## A Practical Guide to Pseudospectral MethodsDuring the past two decades, pseudospectral methods have emerged as successful, and often superior, alternatives to better known computational procedures, such as finite difference and finite element methods of numerical solution, in several key application areas. These areas include computational fluid dynamics, wave motion, and weather forecasting. This book explains how, when and why this pseudospectral approach works. In order to make the subject accessible to students as well as researchers and engineers, the author presents the subject using illustrations, examples, heuristic explanations, and algorithms rather than rigorous theoretical arguments. This book will be of interest to graduate students, scientists, and engineers interested in applying pseudospectral methods to real problems. |

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### Contents

Introduction to spectral methods via orthogonal functions | 4 |

Introduction to PS methods via finite differences | 14 |

Key properties of PS approximations | 36 |

PS variations and enhancements | 83 |

PS methods in polar and spherical geometries | 101 |

Comparisons of computational cost for FD and | 118 |

Applications for spectral methods | 127 |

Appendices | 153 |

B Tau Galerkin and collocation PS implementations | 161 |

Codes for algorithm to find FD weights | 167 |

E Potential function estimate for polynomial interpolation | 173 |

G Stability domains for some ODE solvers | 197 |

H Energy estimates | 210 |

217 | |

229 | |

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### Common terms and phrases

analytic applied approach arise basis functions boundary conditions calculation Chebyshev node Chebyshev polynomials Chebyshev-PS closed-form expressions codes complex computational convergence convolutions curves data vectors derivative described DFT matrix differentiation DIMENSION discrete discussed eigenvalues equi-spaced grid errors Example expansion exponentially FD approximations FD methods FD schemes FD stencil finite formulas Fornberg Fourier modes Fourier-PS method Galerkin Gaussian quadrature governing equation grid densities gridpoints heat equation implementation input instabilities integration interface Jacobi polynomials KdV equation Kreiss L2 norm leapfrog Lebesgue constants Legendre polynomials linear matrix x vector node locations nonlinear wave numerical solution obtained ODE solvers order of accuracy Orszag orthogonal parameter PDEs periodic PS method polynomial interpolation pseudospectral requirement RK methods satisfy Section shows smooth solving space spatial spectral methods spurious stability condition stability domains stencils SUBROUTINE Table techniques tion transform trigonometric truncated values variable coefficients wave equation weights zero