Fourier Analysis Methods
Fourier analysis is based on the idea that a function defined on a finite interval can be expanded in a series of cosine or sine functions. In the context of numerical simulation of the shallow water equations, Fourier analysis is used to study the ability of various numerical schemes to accurately simulate waves of different wave lengths. Fourier analysis is typically applied to the linearized form of the shallow water equations (2.37) to (2.40). Linearization uncouples the different wave lengths from one another and allows the study of only one wave length at a time. This method provides stability criteria and an accuracy measure for the amplitude and phase characteristics of each wave length. The technique to obtain the two latter is outlined in Section 3.2. These amplitude and phase portraits have also been used to predict the ability of a scheme to suppress node-to-node oscillations. This is more thoroughly discussed in Chapter 7. The method has mostly been used on uniform one-dimensional meshes, although some applications to uniform, orthogonal two-dimensional meshes exist (Sobey, 1970; Kinnmark and Gray, 1984a). The method is however more general. It can be applied to the case of meshes with variable node-spacing, especially useful for finite element approximations, and for the case of variable parameters.
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