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The Tetrahedral Scheme

The tetrahedral scheme in (3+1)D [11] is somewhat similar to the hexagonal scheme in (2+1)D, in that the grid is divided evenly into two sets of points, at which updating is performed using ``mirror-image'' stencils. It is different, however, because grid points can easily be indexed with reference to a regular cubic lattice; the hexagonal scheme operates on a rectangular grid in stretched or transformed coordinates. In fact, a tetrahedral scheme can be obtained directly from an octahedral scheme simply by removing half of the grid points it employs; as such, any given grid point in the tetrahedral scheme has four nearest neighbors. As usual, we assume the nearest-neighbor grid spacing to be $ \Delta$. See Figure 12(a) for a representation of the numerical grid.

As per the hexagonal scheme, we will view this as a vectorized scheme operating on two distinct sub grids, labelled 1 and 2 in Figure 12(a). The two grid functions $ U_{1,i,j,k}(n)$ and $ U_{2,i+1,j+1,k+1}(n)$ are defined for integers $ i$, $ j$ and $ k$ all even such that $ (i+j+k)/2$ is also even. $ U_{1,i,j,k}$ will be used to approximate a continuous function $ u_{1}$ at the point with coordinates $ x=i\Delta/\sqrt{3}$, $ y=j\Delta/\sqrt{3}$ and $ z=k\Delta/\sqrt{3}$, and $ U_{2,i+1,j+1,k+1}$ approximates $ u_{2}$ at coordinates $ x=(i+1)\Delta/\sqrt{3}$, $ y=(j+1)\Delta/\sqrt{3}$ and $ z=(k+1)\Delta/\sqrt{3}$. The numerical scheme can then be written as \begin{subequations}
\begin{eqnarray}
U_{1,i,j,k}(n+1)+U_{1,i,j,k}(n-1) &=& \fra...
...\left(1-3\lambda^{2}\right)U_{2,i+1,j+1,k+1}(n)
\end{eqnarray}\end{subequations}
As for the hexagonal scheme, we may check consistency of this system with the wave equation by treating the grid functions as samples of continuous functions $ u_{1}$ and $ u_{2}$ and expanding (32) in terms of partial derivatives; both grid functions updated according to this scheme will approximate the solution to the wave equation on their respective grids.

Determining the stability condition proceeds as in the hexagonal scheme; taking spatial Fourier transforms of (32) gives a vector spectral update equation of the form (10), with $ {\bf B}_{\mbox{{\scriptsize\boldmath $\beta$}}}$ given by

$\displaystyle {\bf B}_{\mbox{{\scriptsize\boldmath$\beta$}}} = \begin{bmatrix}-...
...i_{\mbox{{\scriptsize\boldmath$\beta$}}}^{*}&-2(1-3\lambda^{2})\\ \end{bmatrix}$    

with

$\displaystyle \psi_{\mbox{{\scriptsize\boldmath$\beta$}}} = 2\Big(e^{j\Delta\be...
...)+e^{-j\Delta\beta_{x}/\sqrt{3}}\cos(\Delta(\beta_{y}-\beta_{z})/\sqrt{3})\Big)$    

$ {\bf B}_{\mbox{{\scriptsize\boldmath $\beta$}}}$ is again Hermitian, and has eigenvalues
$\displaystyle \Lambda_{\mbox{{\scriptsize\boldmath$\beta$}},1}$ $\displaystyle =$ $\displaystyle -2\left(1-3\lambda^{2}\right)+\frac{3}{2}\lambda^{2}\vert\psi_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert$  
$\displaystyle \Lambda_{\mbox{{\scriptsize\boldmath$\beta$}},2}$ $\displaystyle =$ $\displaystyle -2\left(1-3\lambda^{2}\right)-\frac{3}{2}\lambda^{2}\vert\psi_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert$  

The stability condition can thus be written as

$\displaystyle \left\vert-2\left(1-3\lambda^{2}\right)\pm\frac{3}{2}\lambda^{2}\vert\psi_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert\right\vert\leq 2$ (31)

$ \psi_{\mbox{{\scriptsize\boldmath $\beta$}}}$ can be shown to take on a maximum of 4, and a minimum of 0, and it then follows that (33) will be satisfied if and only if $ \lambda\leq
1/\sqrt{3}$, the same bound as obtained for the cubic rectilinear and octahedral schemes. The bound is the same as the bound for passivity of a tetrahedral mesh. We note that as for these other schemes, the grid permits a subdivision into mutually exclusive subschemes at this stability limit--see Figure 12(a). By a simple comparison with the hexagonal scheme, we can obtain the four spectral amplification factors by

$\displaystyle G_{\mbox{{\scriptsize\boldmath$\beta$}},1,\pm} = \frac{1}{2}\left...
...ta$}},2}\pm\sqrt{\Lambda_{\mbox{{\scriptsize\boldmath$\beta$}},2}^{2}-4}\right)$    

it is easy to see that parasitic modes (characterized by the amplification factors $ G_{\mbox{{\scriptsize\boldmath $\beta$}},1,\pm}$) will be present in the tetrahedral scheme, due to the nonuniformity of updating on the numerical grid. The numerical dispersion characteristics of the dominant modes with amplification factors $ G_{\mbox{{\scriptsize\boldmath $\beta$}},2,\pm}$ are shown in planar and spherical cross-sections in Figure 12(b) and (c).

The computational and add densities of this scheme, in general, are

$\displaystyle \rho_{tetr} = \frac{3\sqrt{3}v_{0}}{8\Delta^{4}}\hspace{0.5in}\sigma_{tetr} = \frac{15\sqrt{3}v_{0}}{8\Delta^{4}}$    

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

$\displaystyle \rho^{s}_{tetr} = \frac{9\gamma}{16\Delta^{4}}\hspace{0.5in}\sigma^{s}_{tetr} = \frac{9\gamma}{4\Delta^{4}}$    

at the stability limit $ v_{0} = \sqrt{3}\gamma$. \begin{figure}[h]
\begin{center}
\begin{picture}(550,450)
\par
\put(-5,0){\eps...
...ert$, and $\vert\beta_{z}\vert$\ all less than $\pi/(2\Delta)$.}}
\end{figure}


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