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
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

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

with

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

${\bf B}_{\mbox{{\scriptsize\boldmath $\beta$}}}$ is again Hermitian, and has eigenvalues

$$\Lambda_{\mbox{{\scriptsize\boldmath$\beta$}},1} = -2\left(1-3\lambda^{2}\right)+\frac{3}{2}\lambda^{2}\vert\psi_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert$$

$$\Lambda_{\mbox{{\scriptsize\boldmath$\beta$}},2} = -2\left(1-3\lambda^{2}\right)-\frac{3}{2}\lambda^{2}\vert\psi_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert$$

The stability condition can thus be written as

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

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

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

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

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

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

at the stability limit $v_{0} = \sqrt{3}\gamma$.