Type III: Mixed Mesh

We set

$$Z_{x^{-},i+\frac{1}{2},j} = Z_{x^{+},i+\frac{1}{2},j} = Z_{y^{-},i,j+\frac{1}{2}} = Z_{y^{+},i,j+\frac{1}{2}} = Z_{const}$$

where $Z_{const}$ is some positive constant, and then choose

$$Y_{c,i,j} = 2v_{0}c_{i,j}-\frac{4}{Z_{const}}$$

$$Z_{c,i+\frac{1}{2},j} = 2v_{0}l_{i+\frac{1}{2},j}-2Z_{const}$$

$$Z_{c,i,j+\frac{1}{2}} = 2v_{0}l_{i,j+\frac{1}{2}}-2Z_{const}$$

The optimal value of $Z_{const}$ is easily shown to be

$$Z_{const} = \sqrt{\frac{2\min_{2(k+p)\hspace{0.05in}{\rm odd}}l_{k,p}}{\min_{i,j}c_{i,j}}}$$

and this leads to the constraint

$$v_{0}\geq \sqrt{\frac{2}{\min_{2(k+p)\hspace{0.05in}{\rm odd}}(l_{k,p})\min_{i,j}(c_{i,j})}}$$

for $i$, $j$ integer and $k$ and $p$ half-integer. As in (1+1)D, this bound is inferior to those obtained using the type I and type II meshes. Because the immittance settings are simpler, however, this form may be preferable, from a programming standpoint.