Contents PREFACE Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 l.1 Series expansions.·.······· 1 1.2 First Example 2 1.3 Comparison with finite element methods 4 l.4 Comparisons with Finite Differences.·..·.····.······ 6 1.5 Parallel Computers.......·················· 1.6 Choice of basis functions.····················· 9 1.7 Boundary conditions........... 10 1.8 Non-Interpolating and Pseudospectral..·.·.·.····. 12 1.9 Nonlinearity........。。·...·。············· 13 l.l0Time-dependent problems.·.·.·.················ 15 1.11 FAQ:Frequently Asked Questions.....·.·····.···.· 16 1.12 The Chrysalis 1 2 Chebyshev Fourier Series 19 19 2.2 Fourier series 20 2.3 Orders of Convergence..·.···.·.·············· 25 2.4 Convergence Order.......················ 27 2.5 Assumption of Equal Errors 31 2.6 Darboux's Principle.················ 32 2.7 Why Taylor Series Fail................... 35 2.8 Location of Singularities..·.·.·.·... 36 2.8.1 Corner Singularities Compatibility Conditions 37 2.9 FACE:Integration-by-Parts Bound·.·.·...·.·.········ 41 2.l0 Asymptotic Calculation of Fourier Coefficients.·..·.·.·.. 5 2.11 Convergence Theory:Chebyshev Polynomials 6 2.12 Last Coefficient Rule-of-Thumb.................... 50 2.l3 Convergence Theory for Legendre Polynomials.·.·.····.. 2.l4 Quasi-Sinusoidal Rule of Thumb..,.·.,.················· 54 2.l5 Witch of Agnesi Rule--of-Thumb.··...···········:.·····. 2.16 Boundary Layer Rule-of-Thumb 57 iContents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.8 Non-Interpolating and Pseudospectral . . . . . . . . . . . . . . . . . . . . . . 12 1.9 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.10 Time-dependent problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.11 FAQ: Frequently Asked Questions . . . . . . . . . . . . . . . . . . . . . . . . 16 1.12 The Chrysalis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Chebyshev & Fourier Series 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Orders of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 Convergence Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 Assumption of Equal Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 Darboux’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7 Why Taylor Series Fail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.8 Location of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.8.1 Corner Singularities & Compatibility Conditions . . . . . . . . . . . 37 2.9 FACE: Integration-by-Parts Bound . . . . . . . . . . . . . . . . . . . . . . . . 41 2.10 Asymptotic Calculation of Fourier Coefficients . . . . . . . . . . . . . . . . . 45 2.11 Convergence Theory: Chebyshev Polynomials . . . . . . . . . . . . . . . . . 46 2.12 Last Coefficient Rule-of-Thumb . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.13 Convergence Theory for Legendre Polynomials . . . . . . . . . . . . . . . . . 51 2.14 Quasi-Sinusoidal Rule of Thumb . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.15 Witch of Agnesi Rule–of–Thumb . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.16 Boundary Layer Rule-of-Thumb . . . . . . . . . . . . . . . . . . . . . . . . . 57 ii