Explores spectral methods built around MATLAB programmes. Along with finite differences and finite elements, spectral methods are one of the three main technologies for solving partial differential equations on computers. Since spectral methods involve significant linear algebra and graphics they are very suitable for the high level programming of MATLAB. This hands-on introduction is built around 40 short and powerful MATLAB programmes, which the reader can download from the World Wide Web. The book presents the key ideas along with many figures, examples and short, elegant MATLAB programmes for readers to adapt to their own needs. It covers ODE and PDE boundary value problems, eigenvalues and pseudospectra, linear and nonlinear waves, and numerical quadrature.
Lloyd N. Trefethen is Professor of Numerical Analysis at Oxford University, UK. His previous book, Numerical Linear Algebra (with David Bau, III) is one of SIAM's bestsellers.
Preface Acknowledgments A Note on the MATLAB Programs Chapter 1: Differentiation Matrices Chapter 2: Unbounded Grids: The Semidiscrete Fourier Transform Chapter 3: Periodic Grids: The DFT and FFT Chapter 4: Smoothness and Spectral Accuracy Chapter 5: Polynomial Interpolation and Clustered Grids Chapter 6: Chebyshev Differentiation Matrices Chapter 7: Boundary Value Problems Chapter 8: Chebyshev Series and the FFT Chapter 9: Eigenvalues and Pseudospectra Chapter 10: Time-Stepping and Stability Regions Chapter 11: Polar Coordinates Chapter 12: Integrals and Quadrature Formulas Chapter 13: More About Boundary Conditions Chapter 14: Fourth-Order Problems Afterword Bibliography Index
'A brief, intuitive, and excellent introductory textbook of spectral methods. Readers with basic experience of finite difference methods can easily get to the core knowledge of spectral methods through this book. Those MATLAB scripts for examples in the book are very useful for both teaching and research.' Professor Tzyy-Leng Allen Horng, Department of Applied Mathematics, Feng Chia University, Taichung, Taiwan 'This book is a very nice addition to the collection of books on spectral methods, from a totally different angle. It should attract more students and researchers to the powerful spectral methods.' Chi-Wang Shu, Mathematics of Computation 'What a great book! I sat down to read through it last month, and it's so inspiringly lean with such great examples that I can't resist using it this term for 18.336 [Numerical Methods for PDEs].' Peter Mucha, Massachusetts Institute of Technology 'This is a charming book, beautifully written, easy to understand without sacrificing accuracy. The idea of using MATLAB is brilliant and will appeal to the students and the other readers.' David Gottlieb, Ford Foundation Professor of Applied Mathematics, Brown University 'Fascinating mathematics, intriguing graphics, and beautiful MATLAB codes.' Cleve Moler, Chairman and Chief Scientist, The MathWorks, Inc. "The book's succinct style and the understanding that is gained by parsing and experimenting with MATLAB routines allow for a large number of fundamental concepts to be conveyed in a few pages... I recommend this book highly; it is clear, concise, and dense with interesting and exciting examples. It is an appropriate text for an introductory class on the subject and also includes many ideas and explanations that are appealing to a broader research audience." -Paul Milewski, University of Wisconsin, SIAM Review, Vol. 43, No. 3, pp.541-581.
