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If we are interested in using a grid of doubled density over a particular region of the problem domain (in order to surround a particular feature or an irregular part of the boundary), then we are faced with corners, and must develop special scattering junctions for them. An example of an irregular partitioning of the problem domain into two regions, $ I$ and $ II$, is shown in Figure 4.36(a).

Figure 4.36: (a) A particular grid arrangement between a rectilinear mesh $ I$ and one of doubled grid density $ II$, (b) a scattering junction at a corner point (labelled $ C$), and (c) a scattering junction at another type of corner point (labelled $ C'$).
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Boundary points (labelled $ B$) were treated previously, and it was found that waveguides connected to points $ B$ which lie along the boundary must have their admittances set to one-half what they would be in the interior of region $ I$ (i.e., to $ \frac{1}{2v_{0}l}$, where $ l$ is the inductance at the center of the particular boundary waveguide).

There are two types of corners which can arise in an irregular domain decomposition of this kind: those which are concave with respect to region $ I$ (the grid point at such a corner is labelled $ C$ in Figure 4.36(a)), and those which are concave with respect to region $ II$ (labelled $ C'$). There are obviously four possible orientations for each type of corner, though, by symmetry, we need only treat the type shown. Because the admittances of the boundary waveguides (represented by thick lines connecting boundary points) are now prescribed, then for a corner junction the linking admittances are fixed; only the self-loop admittance may be varied. Scattering junctions corresponding to corners of type $ C$ and $ C'$ are shown in Figure 4.36(b) and (c).

Suppose we have a corner point of type $ C$ located at coordinates $ (0,0)$. Then, from the results previously given in this section, the admittances of the five waveguides connecting this corner to its neighbors will be

$\displaystyle Y_{y^{+},0,0} = \frac{1}{v_{0}l_{0,\frac{1}{2}}}\hspace{0.3in}Y_{...
...}\hspace{0.3in}Y_{x^{+}y^{-},0,0} = \frac{1}{v_{0}l_{\frac{1}{4},-\frac{1}{4}}}$    

$\displaystyle Y_{x^{+},0,0} = \frac{1}{2v_{0}l_{\frac{1}{2},0}}\hspace{0.5in}Y_{y^{-},0,0} = \frac{1}{2v_{0}l_{0,-\frac{1}{2}}}$    

and the difference equation relating the corner junction voltage $ U_{J,0,0}$ to those of its neighbors is

$\displaystyle \hspace{-0.4in}\frac{Y_{J,0,0}}{2}\big(U_{J,0,0}(n+1)+U_{J,0,0}(n-1)\big)$ $\displaystyle =$   $\displaystyle \frac{1}{v_{0}l_{-\frac{1}{2},0}}U_{J,-1,0}+\frac{1}{v_{0}l_{0,\frac{1}{2}}}U_{J,0,1}\notag$    
  $\displaystyle \hspace{0.2in}$   $\displaystyle \hspace{-0.05in}+\frac{1}{2v_{0}l_{\frac{1}{2},0}}U_{J,1,0}+\frac{1}{2v_{0}l_{0,-\frac{1}{2}}}U_{J,0,-1}\notag$    
  $\displaystyle \hspace{0.2in}$   $\displaystyle \hspace{-0.05in}+\frac{1}{v_{0}l_{\frac{1}{4},-\frac{1}{4}}}U_{J,\frac{1}{2},-\frac{1}{2}}+Y_{c,0,0}U_{J,0,0}$    

where we have yet not specified $ Y_{c,0,0}$ or $ Y_{J,0,0}$. A Taylor expansion about $ (0,0)$ gives, in terms of the continuous variable $ u$ and neglecting higher-order terms,

$\displaystyle \frac{4Y_{J,0,0}}{7v_{0}}\frac{\partial^{2}u}{\partial t^{2}}$ $\displaystyle =$   $\displaystyle \left(\frac{\partial}{\partial x}\left(\frac{1}{l}\right)-\frac{1...
...}{\partial x}\left(\frac{1}{l}\right)\right)\frac{\partial u}{\partial y}\notag$    
      $\displaystyle + \frac{1}{l}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\p...
...{\partial y^{2}}\right)+ \frac{2}{7l}\frac{\partial^{2}u}{\partial x\partial y}$    

We can thus conclude that the differencing occurring at the corner junction is not consistent with the lossless source-free parallel-plate system, regardless of our choice of the self-loop admittance. The coefficients of the spatial first derivative terms are incorrect, and there is an extra mixed-derivative term; neither vanishes in the limit as $ \Delta\rightarrow 0$ (although if $ l$ is constant in a neighborhood surrounding the corner, then the first derivative terms vanish).

The corner junction is, however, still lossless, as are all junctions in this waveguide network, and hence a simulation of the parallel-plate system using such a mesh will be stable, regardless of inconsistencies at the corners. It is possible to argue, loosely speaking, that if the number of corners in the interface does not grow as the grid spacing is decreased (an example of this would be an enclosed rectangular doubled density grid, for which the number of corners will be four, independently of the grid spacing), then the error at the corners will become negligible. Although we will make no attempt to prove this, the simulation which we will present later in this section concurs readily with this assertion; indeed, in all tests we have run, any anomalous scattering at the corners is certainly far less important than the first-order scattering error (i.e., numerical reflection) along the interface itself. In the interest, however, of making any scattering error at the corner junctions as small as possible, we should set, at a corner with coordinates $ C$ (in order that the wave speed at the corner is correct, in a gross sense)

$\displaystyle Y_{c,C} = \frac{7}{4}v_{0}c_{C} - \frac{4}{v_{0}l_{C}}$    

where $ l_{C}$ and $ c_{C}$ are the inductance and capacitance at a corner point $ C$. The stability requirement at such a corner is then

$\displaystyle v_{0}\geq \sqrt{\frac{16}{7l_{C}c_{C}}}$   $\displaystyle \mbox{{\rm Corner $C$}}$    

which is, like the stability condition at boundary points $ B$, only marginally worse than the CFL bound, and mitigated by the somewhat worse bound to be found in region $ II$ (due to the decreased inter-junction spacing). See the discussion earlier in this section.

A similar argument follows for points $ C'$, and we also ideally have $ l$ constant in the neighborhood of such points. The setting of the self-loop admittance $ Y_{c,C'}$ should be

$\displaystyle Y_{c,C'} = \frac{5}{4}v_{0}c_{C'} - \frac{4}{v_{0}l_{C'}}$    

The resulting stability bound is

$\displaystyle v_{0}\geq \sqrt{\frac{16}{5l_{C'}c_{C'}}}$   $\displaystyle \mbox{{\rm Corner $C'$}}$    

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Next: Simulation Up: Doubled Grid Density Across Previous: Doubled Grid Density Across
Stefan Bilbao 2002-01-22