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Table of Contents

1. Introduction.- 1.1. Historical Background.- 1.2. Some Examples of Spectral Methods.- 1.3. The Equations of Fluid Dynamics.- 1.4. Spectral Accuracy for a Two-Dimensional Fluid Calculation.- 1.5. Three-Dimensional Applications in Fluids.- 2. Spectral Approximation.- 2.1. The Fourier System.- 2.2. Orthogonal Polynomials in ( — 1, 1).- 2.3. Legendre Polynomials.- 2.4. Chebyshev Polynomials.- 2.5. Generalizations.- 3. Fundamentals of Spectral Methods for PDEs.- 3.1. Spectral Projection of the Burgers Equation.- 3.2. Convolution Sums.- 3.3. Boundary Conditions.- 3.4. Coordinate Singularities.- 3.5. Two-Dimensional Mapping.- 4. Temporal Discretization.- 4.1. Introduction.- 4.2. The Eigenvalues of Basic Spectral Operators.- 4.3. Some Standard Schemes.- 4.4. Special Purpose Schemes.- 4.5. Conservation Forms.- 4.6. Aliasing.- 5. Solution Techniques for Implicit Spectral Equations.- 5.1. Direct Methods.- 5.2. Fundamentals of Iterative Methods.- 5.3. Conventional Iterative Methods.- 5.4. Multidimensional Preconditioning.- 5.5. Spectral Multigrid Methods.- 5.6. A Semi-Implicit Method for the Navier—Stokes Equations.- 6. Simple Incompressible Flows.- 6.1. Burgers Equation.- 6.2. Shear Flow Past a Circle.- 6.3. Boundary-Layer Flows.- 6.4. Linear Stability.- 7. Some Algorithms for Unsteady Navier—Stokes Equations.- 7.1. Introduction.- 7.2. Homogeneous Flows.- 7.3. Inhomogeneous Flows.- 7.4. Flows with Multiple Inhomogeneous Directions.- 7.5. Mixed Spectral/Finite-Difference Methods.- 8. Compressible Flow.- 8.1. Introduction.- 8.2. Boundary Conditions for Hyperbolic Problems.- 8.3. Basic Results for Scalar Nonsmooth Problems.- 8.4. Homogeneous Turbulence.- 8.5. Shock-Capturing.- 8.6. Shock-Fitting.- 8.7. Reacting Flows.- 9. Global Approximation Results.- 9.1. FourierApproximation.- 9.2. Sturm—Liouville Expansions.- 9.3. Discrete Norms.- 9.4. Legendre Approximations.- 9.5. Chebyshev Approximations.- 9.6. Other Polynomial Approximations.- 9.7. Approximation Results in Several Dimensions.- 10. Theory of Stability and Convergence for Spectral Methods.- 10.1. The Three Examples Revisited.- 10.2. Towards a General Theory.- 10.3. General Formulation of Spectral Approximations to Linear Steady Problems.- 10.4. Galerkin, Collocation and Tau Methods.- 10.5. General Formulation of Spectral Approximations to Linear Evolution Equations.- 10.6. The Error Equation.- 11. Steady, Smooth Problems.- 11.1. The Poisson Equation.- 11.2. Advection-Diffusion Equation.- 11.3. Navier—Stokes Equations.- 11.4. The Eigenvalues of Some Spectral Operators.- 12. Transient, Smooth Problems.- 12.1. Linear Hyperbolic Equations.- 12.2. Heat Equation.- 12.3. Advection-Diffusion Equation.- 13. Domain Decomposition Methods.- 13.1. Introduction.- 13.2. Patching Methods.- 13.3. Variational Methods.- 13.4. The Alternating Schwarz Method.- 13.5. Mathematical Aspects of Domain Decomposition Methods.- 13.6. Some Stability and Convergence Results.- Appendices.- A. Basic Mathematical Concepts.- B. Fast Fourier Transforms.- C. Jacobi—Gauss—Lobatto Roots.- References.