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Type I: Voltage-centered Network

In order to satisfy (5.10), with $ Y_{J}$ defined by (5.6) we set

$\displaystyle Y_{x^{-},i}=Y_{x^{+},i} = \frac{(\rho A)_{i}}{2\mu}\hspace{0.5in}Y_{t,i} = \frac{(\rho A)_{i}}{\mu}\hspace{0.5in}Y_{c,i} = 0$    

Now, we have, recalling (5.7) and (5.8),

$\displaystyle \tilde{Z}_{J,i} = \frac{2\mu}{(\rho A)_{i+1}}+\frac{2\mu}{(\rho A)_{i-1}}+\frac{4\mu}{(\rho A)_{i}}+\tilde{Z}_{c,i}$    

and in order to satisfy (5.9), we set

$\displaystyle \tilde{Z}_{c,i} = \left(\frac{1}{2\mu (EI)_{i+1}}+\frac{1}{2\mu (...
...u}{(\rho A)_{i+1}}+\frac{2\mu}{(\rho A)_{i-1}}+\frac{4\mu}{(\rho A)_{i}}\right)$    

We have written here $ h_{i}\triangleq h(i\Delta)$, where $ h$ is either of $ \rho A$ or $ EI$. In this case, $ \tilde{Z}_{c,i}$ will be positive if

$\displaystyle \mu\leq \frac{1}{2}\min_{i}\sqrt{\frac{(\rho A)_{i}}{(EI)_{i}}}$ (5.12)

and we have an upper bound on the time step in terms of the grid spacing. In the constant-coefficient case, we note that this is the same requirement as for the stability of scheme (5.3).

Stefan Bilbao 2002-01-22