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The Cubic Rectilinear Scheme

This is the simplest scheme for the (3+1)D wave equation. The grid points, indexed by $ i$, $ j$ and $ k$ are located at coordinates $ (x,y,z) = (i\Delta, j\Delta, k\Delta)$. The finite difference scheme is written as

\begin{displaymath}\begin{split}U_{i,j,k}(n+1)+U_{i,j,k}(n-1) &= \lambda^{2}\Big...
...)\\ &\quad +\left(2-6\lambda^{2}\right)U_{i,j,k}(n) \end{split}\end{displaymath} (27)

If the grid points are located at the corners of a cubic lattice, then updating the scheme requires access to the grid function at the six neighboring corners; see Figure 8(a). The stability analysis is very similar to that of the (2+1)D rectilinear scheme, except that we now have a 3-tuple of spatial frequencies, $ \beta$$ = [\beta_{x}, \beta_{y}, \beta_{z}]^{T}$. The amplification polynomial equation is again of the form of (5), with

$\displaystyle B_{\mbox{{\scriptsize\boldmath$\beta$}}} = -2\left(1+\lambda^{2}\...

and thus

$\displaystyle F_{\mbox{{\scriptsize\boldmath$\beta$}}} = \cos(\beta_{x}\Delta)+\cos(\beta_{y}\Delta)+\cos(\beta_{x}\Delta)-3$    

Because $ F_{\mbox{{\scriptsize\boldmath $\beta$}}}$ is multilinear in the cosines, it is simple to show that

$\displaystyle \max_{\mbox{{\scriptsize\boldmath$\beta$}}}F_{\mbox{{\scriptsize\...
...ox{{\scriptsize\boldmath$\beta$}}}F_{\mbox{{\scriptsize\boldmath$\beta$}}} = -6$    

and so, from (9),

$\displaystyle \lambda\leq\frac{1}{\!\!\sqrt{3}}$   (for Von Neumann stability)    

When $ \lambda = 1/\sqrt{3}$, the amplification factors become degenerate and linear growth of the solution may occur for $ \beta_{x} = \beta_{y} = \beta_{z} = 0$, and for $ \vert\beta_{x}\vert=\vert\beta_{y}\vert=\vert\beta_{z}\vert = \pi/\Delta$. The computational and add densities are

$\displaystyle \rho_{cub} = \frac{v_{0}}{\Delta^{4}}\hspace{0.5in}\sigma_{cub} = \frac{7v_{0}}{\Delta^{4}}$    

for $ v_{0}>\sqrt{3}\gamma$, and

$\displaystyle \rho^{s}_{cub} = \frac{\gamma}{2\Delta^{4}}\hspace{0.5in}\sigma^{s}_{cub} = \frac{3\gamma}{\Delta^{4}}$    

at the stability limit $ v_{0} = \sqrt{3}\gamma$. At this limit, the scheme may, like the (2+1)D scheme, be divided into two mutually exclusive subschemes. See Figure 8(b) and (c) for plots of the numerical dispersion properties of the cubic rectilinear scheme. \begin{figure}[h]
...cent from the ideal value of 1 which is obtained at spatial DC.}}

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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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