Chebyshev and Fourier Spectral Methods / Edition 2

Chebyshev and Fourier Spectral Methods / Edition 2

by John Philip Boyd, J. P. Boyd

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ISBN-10: 0486411834

ISBN-13: 9780486411835

Pub. Date: 12/03/2001

Publisher: Dover Publications

Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals,


Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.

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Dover Publications
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Dover Books on Mathematics Series
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6.40(w) x 9.10(h) x 1.40(d)

Table of Contents

  Preface; Acknowledgments; Errata and Extended-Bibliography
1. Introduction
  1.1 Series expansions
  1.2 First example
  1.3 Comparison with finite element methods
  1.4 Comparisons with finite differences
  1.5 Parallel computers
  1.6 Choice of basis functions
  1.7 Boundary conditions
  1.8 Non-Interpolating and Pseudospectral
  1.9 Nonlinearity
  1.10 Time-dependent problems
  1.11 FAQ: frequently asked questions
  1.12 The chrysalis
2. Chebyshev & Fourier series
  2.1 Introduction
  2.2 Fourier series
  2.3 Orders of convergence
  2.4 Convergence order
  2.5 Assumption of equal errors
  2.6 Darboux's principle
  2.7 Why Taylor series fail
  2.8 Location of singularities
    2.8.1 Corner singularities & compatibility conditions
  2.9 FACE: Integration-by-Parts bound
  2.10 Asymptotic calculation of Fourier coefficients
  2.11 Convergence theory: Chebyshev polynomials
  2.12 Last coefficient rule-of-thumb
  2.13 Convergence theory for Legendre polynomials
  2.14 Quasi-Sinusoidal rule of thumb
  2.15 Witch of Agensi rule-of-thumb
  2.16 Boundary layer rule-of-thumb
3. Galerkin & Weighted residual methods
  3.1 Mean weighted residual methods
  3.2 Completeness and boundary conditions
  3.3 Inner product & orthogonality
  3.4 Galerkin method
  3.5 Integration-by-Parts
  3.6 Galerkin method: case studies
  3.7 Separation-of-Variables & the Galerkin method
  3.8 Heisenberg Matrix mechanics
  3.9 The Galerkin method today
4. Interpolation, collocation & all that
  4.1 Introduction
  4.2 Polynomial interpolation
  4.3 Gaussian integration & pseudospectral grids
  4.4 Pseudospectral Is Galerkin method via Quadrature
  4.5 Pseudospectral errors
5. Cardinal functions
  5.1 Introduction
  5.2 Whittaker cardinal or "sinc" functions
  5.3 Trigonometric interpolation
  5.4 Cardinal functions for orthogonal polynomials
  5.5 Transformations and interpolation
6. Pseudospectral methods for BVPs
  6.1 Introduction
  6.2 Choice of basis set
  6.3 Boundary conditions: behavioral & numerical
  6.4 "Boundary-bordering"
  6.5 "Basis Recombination"
  6.6 Transfinite interpolation
  6.7 The Cardinal function basis
  6.8 The interpolation grid
  6.9 Computing basis functions & derivatives
  6.10 Higher dimensions: indexing
  6.11 Higher dimensions
  6.12 Corner singularities
  6.13 Matrix methods
  6.14 Checking
  6.15 Summary
7. Linear eigenvalue problems
  7.1 The No-brain method
  7.2 QR/QZ Algorithm
  7.3 Eigenvalue rule-of-thumb
  7.4 Four kinds of Sturm-Liouville problems
  7.5 Criteria for Rejecting eigenvalues
  7.6 "Spurious" eigenvalues
  7.7 Reducing the condition number
  7.8 The power method
  7.9 Inverse power method
  7.10 Combining global & local methods
  7.11 Detouring into the complex plane
  7.12 Common errors
8. Symmetry & parity
  8.1 Introduction
  8.2 Parity
  8.3 Modifying the Grid to Exploit parity
  8.4 Other discrete symmetries
  8.5 Axisymmetric & apple-slicing models
9. Explicit time-integration methods
  9.1 Introduction
  9.2 Spatially-varying coefficients
  9.3 The Shamrock principle
  9.4 Linear and nonlinear
  9.5 Example: KdV equation
  9.6 Implicitly-Implicit: RLW & QG
10. Partial summation, the FFT and MMT
  10.1 Introduction
  10.2 Partial summation
  10.3 The fast Fourier transform: theory
  10.4 Matrix multiplication transform
  10.5 Costs of the fast Fourier transform
  10.6 Generalized FFTs and multipole methods
  10.7 Off-grid interpolation
  10.8 Fast Fourier transform: practical matters
  10.9 Summary
11. Aliasing, spectral blocking, & blow-up
  11.1 Introduction
  11.2 Aliasing and Equality-on-the-grid
  11.3 "2 h-Waves" and spectral blocking
  11.4 Aliasing instability: history and remedies
  11.5 Dealiasing and the Orszag two-thirds rule
  11.6 Energy-conserving: constrained interpolation
  11.7 Energy-conserving schemes: discussion
  11.8 Aliasing instability: theory
  11.9 Summary
12. Implicit schemes & the slow manifold
  12.1 Introduction
  12.2 Dispersion and amplitude errors
  12.3 Errors & CFL limit for explicit schemes
  12.4 Implicit time-marching algorithms
  12.5 Semi-implicit methods
  12.6 Speed-reduction rule-of-thumb
  12.7 Slow manifold: meteorology
  12.8 Slow manifold: definition & examples
  12.9 Numerically-induced slow manifolds
  12.10 Initialization
  12.11 The method of multiple scales (Baer-Tribbia)
  12.12 Nonlinear Galerkin methods
  12.13 Weaknesses of the nonlinear Galerkin method
  12.14 Tracking the slow manifold
  12.15 Three parts to multiple scale algorithms
13. Splitting & its cousins
  13.1 Introduction
  13.2 Fractional steps for diffusion
  13.3 Pitfalls in splitting, I: boundary conditions
  13.4 Pitfalls in splitting, II: consistency
  13.5 Operator theory of time-stepping
  13.6 High order splitting
  13.7 Splitting and fluid mechanics
14. Semi-Lagrangian advection
  14.1 Concept of an integrating factor
  14.2 Misuse of integrating factor methods
  14.3 Semi-Lagrangian advection: introduction
  14.4 Advection & method of characteristics
  14.5 Three-level, 2D order semi-implicit
  14.6 Multiply-upstream SL
  14.7 Numerical illustrations & superconvergence
  14.8 Two-level SL/SI algorithms
  14.9 Noninterpolating SL & numerical diffusion
  14.10 Off-grid interpolation
    14.10.1 Off-grid interpolation: generalities
    14.10.2 Spectral off-grid
    14.10.3 Low-order polynomial interpolation
    14.10.4 McGregor's Taylor series scheme
    14.11 Higher order SL methods
    14.12 History and relationships to other methods
  14.13 Summary
15. Matrix-solving methods
  15.1 Introduction
  15.2 Stationary one-step iterations
  15.3 Preconditioning: finite difference
  15.4 Computing iterates: FFT/matrix multiplication
  15.5 Alternative preconditioners
  15.6 Raising the order through preconditioning
  15.7 Multigrid: an overview
  15.8 MRR method
  15.9 Delves-Freeman block-and-diagonal iteration
  15.10 Recursions & formal integration: constant coefficient ODEs
  15.11 Direct methods for separable PDE's
  15.12 Fast interations for almost separable PDEs
  15.13 Positive definite and indefinite matrices
  15.14 Preconditioned Newton flow
  15.15 Summary & proverbs
16. Coordinate transformations
  16.1 Introduction
  16.2 Programming Chebyshev methods
  16.3 Theory of 1-D transformations
  16.4 Infinite and semi-infinite intervals
  16.5 Maps for endpoint & corner singularities
  16.6 Two-dimensional maps & corner branch points
  16.7 Periodic problems & the Arctan/Tan map
  16.8 Adaptive methods
  16.9 Almost-equispaced Kosloff/Tal-Ezer grid
17. Methods for unbounded intervals
  17.1 Introduction
  17.2 Domain truncation
    17.2.1 Domain truncation for rapidly-decaying functions
  17.7 Rational Chebyshev functions: TB subscript n
  17.8 Behavioral versus numerical boundary conditions
  17.9 Strategy for slowly decaying functions
  17.10 Numerical exemples: rational Chebyshev functions
  17.11 Semi-infinite interval: rational Chebyshev TL subscript n
  17.12 Numerical Examples: Chebyshev for semi-infinite interval
  17.13 Strategy: Oscillatory, non-decaying functions
  17.14 Weideman-Cloot Sinh mapping
  17.15 Summary
18. Spherical & Cylindrical geometry
  18.1 Introduction
  18.2 Polar, cylindrical, toroidal, spherical
  18.3 Apparent singularity at the pole
  18.4 Polar coordinates: parity theorem
  18.5 Radial basis sets and radial grids
    18.5.1 One-sided Jacobi basis for the radial coordinate
    18.5.2 Boundary value & eigenvalue problems on a disk
    18.5.3 Unbounded domains including the origin in Cylindrical coordinates
  18.6 Annual domains
  18.7 Spherical coordinates: an overview
  18.8 The parity factoro for scalars: sphere versus torus
  18.9 Parity II: Horizontal velocities & other vector components
  18.10 The Pole problem: spherical coordinates
  18.11 Spherical harmonics: introduction
  18.12 Legendre transforms and other sorrows
    18.12.1 FFT in longitude/MMT in latitude
    18.12.2 Substitutes and accelerators for the MMT
    18.12.3 Parity and Legendre Transforms
    18.12.4 Hurrah for matrix/vector multiplication
    18.12.5 Reduced grid and other tricks
    18.12.6 Schuster-Dilts triangular matrix acceleration
    18.12.7 Generalized FFT: multipoles and all that
    18.12.8 Summary
  18.13 Equiareal resolution
  18.14 Spherical harmonics: limited-area models
  18.15 Spherical harmonics and physics
  18.16 Asymptotic approximations, I
  18.17 Asymptotic approximations, II
  18.18 Software: spherical harmonics
  18.19 Semi-implicit: shallow water
  18.20 Fronts and topography: smoothing/filters
    18.20.1 Fronts and topography
    18.20.2 Mechanics of filtering
    18.20.3 Spherical splines
    18.20.4 Filter order
    18.20.5 Filtering with spatially-variable order
    18.20.6 Topographic filtering in meteorology
  18.21 Resolution of spectral models
  18.22 Vector harmonics & Hough functions
  18.23 Radial/vertical coordinate: spectral or non-spectral?
    18.23.1 Basis for Axial coordinate in cylindrical coordinates
    18.23.2 Axial basis in toroidal coordinates
    18.23.3 Vertical/radial basis in spherical coordinates
  18.24 Stellar convection in a spherical annulus: Glatzmaier (1984)
  18.25 Non-tensor grids: icosahedral, etc.
  18.26 Robert basis for the sphere
  18.27 Parity-modified latitudinal Fourier series
  18.28 Projective filtering for latitudinal Fourier series
  18.29 Spectral elements on the sphere
  18.30 Spherical harmonics besieged
  18.31 Elliptic and elliptic cylinder coordinates
  18.32 Summary
19. Special tricks
  19.1 Introduction
  19.2 Sideband truncation
  19.3 Special basis functions, I: corner singularities
  19.4 Special basis functions, II: wave scattering
  19.5 Weakly nonlocal solitary waves
  19.6 Root-finding by Chebyshev polynomials
  19.7 Hilbert transform
  19.8 Spectrally-accurate quadrature methods
    19.8.1 Introduction: Gaussian and Clenshaw-Curtis quadrature
    19.8.2 Clenshaw-Curtis adaptivity
    19.8.3 Mechanics
    19.8.4 Integration of periodic functions and the trapezoidal rule
    19.8.5 Infinite intervals and the trapezoidal rule
    19.8.6 Singular integrands
    19.8.7 Sets and solitaries
20. Symbolic calculations
  20.1 Introduction
  20.2 Strategy
  20.3 Examples
  20.4 Summary and open problems
21. The Tau-method
  21.1 Introduction
  21.2 tau-Approximation for a rational function
  21.3 Differential equations
  21.4 Canonical polynomials
  21.5 Nomenclature
22. Domain decomposition methods
  22.1 Introduction
  22.2 Notation
  22.3 Connecting the subdomains: patching
  22.4 Weak coupling of elemental solutions
  22.5 Variational principles
  22.6 Choice of basis & grid
  22.7 Patching versus variational formalism
  22.8 Matrix inversion
  22.9 The influence matrix method
  22.10 Two-dimensional mappings & sectorial elements
  22.11 Prospectus
23. Books and reviews
  A. A bestiary of basis functions
    A.1 Trigonometric basis functions: Fourier series
    A.2 Chebyshev polynomials T subscript n (x)
    A.3 Chebyshev polynomials of the second kind: U subscript n (x)
    A.4 Legendre polynomials: P subscript n (x)
    A.5 Gegenbauer polynomials
    A.6 Hermite polynomials: H subscript n (x)
    A.7 Rational Chebyshev functions: TB subscript n (y)
    A.8 Laguerre polynomials: L subscript n (x)
    A.9 Rational Chebyshev functions: TL subscript n (y)
    A.10 Graphs of convergence domains in the complex plane
  B. Direct matrix-solvers
    B.1 Matrix factorizations
    B.2 Banded matrix
    B.3 Matrix-of-matrices theorem
    B.4 Block-banded elimination: the "Lindzen-Kuo" algorithm
    B.5 Block and "bordered" matrices
    B.6 Cyclic banded matrices (periodic boundary conditions)
    B.7 Parting shots
  C. Newton iteration
    C.1 Introduction
    C.2 Examples
    C.3 Eigenvalue problems
    C.4 Summary
  D. The continuation method
    D.1 Introduction
    D.2 Examples
    D.3 Initialization strategies
    D.4 Limit Points
    D.5 Bifurcation points
    D.6 Pseudoarclength continuation
  E. Change-of-Coordinate derivative transformations
  F. Cardinal functions
    F.1 Introduction
    F.2 General Fourier series: endpoint grid
    F.3 Fourier Cosine series: endpoint grid
    F.4 Fourier Sine series: endpoint grid
    F.5 Cosine cardinal functions: interior grid
    F.6 Sine cardinal functions: interior grid
    F.7 Sinc(x): Whittaker cardinal function
    F.8 Chebyshev Gauss-Lobatto ("endpoints")
    F.9 Chebyshev polynomials: interior or "roots" grid
    F.10 Legendre polynomials: Gauss-Lobatto grid
  G. Transformation of derivative boundary conditions
  Glossary; Index; References

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