Next: Numerical Phase Velocity
Up: Von Neumann Analysis of
Previous: Multistep 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 [176], 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

(A.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

(A.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 (A.11). We thus require that the eigenvalues of
, namely
for
, which are the elements on the diagonal of
, all satisfy

(A.13) 
At frequencies
for which any of the eigenvalues satisfies (A.12) with equality, then we may again have the same problem with mild linear growth in the solution.
Next: Numerical Phase Velocity
Up: Von Neumann Analysis of
Previous: Multistep Schemes
Stefan Bilbao
20020122