谱方法和高精度算法及应用

Contents
Preface
Chapter 1 Preliminaries 1
1.1 Some basic ideas of spectral methods 2
1.2 Orthogonal polynomials 6
1.3 Chebyshev and Legendre polynomials 15
1.4 Jacobi polynomials and generalized Jacobi polynomials 23
1.5 Fast Fourier transform 27
1.6 Several popular time discretization methods 38
1.7 Iterative methods and preconditioning 48
1.8 Error estimates of polynomial approximations 61
Chapter 2 Spectral-Collocation Methods 68
2.1 Differentiation matrices for polynomial basis functions 69
2.2 Differentiation matrices for Fourier collocation methods 79
2.3 Eigenvalues of Chebyshev collocation operators 84
2.4 Chebyshev collocation method for two-point BVPs 91
2.5 Collocation method in the weak form and preconditioning 99
Chapter 3 Spectral-Galerkin Methods 105
3.1 General setup 105
3.2 Legendre-Galerkin method 109
3.3 Chebyshev-Galerkin method 114
3.4 Chebyshev-Legendre Galerkin method 118
3.5 Preconditioned iterative method 121
3.6 Spectral-Galerkin methods for higher-order equations 126
3.7 Error estimates 131
Chapter 4 Spectral Methods in Unbounded Domains 143
4.1 Hermite spectral methods 144
4.2 Laguerre spectral methods 158
4.3 Spectral methods using rational functions 170
4.4 Error estimates in unbounded domains 177
Chapter 5 Some applications in one space dimension 183
5.1 Pseudospectral methods for boundary layer problems 184
5.2 Pseudospectral methods for Fredholm integral equations 190
5.3 Chebyshev spectral methods for parabolic equations 196
5.4 Fourier spectral methods for the KdV equation 204
5.5 Fourier method and filters 214
5.6 Essentially non-oscillatory spectral schemes 222
Chapter 6 Spectral methods in Multi-dimensional Domains 231
6.1 Spectral-collocation methods in rectangular domains 233
6.2 Spectral-Galerkin methods in rectangular domains 237
6.3 Spectral-Galerkin methods in cylindrical domains 243
6.4 A fast Poisson Solver using finite differences 247
Chapter 7 Some applications in multi-dimensions 256
7.1 Spectral methods for wave equations 257
7.2 Laguerre-Hermite method for Schrodinger equations 264
7.3 Spectral approximation of the Stokes equations 276
7.4Spectral-projection method for Navier-Stokes equations 282
7.5Axisymmetric flows in a cylinder 288
Appendix A Some online software 299
A.l MATLAB Differentiation Matrix Suite 300
A.2 PseudoPack 308
Bibliography 313
Index 323