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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 Fouriertransformed 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 twostep 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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