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

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

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

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

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

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