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