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Appendix A: Finite Difference Schemes for the Wave Equation

Appendix A serves a dual purpose. First, it is intended as a review of the basics of the spectral or *Von Neumann* analysis of finite difference schemes; this analysis is quite powerful and revealing if the underlying physical model problem is linear and shift-invariant. We pay close attention to the *numerical stability* conditions that can be arrived at through a straightforward application of these spectral methods. For this review portion of the appendix, we depend primarily on the excellent text by Strikwerda [176]. (For the reader with no prior exposure to the analysis of finite difference methods, §A.1 could well serve as point of departure, before jumping directly into network and scattering theory in Chapter 2.) We then systematically revisit all of the large variety of forms of DWN for the two- and three-dimensional wave equation, applying this spectral analysis to the equivalent difference schemes. The comparison of the Von Neumann numerical stability conditions with the passivity conditions on the associated mesh structures yields somewhat surprising results, for the so-called triangular (**§A.2.3**) and interpolated meshes (**§A.2.2**, **§A.3.3**) structures; indeed, the conditions do not coincide in these cases, leaving us with some fundamental and puzzling questions about the nature of this discrepancy. In addition, we also introduce some techniques for rigorously analyzing certain vector-type schemes (the hexagonal scheme, in **§A.2.4** and the tetrahedral scheme, in **§A.3.4**), and look at a theoretical means of obtaining optimally direction-independent numerical dispersion properties for certain schemes for which we have free parameters at our disposal (the interpolated schemes in **§A.2.2** and **§A.3.3**). Throughout the appendix we pay particular attention to evaluating the relative memory requirements and computational efficiency of the various schemes, and provide numerical dispersion error plots for all the schemes.

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Stefan Bilbao
2002-01-22