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Numerical Phase Velocity

For a given amplification factor $ G_{\mbox{{\scriptsize\boldmath $\beta$}}}$, the numerical phase velocity at frequency $ \beta$ is defined by

$\displaystyle v_{\mbox{{\scriptsize\boldmath$\beta$}}, phase} = \left\vert\frac...
...\boldmath$\beta$}}}\vert)}{i\Vert\mbox{\boldmath$\beta$}\Vert _{2}T}\right\vert$    

where $ \Vert$$ \beta$$ \Vert _{2}$ is the Euclidean norm of the vector $ \beta$. This expression gives the speed of propagation for a plane wave of wavenumber $ \beta$, according to the numerical scheme of which $ G_{\mbox{{\scriptsize\boldmath $\beta$}}}$ is an amplification factor. For the wave equation model problem, the speed of any plane wave solution will simply be $ \gamma$, 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 equation2.

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