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One-step Schemes

Consider the following one-step explicit difference scheme, which relates values of the grid function $ U_{{\bf m}}(n+1)$ to values at the previous time step:

$\displaystyle U_{{\bf m}}(n+1) = \sum_{{\bf k}\in \mathbb{K}}\alpha_{{\bf k}}U_{{\bf m - k}}(n)$    

where $ \mathbb{K}$ is some subset of $ \mathbb{Z}^{N}$, and the parameters $ \alpha_{{\bf k}}$ are constants; it is initialized by setting $ U_{{\bf m}}(0)$ equal to some function $ U_{{\bf m},0}$ (assumed to be in $ L_{2}$). Taking the spatial Fourier transform of this recursion gives
$\displaystyle \hat{U}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n+1)$ $\displaystyle =$ $\displaystyle \left(\sum_{{\bf k}\in \mathbb{K}}\alpha_{{\bf k}}e^{-j\Delta{\bf...
...oldmath$\beta$}}}\right)\hat{U}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n)\notag$ (2)
$\displaystyle \hbox{ }$ $\displaystyle =$ $\displaystyle \hspace{0.5in}G_{\mbox{{\scriptsize\boldmath$\beta$}}}\hspace{0.44in}\hat{U}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n)$ (3)

$ G_{\mbox{{\scriptsize\boldmath $\beta$}}}$ so defined is called the spectral amplification factor for a one-step finite difference scheme. (2) implies that we have, in particular, that

$\displaystyle \hat{U}_{\mbox{{\scriptsize\boldmath$\beta$}}}(n+1) = G_{\mbox{{\...
...ptsize\boldmath$\beta$}}}^{n+1}\hat{U}_{\mbox{{\scriptsize\boldmath$\beta$}},0}$ (4)

where $ \hat{U}_{\mbox{{\scriptsize\boldmath $\beta$}},0}$ is the spatial Fourier transform of the initial condition $ U_{{\bf m},0}$. (3) further implies that

$\displaystyle \Vert\hat{U}(n+1)\Vert _{2} \leq \left(\max_{\mbox{{\scriptsize\b...{{\scriptsize\boldmath$\beta$}}}\vert\right)^{n+1}\Vert\hat{U}_{0}\Vert _{2}$    

and finally, through Parseval's relation, that

$\displaystyle \Vert U(n+1)\Vert _{2} \leq \left(\max_{\mbox{{\scriptsize\boldma...
...G_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert\right)^{n+1}\Vert U_{0}\Vert _{2}$    

If the $ \alpha_{{\bf k}}$ which define the difference scheme are independent of the grid spacing and the time step, then such a difference scheme is called stable if

$\displaystyle \max_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert G_{\mbox{{\scriptsize\boldmath$\beta$}}}\vert\leq 1$    

The $ L_{2}$ norm of the solution to the difference equation will thus not increase as the simulation progresses.
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``Spectral Analysis of Finite Difference Meshes'', by .
Copyright © 2005-12-28 by Julius O. Smith III<jos_email.html>
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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