Chebyshev and Fourier Spectral Methods

1 Chebyshev and Fourier Spectral MethodsSecond EditionJohn P. BoydUniversity of MichiganAnn Arbor, Michigan 48109-2143email: jpboyd/2000 DOVER Publications, East 2nd StreetMineola, New York 115011 DedicationTo Marilyn, Ian, and Emma A computation is a temptation that should be resisted aslong as possible. J. P. Boyd, paraphrasing T. S. EliotiContentsPREFACExAcknowledgmentsxiv Errata and Extended-Bibliographyxvi1 with finite element with Finite of basis and Time-dependent FAQ: Frequently Asked The 172 Chebyshev & Fourier of of Equal s Taylor Series of Singularities & Compatibility : Integration-by-Parts Asymptotic Calculation of Fourier Convergence Theory.

2 Chebyshev Last Coefficient Convergence Theory for Legendre Quasi-Sinusoidal Rule of Witch of Agnesi Rule of Boundary Layer 57iiCONTENTSiii3 Galerkin & Weighted Residual Weighted Residual and Boundary Product & Method: Case & the Galerkin Matrix Galerkin Method 804 Interpolation, Collocation & All Integration & Pseudospectral Is Galerkin Method via 935 Cardinal Cardinal or Sinc Functions for Orthogonal and 1076 Pseudospectral Methods for of Basis Conditions: Behavioral & Boundary-Bordering.

3 Basis Recombination .. Cardinal Function Interpolation Basis Functions & Higher Dimensions: Higher Corner Matrix 1237 Linear Eigenvalue No-Brain Kinds of Sturm-Liouville for Rejecting Spurious the Condition Power Power Combining Global & Local Detouring into the Complex Common 1558 Symmetry & the Grid to Exploit Discrete & Apple-Slicing 1709 Explicit Time-Integration Shamrock and : KdV : RLW & 18110 Partial Summation, the FFT and Partial The Fast Fourier Transform: Matrix Multiplication Costs of the Fast Fourier Generalized FFTs and Multipole Off-Grid Fast Fourier Transform: Practical 20011 Aliasing, Spectral Blocking, & Aliasing and 2 h-Waves and Spectral Aliasing Instability: History and Dealiasing and the Orszag Two-Thirds Energy-Conserving: Constrained Energy-Conserving Schemes: Aliasing Instability.

4 21812 Implicit Schemes & the Slow Dispersion and Amplitude Errors & CFL Limit for Explicit Implicit Time-Marching Semi-Implicit Speed-Reduction Slow Manifold: Slow Manifold: Definition & Numerically-Induced Slow The Method of Multiple Scales(Baer-Tribbia).. Galerkin of the Nonlinear Galerkin the Slow Parts to Multiple Scale 24913 Splitting & Its Fractional Steps for Pitfalls in Splitting, I: Boundary Pitfalls in Splitting, II: Operator Theory of High Order Splitting and Fluid 26214 Semi-Lagrangian Concept of an Integrating Misuse of Integrating Factor Semi-Lagrangian Advection: Advection & Method of Three-Level, 2D Order Multiply-Upstream Numerical Illustrations & Two-Level SL/SI Noninterpolating SL & Numerical Off-Grid Interpolation.

5 Spectral Low-order Polynomial McGregor s Taylor Series Higher Order SL and Relationships to Other 28915 Matrix-Solving Stationary One-Step Preconditioning: Finite Computing Iterates: FFT/Matrix Alternative Raising the Order Through Multigrid: An MRR Delves-Freeman Block-and-Diagonal & Formal Integration: Constant Coefficient Direct Methods for Separable PDE Iterations for Almost Separable Definite and Indefinite Newton & 322viCONTENTS16 Coordinate Programming Chebyshev Theory of 1-D Infinite and Semi-Infinite Maps for Endpoint & Corner Two-Dimensional Maps & Corner Branch Periodic Problems & the Arctan/Tan Adaptive Almost-Equispaced Kosloff/Tal-Ezer 33417 Methods for Unbounded Domain Domain Truncation for Rapidly-decaying Domain Truncation for Slowly-Decaying Domain Truncation for

6 Time-Dependent Wave Propagation:Sponge Whittaker Cardinal or Sinc Hermite Semi-Infinite Interval: Laguerre New Basis Sets via Change of Rational Chebyshev Behavioral versus Numerical Boundary Strategy for Slowly Decaying Examples: Rational Chebyshev Semi-Infinite Interval: Rational Examples: Chebyshev for Semi-Infinite : Oscillatory, Non-Decaying Sinh 37718 Spherical & Cylindrical Polar, Cylindrical, Toroidal, Apparent Singularity at the Polar Coordinates: Parity Radial Basis Sets and Radial One-Sided Jacobi Basis for the Radial Boundary Value & Eigenvalue Problems on a Unbounded Domains Including the Origin in Cylindrical Coordinates Annular Spherical Coordinates: An The Parity Factor for Scalars: Sphere versus Parity II: Horizontal Velocities & Other Vector Pole Problem: Spherical Spherical Harmonics.

7 Transforms and Other FFT in Longitude/MMT in Substitutes and Accelerators for the Parity and Legendre Hurrah for Matrix/Vector Reduced Grid and Other Schuster-Dilts Triangular Matrix Generalized FFT: Multipoles and All Harmonics: Limited-Area Harmonics and Approximations, Approximations, : Spherical : Shallow and Topography: Fronts and Mechanics of Spherical Filter Filtering with Spatially-Variable Topographic Filtering in of Spectral Harmonics & Hough Coordinate: Spectral or Non- Spectral ?

8 Basis for Axial Coordinate in Cylindrical Axial Basis in Toroidal Vertical/Radial Basis in Spherical Convection in a Spherical Annulus: Glatzmaier (1984).. Grids: Icosahedral, Basis for the Latitudinal Fourier Filtering for Latitudinal Fourier Elements on the Harmonics and Elliptic Cylinder 44019 Special Sideband Special Basis Functions, I: Corner Special Basis Functions, II: Wave Weakly Nonlocal Solitary Root-Finding by Chebyshev Hilbert Spectrally-Accurate Quadrature Introduction: Gaussian and Clenshaw-Curtis Clenshaw-Curtis Integration of Periodic Functions and the Trapezoidal Infinite Intervals and the Trapezoidal Singular Sets and 460viiiCONTENTS20 Symbolic Summary and Open 47221 The -Approximation for a Rational Differential Canonical 47822 Domain Decomposition Connecting the Subdomains.

9 Weak Coupling of Elemental Variational Choice of Basis & Patching versus Variational Matrix The Influence Matrix Mappings & Sectorial 49223 Books and Reviews494A A Bestiary of Basis Trigonometric Basis Functions: Fourier Chebyshev Polynomials:Tn(x).. Chebyshev Polynomials of the Second Kind:Un(x).. Legendre Polynomials:Pn(x).. Gegenbauer Hermite Polynomials:Hn(x).. Rational Chebyshev Functions:TBn(y).. Laguerre Polynomials:Ln(x).. Rational Chebyshev Functions:TLn(y).. Graphs of Convergence Domains in the Complex 511B Direct Matrix Banded Matrix-of-Matrices Block-Banded Elimination: the Lindzen-Kuo Block and Bordered Cyclic Banded Matrices (Periodic Boundary Conditions).

10 Parting 524 CONTENTSixC Newton Eigenvalue 534D The Continuation Initialization Limit Bifurcation Pseudoarclength 546E Change-of-Coordinate Derivative Transformations550F Cardinal Fourier Series: Endpoint Cosine Series: Endpoint Sine Series: Endpoint Cardinal Functions: Interior Cardinal Functions: Interior (x): Whittaker cardinal Gauss-Lobatto ( Endpoints ).. Polynomials: Interior or Roots Legendre Polynomials: Gauss-Lobatto 572G Transformation of Derivative Boundary Conditions575 Glossary577 Index586 References595 Preface[Preface to the First Edition (1988)]The goal of this book is to teach Spectral Methods for solving boundary value, eigen-value and time-dependent problems.