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This chapter discusses the fundamental concepts that are further developed in the remainder of the book, in the context of a one-dimensional formulation. In doing so, the principles and underlying theory behind the construction of the spectral/hp element method are illustrated. Topics covered include method of weighted residuals, Galerkin formulation, one-dimensional expansion bases, elemental operations, error estimates, and implementation of a one-dimensional spectral/hp element. The chapter ends with exercises aimed at developing a one-dimensional spectral element solver.

This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.

This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.

Chapter 9 studies S. Dahlke studies wavelet methods. This is the first presentation in this generality and is appropriate for nonlinear problems and bifurcation for elliptic PDEs. As a consequence of the difficulties with evaluating nonlinear functionals and operators with wavelet arguments, general quasilinear, and fully nonlinear problems are limited, and excluded, respectively. With this exception, the whole spectrum of corresponding wavelet methods is shown to be stable and convergent. Again the corresponding linearized operator has to be boundedly invertible. This chapter finishes with adaptive wavelet methods. In contrast to Chapter 6, nonlinear approximation, and wavelet matrix compression are employed for adaptive descent iterations.

This chapter provides an overview of unsupervised learning algorithms that can be viewed as spectral methods for linear and nonlinear dimensionality reduction. Spectral methods have recently emerged as a powerful tool for nonlinear dimensionality reduction and manifold learning. These methods are able to reveal low-dimensional structure in high-dimensional data from the top or bottom eigenvectors of specially constructed matrices. To analyze data that lie on a low-dimensional submanifold, the matrices are constructed from sparse weighted graphs whose vertices represent input patterns and whose edges indicate neighborhood relations. The main computations for manifold learning are based on tractable, polynomial-time optimizations, such as shortest-path problems, least-squares fits, semi-definite programming, and matrix diagonalization.

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is not invariant, on it. The concepts of a vector space, Lie algebra, and inner product are reviewed. A hamiltonian dynamical system is derived from each metric Lie algebra. The Virasoro algebra (famous in string theory) is shown to lead to the KdV equation; and in a limiting case, to the Burgers equation for shocks. A hamiltonian formalism for two-dimensional Euler equations is then developed in detail. A discretization of these equations (using a spectral method) is then developed using mathematical ideas from quantum mechanics. Then a hamiltonian formalism for the full three-dimensional Euler equations is developed. The Clebsch variables which provide canonical pairs for fluid dynamics are then explained, in analogy to angular momentum.