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A Peek at Stability of Finite Difference Schemes

Let's look again at the difference scheme we derived for the 1-D wave eq, with the special time/space step $ c=T/\Delta$ :

\begin{eqnarray*}
u_{k}^{n+1} = u_{k-1}^{n}+u_{k+1}^{n}-u_{k}^{n-1}
\end{eqnarray*}

The velocity sample $ u(k,n)$ is a two-dimensional sequence with a time index and a spatial coordinate index.

Suppose we now take the DTFT with respect to the spatial index $ k$ :

\begin{eqnarray*}
\sum_{k=-\infty}^{\infty}u_{k}^{n+1}e^{-j\omega k\Delta} = \sum_{k=-\infty}^{\infty}(u_{k-1}^{n}+u_{k+1}^{n}-u_{k}^{n-1})e^{-j\omega k\Delta}
\end{eqnarray*}

or

\begin{eqnarray*}
U^{n+1}(\omega)= (e^{-j\omega\Delta}+e^{j\omega\Delta})U^{n}(\omega)-U^{n-1}(\omega)
\end{eqnarray*}

where here $ U^{n}(\omega)$ is the spatial spectrum of the solution at time $ n$ , and $ \omega$ is the spatial frequency variable. We can also write this in vector form as:

\begin{eqnarray*}
\left[\begin{array}{c} U^{n+1}(\omega) \\ [2pt] U^{n}(\omega) \end{array}\right] &=&
\left[\begin{array}{cc} 2\cos\omega\Delta & -1 \\ [2pt] 1 & 0 \end{array}\right]
\left[\begin{array}{c} U^{n}(\omega) \\ [2pt] U^{n-1}(\omega) \end{array}\right]
\end{eqnarray*}

Note that the state of the system is completely determined by $ U^n(\cdot)$ and $ U^{n-1}(\cdot)$ .


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``Discrete-Time Lumped Models'', by Stefan Bilbao and Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Stefan Bilbao and Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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