Along with finite differences and finite elements, spectral methods are one of the three main methodologies for solving partial differential equations on computers. This book provides a detailed presentation of basic spectral algorithms, as well as a systematical presentation of basic convergence theory and error analysis for spectral methods. Readers of this book will be exposed to a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains. The book contains a large number of figures which are designed to illustrate various concepts stressed in the book. A set of basic matlab codes has been made available online to help the readers to develop their own spectral codes for their specific applications.
Jie Shen: Ph.D., Numerical Analysis, Universite de Paris-Sud, Orsay, France, 1987; B.S., Computational Mathematics, Peking University, China, 1982.
Professor of Mathematics at Purdue University; Guest Professorships in Shanghai University and Xiamen University; Member of editorial boards for numerous top research journals.
Tao Tang: Ph.D., Applied Mathematics, University of Leeds, 1989;
Computational Mathematics, Peking University, China, 1984.
Head and Chair Professor of Hong Kong Baptist University; Cheung Kong Chair Professor under Ministry of Education of China; Winner of a Leslie Fox Prize in 1988 and a Feng Kang Prize in Scientific Computing in 2003; Member of editorial boards for numerous top research journals.
Lilian Wang: Ph.D, Computational Mathematics, Shanghai University, China 2000; B.S., Mathematics Education, Hunan University of Science and Technology, China, 1995.
Assistant Professor of Mathematics, Nanyang Technological University, Singapore. A prolific researcher with over twenty research papers in top journals.
Introduction.- Fourier Spectral Methods for Periodic Problems.- Orthogonol Polynomials and Related Approximation Results.- Second-Order Two-Point Boundary Value Problems.- Integral Equations.- High-Order Differential Equations.- Problems in Unbounded Domains.- Multi-Dimensional Domains.- Mathematical Preliminaries.- Basic iterative methods.- Basic time discretization schemes.- Instructions for routines in Matlab.