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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 | |