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.
and#160; Preface; Acknowledgments; Errata and Extended-Bibliography
1. Introduction
and#160; 1.1 Series expansions
and#160; 1.2 First example
and#160; 1.3 Comparison with finite element methods
and#160; 1.4 Comparisons with finite differences
and#160; 1.5 Parallel computers
and#160; 1.6 Choice of basis functions
and#160; 1.7 Boundary conditions
and#160; 1.8 Non-Interpolating and Pseudospectral
and#160; 1.9 Nonlinearity
and#160; 1.10 Time-dependent problems
and#160; 1.11 FAQ: frequently asked questions
and#160; 1.12 The chrysalis
2. Chebyshev and Fourier series
and#160; 2.1 Introduction
and#160; 2.2 Fourier series
and#160; 2.3 Orders of convergence
and#160; 2.4 Convergence order
and#160; 2.5 Assumption of equal errors
and#160; 2.6 Darboux's principle
and#160; 2.7 Why Taylor series fail
and#160; 2.8 Location of singularities
and#160;and#160;and#160; 2.8.1 Corner singularities and compatibility conditions
and#160; 2.9 FACE: Integration-by-Parts bound
and#160; 2.10 Asymptotic calculation of Fourier coefficients
and#160; 2.11 Convergence theory: Chebyshev polynomials
and#160; 2.12 Last coefficient rule-of-thumb
and#160; 2.13 Convergence theory for Legendre polynomials
and#160; 2.14 Quasi-Sinusoidal rule of thumb
and#160; 2.15 Witch of Agensi rule-of-thumb
and#160; 2.16 Boundary layer rule-of-thumb
3. Galerkin and Weighted residual methods
and#160; 3.1 Mean weighted residual methods
and#160; 3.2 Completeness and boundary conditions
and#160; 3.3 Inner product and orthogonality
and#160; 3.4 Galerkin method
and#160; 3.5 Integration-by-Parts
and#160; 3.6 Galerkin method: case studies
and#160; 3.7 Separation-of-Variables and the Galerkin method
and#160; 3.8 Heisenberg Matrix mechanics
and#160; 3.9 The Galerkin method today
4. Interpolation, collocation and all that
and#160; 4.1 Introduction
and#160; 4.2 Polynomial interpolation
and#160; 4.3 Gaussian integration and pseudospectral grids
and#160; 4.4 Pseudospectral Is Galerkin method via Quadrature
and#160; 4.5 Pseudospectral errors
5. Cardinal functions
and#160; 5.1 Introduction
and#160; 5.2 Whittaker cardinal or "sinc" functions
and#160; 5.3 Trigonometric interpolation
and#160; 5.4 Cardinal functions for orthogonal polynomials
and#160; 5.5 Transformations and interpolation
6. Pseudospectral methods for BVPs
and#160; 6.1 Introduction
and#160; 6.2 Choice of basis set
and#160; 6.3 Boundary conditions: behavioral and numerical
and#160; 6.4 "Boundary-bordering"
and#160; 6.5 "Basis Recombination"
and#160; 6.6 Transfinite interpolation
and#160; 6.7 The Cardinal function basis
and#160; 6.8 The interpolation grid
and#160; 6.9 Computing basis functions and derivatives
and#160; 6.10 Higher dimensions: indexing
and#160; 6.11 Higher dimensions
and#160; 6.12 Corner singularities
and#160; 6.13 Matrix methods
and#160; 6.14 Checking
and#160; 6.15 Summary
7. Linear eigenvalue problems
and#160; 7.1 The No-brain method
and#160; 7.2 QR/QZ Algorithm
and#160; 7.3 Eigenvalue rule-of-thumb
and#160; 7.4 Four kinds of Sturm-Liouville problems
and#160; 7.5 Criteria for Rejecting eigenvalues
and#160; 7.6 "Spurious" eigenvalues
and#160; 7.7 Reducing the condition number
and#160; 7.8 The power method
and#160; 7.9 Inverse power method
and#160; 7.10 Combining global and local methods
and#160; 7.11 Detouring into the complex plane
and#160; 7.12 Common errors
8. Symmetry and parity
and#160; 8.1 Introduction
and#160; 8.2 Parity
and#160; 8.3 Modifying the Grid to Exploit parity
and#160; 8.4 Other discrete symmetries
and#160; 8.5 Axisymmetric and apple-slicing models
9. Explicit time-integration methods
and#160; 9.1 Introduction
and#160; 9.2 Spatially-varying coefficients
and#160; 9.3 The Shamrock principle
and#160; 9.4 Linear and nonlinear
and#160; 9.5 Example: KdV equation
and#160; 9.6 Implicitly-Implicit: RLW and QG
10. Partial summation, the FFT and MMT
and#160; 10.1 Introduction
and#160; 10.2 Partial summation
and#160; 10.3 The fast Fourier transform: theory
and#160; 10.4 Matrix multiplication transform
and#160; 10.5 Costs of the fast Fourier transform
and#160; 10.6 Generalized FFTs and multipole methods
and#160; 10.7 Off-grid interpolation
and#160; 10.8 Fast Fourier transform: practical matters
and#160; 10.9 Summary
11. Aliasing, spectral blocking, and blow-up
and#160; 11.1 Introduction
and#160; 11.2 Aliasing and Equality-on-the-grid
and#160; 11.3 "2 h-Waves" and spectral blocking
and#160; 11.4 Aliasing instability: history and remedies
and#160; 11.5 Dealiasing and the Orszag two-thirds rule
and#160; 11.6 Energy-conserving: constrained interpolation
and#160; 11.7 Energy-conserving schemes: discussion
and#160; 11.8 Aliasing instability: theory
and#160; 11.9 Summary
12. Implicit schemes and the slow manifold
and#160; 12.1 Introduction
and#160; 12.2 Dispersion and amplitude errors
and#160; 12.3 Errors and CFL limit for explicit schemes
and#160; 12.4 Implicit time-marching algorithms
and#160; 12.5 Semi-implicit methods
and#160; 12.6 Speed-reduction rule-of-thumb
and#160; 12.7 Slow manifold: meteorology
and#160; 12.8 Slow manifold: definition and examples
and#160; 12.9 Numerically-induced slow manifolds
and#160; 12.10 Initialization
and#160; 12.11 The method of multiple scales (Baer-Tribbia)
and#160; 12.12 Nonlinear Galerkin methods
and#160; 12.13 Weaknesses of the nonlinear Galerkin method
and#160; 12.14 Tracking the slow manifold
and#160; 12.15 Three parts to multiple scale algorithms
13. Splitting and its cousins
and#160; 13.1 Introduction
and#160; 13.2 Fractional steps for diffusion
and#160; 13.3 Pitfalls in splitting, I: boundary conditions
and#160; 13.4 Pitfalls in splitting, II: consistency
and#160; 13.5 Operator theory of time-stepping
and#160; 13.6 High order splitting
and#160; 13.7 Splitting and fluid mechanics
14. Semi-Lagrangian advection
and#160; 14.1 Concept of an integrating factor
and#160; 14.2 Misuse of integrating factor methods
and#160; 14.3 Semi-Lagrangian advection: introduction
and#160; 14.4 Advection and method of characteristics
and#160; 14.5 Three-level, 2D order semi-implicit
and#160; 14.6 Multiply-upstream SL
and#160; 14.7 Numerical illustrations and superconvergence
and#160; 14.8 Two-level SL/SI algorithms
and#160; 14.9 Noninterpolating SL and numerical diffusion
and#160; 14.10 Off-grid interpolation
and#160;and#160;and#160; 14.10.1 Off-grid interpolation: generalities
and#160;and#160;and#160; 14.10.2 Spectral off-grid
and#160;and#160;and#160; 14.10.3 Low-order polynomial interpolation
and#160;and#160;and#160; 14.10.4 McGregor's Taylor series scheme
and#160;and#160;and#160; 14.11 Higher order SL methods
and#160;and#160;and#160; 14.12 History and relationships to other methods
and#160; 14.13 Summary
15. Matrix-solving methods
and#160; 15.1 Introduction
and#160; 15.2 Stationary one-step iterations
and#160; 15.3 Preconditioning: finite difference
and#160; 15.4 Computing iterates: FFT/matrix multiplication
and#160; 15.5 Alternative preconditioners
and#160; 15.6 Raising the order through preconditioning
and#160; 15.7 Multigrid: an overview
and#160; 15.8 MRR method
and#160; 15.9 Delves-Freeman block-and-diagonal iteration
and#160; 15.10 Recursions and formal integration: constant coefficient ODEs
and#160; 15.11 Direct methods for separable PDE's
and#160; 15.12 Fast interations for almost separable PDEs
and#160; 15.13 Positive definite and indefinite matrices
and#160; 15.14 Preconditioned Newton flow
and#160; 15.15 Summary and proverbs
16. Coordinate transformations
and#160; 16.1 Introduction
and#160; 16.2 Programming Chebyshev methods
and#160; 16.3 Theory of 1-D transformations
and#160; 16.4 Infinite and semi-infinite intervals
and#160; 16.5 Maps for endpoint and corner singularities
and#160; 16.6 Two-dimensional maps and corner branch points
and#160; 16.7 Periodic problems and the Arctan/Tan map
and#160; 16.8 Adaptive methods
and#160; 16.9 Almost-equispaced Kosloff/Tal-Ezer grid
17. Methods for unbounded intervals
and#160; 17.1 Introduction
and#160; 17.2 Domain truncation
and#160;and#160;and#160; 17.2.1 Domain truncation for rapidly-decaying functions
and#160;and#160;andnbsp
and#160; 17.7 Rational Chebyshev functions: TB subscript n
and#160; 17.8 Behavioral versus numerical boundary conditions
and#160; 17.9 Strategy for slowly decaying functions
and#160; 17.10 Numerical exemples: rational Chebyshev functions
and#160; 17.11 Semi-infinite interval: rational Chebyshev TL subscript n
and#160; 17.12 Numerical Examples: Chebyshev for semi-infinite interval
and#160; 17.13 Strategy: Oscillatory, non-decaying functions
and#160; 17.14 Weideman-Cloot Sinh mapping
and#160; 17.15 Summary
18. Spherical and Cylindrical geometry
and#160; 18.1 Introduction
and#160; 18.2 Polar, cylindrical, toroidal, spherical
and#160; 18.3 Apparent singularity at the pole
and#160; 18.4 Polar coordinates: parity theorem
and#160; 18.5 Radial basis sets and radial grids
and#160;and#160;and#160; 18.5.1 One-sided Jacobi basis for the radial coordinate
and#160;and#160;and#160; 18.5.2 Boundary value and eigenvalue problems on a disk
and#160;and#160;and#160; 18.5.3 Unbounded domains including the origin in Cylindrical coordinates
and#160; 18.6 Annual domains
and#160; 18.7 Spherical coordinates: an overview
and#160; 18.8 The parity factoro for scalars: sphere versus torus
and#160; 18.9 Parity II: Horizontal velocities and other vector components
and#160; 18.10 The Pole problem: spherical coordinates
and#160; 18.11 Spherical harmonics: introduction
and#160; 18.12 Legendre transforms and other sorrows
and#160;and#160;and#160; 18.12.1 FFT in longitude/MMT in latitude
and#160;and#160;and#160; 18.12.2 Substitutes and accelerators for the MMT
and#160;and#160;and#160; 18.12.3 Parity and Legendre Transforms
and#160;and#160;and#160; 18.12.4 Hurrah for matrix/vector multiplication
and#160;and#160;and#160; 18.12.5 Reduced grid and other tricks
and#160;and#160;and#160; 18.12.6 Schuster-Dilts triangular matrix acceleration
and#160;and#160;and#160; 18.12.7 Generalized FFT: multipoles and all that
and#160;and#160;and#160; 18.12.8 Summary
and#160; 18.13 Equiareal resolution
and#160; 18.14 Spherical harmonics: limited-area models
and#160; 18.15 Spherical harmonics and physics
and#160; 18.16 Asymptotic approximations, I
and#160; 18.17 Asymptotic approximations, II
and#160; 18.18 Software: spherical harmonics
and#160; 18.19 Semi-implicit: shallow water
and#160; 18.20 Fronts and topography: smoothing/filters
and#160;and#160;and#160; 18.20.1 Fronts and topography
and#160;and#160;and#160; 18.20.2 Mechanics of filtering
and#160;and#160;and#160; 18.20.3 Spherical splines
and#160;and#160;and#160; 18.20.4 Filter order
and#160;and#160;and#160; 18.20.5 Filtering with spatially-variable order
and#160;and#160;and#160; 18.20.6 Topographic filtering in meteorology
and#160; 18.21 Resolution of spectral models
and#160; 18.22 Vector harmonics and Hough functions
and#160; 18.23 Radial/vertical coordinate: spectral or non-spectral?
and#160;and#160;and#160; 18.23.1 Basis for Axial coordinate in cylindrical coordinates
and#160;and#160;and#160; 18.23.2 Axial basis in toroidal coordinates
and#160;and#160;and#160; 18.23.3 Vertical/radial basis in spherical coordinates
and#160; 18.24 Stellar convection in a spherical annulus: Glatzmaier (1984)
and#160; 18.25 Non-tensor grids: icosahedral, etc.
and#160; 18.26 Robert basis for the sphere
and#160; 18.27 Parity-modified latitudinal Fourier series
and#160; 18.28 Projective filtering for latitudinal Fourier series
and#160; 18.29 Spectral elements on the sphere
and#160; 18.30 Spherical harmonics besieged
and#160; 18.31 Elliptic and elliptic cylinder coordinates
and#160; 18.32 Summary
19. Special tricks
and#160; 19.1 Introduction
and#160; 19.2 Sideband truncation
and#160; 19.3 Special basis functions, I: corner singularities
and#160; 19.4 Special basis functions, II: wave scattering
and#160; 19.5 Weakly nonlocal solitary waves
and#160; 19.6 Root-finding by Chebyshev polynomials
and#160; 19.7 Hilbert transform
and#160; 19.8 Spectrally-accurate quadrature methods
and#160;and#160;and#160; 19.8.1 Introduction: Gaussian and Clenshaw-Curtis quadrature
and#160;and#160;and#160; 19.8.2 Clenshaw-Curtis adaptivity
and#160;and#160;and#160; 19.8.3 Mechanics
and#160;and#160;and#160; 19.8.4 Integration of periodic functions and the trapezoidal rule
and#160;and#160;and#160; 19.8.5 Infinite intervals and the trapezoidal rule
and#160;and#160;and#160; 19.8.6 Singular integrands
and#160;and#160;and#160; 19.8.7 Sets and solitaries
20. Symbolic calculations
and#160; 20.1 Introduction
and#160; 20.2 Strategy
and#160; 20.3 Examples
and#160; 20.4 Summary and open problems
21. The Tau-method
and#160; 21.1 Introduction
and#160; 21.2 tau-Approximation for a rational function
and#160; 21.3 Differential equations
and#160; 21.4 Canonical polynomials
and#160; 21.5 Nomenclature
22. Domain decomposition methods
and#160; 22.1 Introduction
and#160; 22.2 Notation
and#160; 22.3 Connecting the subdomains: patching
and#160; 22.4 Weak coupling of elemental solutions
and#160; 22.5 Variational principles
and#160; 22.6 Choice of basis and grid
and#160; 22.7 Patching versus variational formalism
and#160; 22.8 Matrix inversion
and#160; 22.9 The influence matrix method
and#160; 22.10 Two-dimensional mappings and sectorial elements
and#160; 22.11 Prospectus
23. Books and reviews
and#160; A. A bestiary of basis functions
and#160;and#160;and#160; A.1 Trigonometric basis functions: Fourier series
and#160;and#160;and#160; A.2 Chebyshev polynomials T subscript n (x)
and#160;and#160;and#160; A.3 Chebyshev polynomials of the second kind: U subscript n (x)
and#160;and#160;and#160; A.4 Legendre polynomials: P subscript n (x)
and#160;and#160;and#160; A.5 Gegenbauer polynomials
and#160;and#160;and#160; A.6 Hermite polynomials: H subscript n (x)
and#160;and#160;and#160; A.7 Rational Chebyshev functions: TB subscript n (y)
and#160;and#160;and#160; A.8 Laguerre polynomials: L subscript n (x)
and#160;and#160;and#160; A.9 Rational Chebyshev functions: TL subscript n (y)
and#160;and#160;and#160; A.10 Graphs of convergence domains in the complex plane
and#160; B. Direct matrix-solvers
and#160;and#160;and#160; B.1 Matrix factorizations
and#160;and#160;and#160; B.2 Banded matrix
and#160;and#160;and#160; B.3 Matrix-of-matrices theorem
and#160;and#160;and#160; B.4 Block-banded elimination: the "Lindzen-Kuo" algorithm
and#160;and#160;and#160; B.5 Block and "bordered" matrices
and#160;and#160;and#160; B.6 Cyclic banded matrices (periodic boundary conditions)
and#160;and#160;and#160; B.7 Parting shots
and#160; C. Newton iteration
and#160;and#160;and#160; C.1 Introduction
and#160;and#160;and#160; C.2 Examples
and#160;and#160;and#160; C.3 Eigenvalue problems
and#160;and#160;and#160; C.4 Summary
and#160; D. The continuation method
and#160;and#160;and#160; D.1 Introduction
and#160;and#160;and#160; D.2 Examples
and#160;and#160;and#160; D.3 Initialization strategies
and#160;and#160;and#160; D.4 Limit Points
and#160;and#160;and#160; D.5 Bifurcation points
and#160;and#160;and#160; D.6 Pseudoarclength continuation
and#160; E. Change-of-Coordinate derivative transformations
and#160; F. Cardinal functions
and#160;and#160;and#160; F.1 Introduction
and#160;and#160;and#160; F.2 General Fourier series: endpoint grid
and#160;and#160;and#160; F.3 Fourier Cosine series: endpoint grid
and#160;and#160;and#160; F.4 Fourier Sine series: endpoint grid
and#160;and#160;and#160; F.5 Cosine cardinal functions: interior grid
and#160;and#160;and#160; F.6 Sine cardinal functions: interior grid
and#160;and#160;and#160; F.7 Sinc(x): Whittaker cardinal function
and#160;and#160;and#160; F.8 Chebyshev Gauss-Lobatto ("endpoints")
and#160;and#160;and#160; F.9 Chebyshev polynomials: interior or "roots" grid
and#160;and#160;and#160; F.10 Legendre polynomials: Gauss-Lobatto grid
and#160; G. Transformation of derivative boundary conditions
and#160; Glossary; Index; References