http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
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. Chrysalis -- 2. Chebyshev and 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 and 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 Agnesi Rule-of-Thumb -- 2.16. Boundary Layer Rule-of-Thumb -- 3. Galerkin and Weighted Residual Methods -- 3.1. Mean Weighted Residual Methods -- 3.2. Completeness and Boundary Conditions -- 3.3. Inner Product and Orthogonality -- 3.4. Galerkin Method -- 3.5. Integration-by-Parts -- 3.6. Galerkin Method: Case Studies -- 3.7. Separation-of-Variables and the Galerkin Method -- 3.8. Heisenberg Matrix Mechanics -- 3.9. Galerkin Method Today -- 4. Interpolation, Collocation and All That -- 4.1. Introduction -- 4.2. Polynomial interpolation -- 4.3. Gaussian Integration and 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 and Numerical -- 6.4. "Boundary-Bordering" -- 6.5. "Basis Recombination" -- 6.6. Transfinite Interpolation -- 6.7. Cardinal Function Basis -- 6.8. Interpolation Grid -- 6.9. Computing Basis Functions and 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. 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. Power Method -- 7.9. Inverse Power Method -- 7.10. Combining Global and Local Methods -- 7.11. Detouring into the Complex Plane -- 7.12. Common Errors -- 8. Symmetry and Parity -- 8.1. Introduction -- 8.2. Parity -- 8.3. Modifying the Grid to Exploit Parity -- 8.4. Other Discrete Symmetries -- 8.5. Axisymmetric and Apple-Slicing Models -- 9. Explicit Time-Integration Methods -- 9.1. Introduction -- 9.2. Spatially-Varying Coefficients -- 9.3. Shamrock Principle -- 9.4. Linear and Nonlinear -- 9.5. Example: KdV Equation -- 9.6. Implicitly-Implicit: RLW and QG -- 10. Partial Summation, the FFT and MMT -- 10.1. Introduction -- 10.2. Partial Summation -- 10.3. 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, and 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 and the Slow Manifold -- 12.1. Introduction -- 12.2. Dispersion and Amplitude Errors -- 12.3. Errors and 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 and Examples -- 12.9. Numerically-Induced Slow Manifolds -- 12.10. Initialization -- 12.11. 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 and 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 and Method of Characteristics -- 14.5. Three-Level, 2D Order Semi-Implicit -- 14.6. Multiply-Upstream SL -- 14.7. Numerical Illustrations and Superconvergence -- 14.8. Two-Level SL/SI Algorithms -- 14.9. Noninterpolating SL and 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 and Formal Integration: Constant Coefficient ODEs -- 15.11. Direct Methods for Separable PDE's -- 15.12. Fast Iterations for Almost Separable PDEs -- 15.13. Positive Definite and Indefinite Matrices -- 15.14. Preconditioned Newton Flow -- 15.15. Summary and 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 and Corner Singularities -- 16.6. Two-Dimensional Maps and Corner Branch Points -- 16.7. Periodic Problems and 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 -- 17.2.2. Domain Truncation for Slowly-Decaying Functions -- 17.2.3. Domain Truncation for Time-Dependent Wave Propagation: Sponge Layers -- 17.3. Whittaker Cardinal or "Sinc" Functions -- 17.4. Hermite functions -- 17.5. Semi-Infinite Interval: Laguerre Functions -- 17.6. New Basis Sets via Change of Coordinate
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 Examples: 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 and 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 and Eigenvalue Problems on a Disk -- 18.5.3. Unbounded Domains Including the Origin in Cylindrical Coordinates -- ^ 18.6. Annular Domains -- 18.7. Spherical Coordinates: An Overview -- 18.8. Parity Factor for Scalars: Sphere versus Torus -- 18.9. Parity II: Horizontal Velocities and Other Vector Components -- 18.10. 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 and 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. Teh 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 and Grid -- 22.7. Patching versus Variational Formalism -- 22.8. Matrix Inversion -- 22.9. Influence Matrix Method -- 22.10. Two-Dimensional Mappings and Sectorial Elements -- 22.11. Prospectus -- 23. Books and Reviews -- 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 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. Blck 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. Eigevalue Problems -- C.4. Summary -- D. 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: Endopint 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.
Includes bibliographical references (pages 595-668) and index.