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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 -element vector of spatially Fourier-transformed functions of . The norm is defined by

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 satisfies an equation of the form

 (11)

for some Hermitian matrix function of , . Because is Hermitian, we may write , for some unitary matrix , and a real diagonal matrix containing the eigenvalues of . As such, we may change variables via , to get

 (12)

The system thus decouples into a system of scalar two-step spectral update equations; because and are related by a unitary transformation, we have , and we may apply stability tests to the uncoupled system (11). We thus require that the eigenvalues of , namely for , which are the elements on the diagonal of , all satisfy

 (13)

At frequencies 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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