Type I: Voltage-centered Network

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

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

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

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

and in order to satisfy (5.9), we set

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

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

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