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Second-Order Finite Difference Scheme

The simplest, and traditional way of discretizing the 1-D wave equation is by replacing the second derivatives by second order differences:

\begin{eqnarray*}
\left.\frac{\partial^{2} u}{\partial t^{2}}\right\vert _{x=k\Delta,t=nT} &\simeq& (u_{k}^{n-1}-2u_{k}^{n}+u_{k}^{n+1})/T^{2} \\
\left.\frac{\partial^{2} u}{\partial x^{2}}\right\vert _{x=k\Delta,t=nT} &\simeq& (u_{k-1}^{n}-2u_{k}^{n}+u_{k+1}^{n})/\Delta^{2}
\end{eqnarray*}

where $ u_{k}^{n}$ is defined as $ u(k\Delta, nT)$ . Here we have sampled the time-space plane in a uniform grid, with a timestep of $ T$ and a space step of $ \Delta$ . The $ u_{k}^{n}$ are the grid variables here. Now, through substitution, the wave equation becomes:

\begin{eqnarray*}
u_{k}^{n-1}-2u_{k}^{n}+u_{k}^{n+1} = \frac{c^{2}T^{2}}{\Delta^{2}}
(u_{k-1}^{n}-2u_{k}^{n}+u_{k+1}^{n})
\end{eqnarray*}


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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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