Spectral Methods for Time-Dependent Problems

Graduate text on spectral methods with applications in fluid dynamics and engineering.

Autorentext
Jan Hesthaven is a Professor of Applied Mathematics at Brown University.

Klappentext
Spectral methods are well-suited to solve problems modeled by time-dependent partial differential equations: they are fast, efficient and accurate and widely used by mathematicians and practitioners. This class-tested introduction, the first on the subject, is ideal for graduate courses, or self-study. The authors describe the basic theory of spectral methods, allowing the reader to understand the techniques through numerous examples as well as more rigorous developments. They provide a detailed treatment of methods based on Fourier expansions and orthogonal polynomials (including discussions of stability, boundary conditions, filtering, and the extension from the linear to the nonlinear situation). Computational solution techniques for integration in time are dealt with by Runge-Kutta type methods. Several chapters are devoted to material not previously covered in book form, including stability theory for polynomial methods, techniques for problems with discontinuous solutions, round-off errors and the formulation of spectral methods on general grids. These will be especially helpful for practitioners.

Zusammenfassung
Spectral methods are useful techniques for solving integral and partial differential equations, many of which appear in fluid mechanics and engineering problems. Based on a graduate course, this 2007 book presents these popular and efficient techniques with both rigorous analysis and extensive coverage of their wide range of applications.

Inhalt
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.

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