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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 2007 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.

519.63 --- 519.6 --- 517.95 --- Numerical methods for solution of partial differential equations --- Computational mathematics. Numerical analysis. Computer programming --- Partial differential equations --- Differential equations, Hyperbolic. --- Differential equations, Partial. --- Spectral theory (Mathematics) --- Spectral theory (Mathematics). --- 517.95 Partial differential equations --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- 519.63 Numerical methods for solution of partial differential equations --- Differential equations, Hyperbolic --- Differential equations, Partial --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Hyperbolic differential equations

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Differential equations, Partial --- Finite differences. --- Finite element method. --- Numerical analysis. --- Numerical solutions. --- Finite differences --- Finite element method --- Numerical analysis --- 517.95 --- 519.63 --- 681.3 *G18 --- Mathematical analysis --- FEA (Numerical analysis) --- FEM (Numerical analysis) --- Finite element analysis --- Isogeometric analysis --- Differences, Finite --- Finite difference method --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 517.95 Partial differential equations --- Partial differential equations --- Numerical solutions --- Equations aux derivees partielles --- Methodes numeriques