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##

Numerical Phase Velocity

For a given amplification factor
, the *numerical phase velocity* at frequency
is defined by

where
is the Euclidean norm of the vector
. This expression gives the speed of propagation for a plane wave of wavenumber , according to the numerical scheme for which
is an amplification factor. For the wave equation model problem, the speed of any plane wave solution will simply be , but the numerical phase velocity will in general be different, and in particular, wave speeds will be directionally-dependent to a certain degree, depending on the type of scheme used. For all these schemes, the numerical phase velocity for at least one of the amplification factors will approach the correct physical velocity near the spatial DC frequency, by *consistency* of the numerical scheme with the wave equation^{}.

Stefan Bilbao
2002-01-22