendre Polynomials: Gauss-Lobatto Grid
Transformation of Derivative Boundary Conditions
Glossary
Index
References