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Vector Schemes

For two of the schemes that we will examine (hexagonal and tetrahedral), it will be necessary to analyze a vectorized system of difference equations. In general, the analysis of vector forms is considerably more difficult; the typical approach will invoke the Kreiss Matrix Theorem [8], which is a set of equivalent conditions which can be used to check the boundedness of a particular amplification matrix. In the general vector case we will be analyzing the evolution of a $ q$-element vector $ \hat{{\bf U}}_{\mbox{{\scriptsize\boldmath $\beta$}}}(n) = [\hat{U}_{1,\mbox{{...
...\beta$}}}(n), \hdots, \hat{U}_{q,\mbox{{\scriptsize\boldmath $\beta$}}}(n)]^{T}$ of spatially Fourier-transformed functions of $ \beta$. The $ L_{2}$ norm is defined by

$\displaystyle \Vert\hat{{\bf U}}(n)\Vert _{2} = \left(\int_{[\pi/\Delta, \pi/\D...
...tsize\boldmath$\beta$}}}(n) d\beta_{1} d\beta_{2}\hdots d\beta_{N}\right)^{1/2}$    

where $ ^{*}$ denotes transpose conjugation.

The schemes for the wave equation that we will examine, however, have a relatively simple form. The column vector of grid spatial frequency spectra $ \hat{{\bf U}}_{\mbox{{\scriptsize\boldmath $\beta$}}}(n)$ satisfies an equation of the form

$\displaystyle \hat{{\bf U}}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n+1)+{\bf B}...
...h$\beta$}}}(n)+\hat{{\bf U}}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n-1)={\bf0}$ (11)

for some Hermitian matrix function of $ \beta$, $ {\bf B}_{\mbox{{\scriptsize\boldmath $\beta$}}}$. Because $ {\bf B}_{\mbox{{\scriptsize\boldmath $\beta$}}}$ is Hermitian, we may write $ {\bf B}_{\mbox{{\scriptsize\boldmath $\beta$}}} = {\bf J}_{\mbox{{\scriptsize\...
...{\scriptsize\boldmath $\beta$}}}{\bf J}_{\mbox{{\scriptsize\boldmath $\beta$}}}$, for some unitary matrix $ {\bf J}_{\mbox{{\scriptsize\boldmath $\beta$}}}$, and a real diagonal matrix $ \Lambda$$ _{\mbox{{\scriptsize\boldmath $\beta$}}}$ containing the eigenvalues of $ {\bf B}_{\mbox{{\scriptsize\boldmath $\beta$}}}$. As such, we may change variables via $ \hat{{\bf V}}_{\mbox{{\scriptsize\boldmath $\beta$}}}(n) = {\bf J}_{\mbox{{\sc...
...ize\boldmath $\beta$}}}\hat{{\bf U}}_{\mbox{{\scriptsize\boldmath $\beta$}}}(n)$, to get

$\displaystyle \hat{{\bf V}}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n+1)+\mbox{\...
...h$\beta$}}}(n)+\hat{{\bf V}}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n-1)={\bf0}$ (12)

The system thus decouples into a system of scalar two-step spectral update equations; because $ \hat{{\bf U}}_{\mbox{{\scriptsize\boldmath $\beta$}}}(n)$ and $ \hat{{\bf V}}_{\mbox{{\scriptsize\boldmath $\beta$}}}(n)$ are related by a unitary transformation, we have $ \Vert\hat{{\bf U}}(n)\Vert _{2} = \Vert\hat{{\bf V}}(n)\Vert _{2}$, and we may apply stability tests to the uncoupled system (11). We thus require that the eigenvalues of $ {\bf B}_{\mbox{{\scriptsize\boldmath $\beta$}}}$, namely $ \Lambda_{\mbox{{\scriptsize\boldmath $\beta$}},j}$ for $ j=1,\hdots,q$, which are the elements on the diagonal of $ \Lambda$$ _{\mbox{{\scriptsize\boldmath $\beta$}}}$, all satisfy

$\displaystyle \max_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert\Lambda_{\mbox{{\scriptsize\boldmath$\beta$}},j}\vert\leq 2$ (13)

At frequencies $ \beta$$ _{0}$ for which any of the eigenvalues satisfies (12) with equality, then we may again have the same problem with mild linear growth in the solution.

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``Spectral Analysis of Finite Difference Meshes'', by .
Copyright © 2005-12-28 by Julius O. Smith III<jos_email.html>
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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