Jan Hesthaven is a Professor of Applied Mathematics at Brown University. Sigal Gottlieb is an Associate Professor at the Department of Mathematics, University of Massachusetts, Dartmouth. David Gottlieb is a Professor in the Division of Applied Mathematics, Brown University.

'The book is excellent and will be valuable for post-graduate students, researchers and scientists working in applied sciences and mainly in the numerical analysis of time-dependent problems. The thoroughness of the exposition, the clarity of the mathematical techniques and the variety of the problems and theoretical results that are presented and rigorously analyzed make this book a primary reference in the advanced numerical analysis of partial differential equations.' Mathematical Reviews

• Introduction; 1. From local to global approximation; 2. Trigonometric polynomial approximation; 3. Fourier spectral methods; 4. Orthogonal polynomials; 5. Polynomial expansions; 6. Polynomial approximations theory for smooth functions; 7. Polynomial spectral methods; 8. Stability of polynomial spectral methods; 9. Spectral methods for non-smooth problems; 10. Discrete stability and time integration; 11. Computational aspects; 12. Spectral methods on general grids; Bibliography.