| 《谱方法和高精度算法及其应用（英文）》对一些新的科学计算法通过深入浅出的方法作了系统的介绍，并着重培养学生对程序设计的兴趣。 This book expands lectur enotes by the authors taught in the past few years in USA， Canada and China．The overall emphasis of these notes is to present basic algorithms together with some applications of spectral methods．The aim is to provide a sufficient background on the implementation and analysis of spectral and high-order methods so that the readers can approach the current research literature with the necessary tools and understanding． It is expected that this book will be a useful supplement for people studying spectral methods on their own．This book iS especially suited to students interested in high-order methods for PDEs．but it will appeal to numerical analysis and mathematically oriented students and researchers in engineering，physics，and related areas． |
| Chapter 1 Preliminaries 1.1 Some basic ideas of spectral methods 1.2 Orthogonal polynomials 1.3 Chebyshev and Legendre polynomials 1.4 Jacobi polynomials and generalized Jacobi polynomials 1.5 Fast Fourier transform 1.6 Several popular time discretization methods 1.7 Iterative methods and preconditioning 1.8 Error estimates of polynomial approximations Chapter 2 Spectral-Collocation Methods 2.1 Differentiation matrices for polynomial basis functions 2.2 Differentiation matrices for Fourier collocation methods 2.3 Eigenvalues of Chebyshev collocation operators 2.4 Chebyshev collocation method for two-point BVPs 2.5 Collocation method in the weak form and preconditioning Chapter 3 Spectral-Galerkin Methods 3.1 General setup 3.2 Legendre-Galerkin method 3.3 Chebyshev-Galerkin method 3.4 Chebyshev-Legendre Galerkin method 3.5 Preconditioned iterative method 3.6 Spectral-Galerkin methods for higher-order equations 3.7 Error estimates Chapter 4 Spectral Methods in Unbounded Domains 4.1 Hermite spectral methods 4.2 Laguerre spectral methods 4.3 Spectral methods using rational functions 4.4 Error estimates in unbounded domains Chapter 5 Some applications in one space dimension 5.1 Pseudospectral methods for boundary layer problems 5.2 Pseudospectral methods for Fredholm integral equations 5.3 Chebyshev spectral methods for parabolic equations 5.4 Fourier spectral methods for the KdV equation 5.5 Fourier method and filters 5.6 Essentially non-oscillatory spectral schemes Chapter 6 Spectral methods in Multi-dimensional Domains 6.1 Spectral-collocation methods in rectangular domains 6.2 Spectral-Galerkin methods in rectangular domains 6.3 Spectral-Galerkin methods in cylindricaldomains 6.4 A fast Poisson Solver using finite differences Chapter 7 Some applications in multi-dimensions 7.1 Spectral methods for wave equations 7.2 Laguerre-Hermite method for Schr?dinger equations 7.3 Spectral approximation of the Stokes equations 7.4 Spectral-projection method for Navier-Stokes equations 7.5 Axisymmetric flows in a cylinder Appendix A Some online software A.1 MATLAB Differentiation Matrix Suite A.2 PseudoPack Bibliography Index |
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