Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.
Author(s): Jan S. Hesthaven, Sigal Gottlieb, David Gottlieb,
Series: Cambridge Monographs on Applied and Computational Mathematics
Publisher: Cambridge University Press
Year: 2007
Language: English
Pages: 285
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Introduction......Page 13
1 From local to global approximation......Page 17
1.1.1 Phase error analysis......Page 21
1.1.2 Finite-order finite difference schemes......Page 22
1.1.3 Infinite-order finite difference schemes......Page 26
1.2 The Fourier spectral method: first glance......Page 28
1.3 Further reading......Page 30
2.1 Trigonometric polynomial expansions......Page 31
2.1.1 Differentiation of the continuous expansion......Page 35
2.2.1 The even expansion......Page 36
2.2.2 The odd expansion......Page 40
2.2.4 Differentiation of the discrete expansions......Page 42
2.3 Approximation theory for smooth functions......Page 46
2.3.1 Results for the continuous expansion......Page 47
2.3.2 Results for the discrete expansion......Page 50
2.4 Further reading......Page 54
3.1 Fourier–Galerkin methods......Page 55
3.2 Fourier–collocation methods......Page 60
3.3 Stability of the Fourier–Galerkin method......Page 64
3.4 Stability of the Fourier–collocation method for hyperbolic problems I......Page 66
3.5 Stability of the Fourier–collocation method for hyperbolic problems II......Page 70
3.6 Stability for parabolic equations......Page 74
3.7 Stability for nonlinear equations......Page 76
3.8 Further reading......Page 77
4 Orthogonal polynomials......Page 78
4.1 The general Sturm–Liouville problem......Page 79
4.2 Jacobi polynomials......Page 81
4.2.1 Legendre polynomials......Page 84
4.2.2 Chebyshev polynomials......Page 86
4.2.3 Ultraspherical polynomials......Page 88
4.3 Further reading......Page 90
5.1 The continuous expansion......Page 91
5.1.1 The continuous legendre expansion......Page 93
5.1.2 The continuous Chebyshev expansion......Page 94
5.2 Gauss quadrature for ultraspherical polynomials......Page 95
5.2.1 Quadrature for Legendre polynomials......Page 98
5.2.2 Quadrature for Chebyshev polynomials......Page 99
5.3 Discrete inner products and norms......Page 100
5.4 The discrete expansion......Page 101
5.4.1 The discrete Legendre expansion......Page 106
5.4.2 The discrete Chebyshev expansion......Page 108
5.4.3 On Lagrange interpolation, electrostatics, and the Lebesgue constant......Page 111
5.5 Further reading......Page 120
6.1 The continuous expansion......Page 121
6.2 The discrete expansion......Page 126
6.3 Further reading......Page 128
7.1 Galerkin methods......Page 129
7.2 Tau methods......Page 135
7.3 Collocation methods......Page 141
7.4 Penalty method boundary conditions......Page 145
8.1 The Galerkin approach......Page 147
8.2 The collocation approach......Page 154
8.3 Stability of penalty methods......Page 157
8.4 Stability theory for nonlinear equations......Page 162
8.5 Further reading......Page 164
9 Spectral methods for nonsmooth problems......Page 165
9.1 The Gibbs phenomenon......Page 166
9.2 Filters......Page 172
9.2.1 A first look at filters and their use......Page 173
9.2.2 Filtering Fourier spectral methods......Page 176
9.2.3 The use of filters in polynomial methods......Page 179
9.2.4 Approximation theory for filters......Page 181
9.3 The resolution of the Gibbs phenomenon......Page 186
9.4 Linear equations with discontinuous solutions......Page 194
9.5 Further reading......Page 198
10 Discrete stability and time integration......Page 199
10.1.1 Eigenvalue analysis......Page 200
10.1.2 Fully discrete analysis......Page 203
10.2 Standard time integration schemes......Page 204
10.2.2 Runge–Kutta schemes......Page 205
10.3.1 SSP theory......Page 209
10.3.2 SSP methods for linear operators......Page 210
10.3.3 Optimal SSP Runge–Kutta methods for nonlinear problems......Page 212
10.4 Further reading......Page 214
11.1 Fast computation of interpolation and differentiation......Page 216
11.1.1 Fast Fourier transforms......Page 217
11.1.2 The even-odd decomposition......Page 219
11.2 Computation of Gaussian quadrature points and weights......Page 222
11.3.1 Finite precision effects in Fourier methods......Page 226
11.3.2 Finite precision in polynomial methods......Page 229
11.4 On the use of mappings......Page 237
11.4.1 Local refinement using Fourier methods......Page 239
11.4.2 Mapping functions for polynomial methods......Page 241
11.5 Further reading......Page 246
12 Spectral methods on general grids......Page 247
12.1 Representing solutions and operators on general grids......Page 248
12.2 Penalty methods......Page 250
12.2.1 Galerkin methods......Page 251
12.2.2 Collocation methods......Page 253
12.2.3 Generalizations of penalty methods......Page 255
12.3 Discontinuous Galerkin methods......Page 258
12.4 Further reading......Page 260
Appendix A Elements of convergence theory......Page 261
B.1.1 The Legendre expansion......Page 264
B.1.4 Operators......Page 266
B.2 Chebyshev polynomials......Page 267
B.2.1 The Chebyshev expansion......Page 268
B.2.3 Special values......Page 269
B.2.4 Operators......Page 270
Bibliography......Page 272
Index......Page 284
