This book presents a systematic development of the fundamental algorithms needed to write spectral methods codes to solve basic problems of mathematical physics: steady potentials, transport, and wave propagation. It shows that only a few fundamental algorithms for interpolation, differentiation and the FFT form the building blocks of any spectral code, even for problems in complex geometries. The algorithms approximate problems in 1D and 2D to show the flexibility of spectral methods, and to make the transition from exploratory to application codes as straightforward as possible. The book serves as a textbook for graduate students and as a starting point for applications scientists.
David Kopriva is Professor of Mathematics at the Florida State University, where he has taught since 1985. He is an expert in the development, implementation and application of high order spectral multi-domain methods for time dependent problems. In 1986 he developed the first multi-domain spectral method for hyperbolic systems, which was applied to the Euler equations of gas dynamics.
Part I Spectral Approximation.- 1 Mathematical Preliminaries.- Part II Basic Algorithms.- 2 Algorithms For Periodic Functions.- 3 Algorithms For Non-Periodic Functions.- Part III Spectral Approximation of PDEs.- 4 Survey of Spectral Approximations.- 5 Spectral Approximation on the Square.- 6 Spectral Methods in Non-Square Geometries.- 7 Spectral Element Methods.- A Miscellaneous Algorithms.