Defining Equations and Centered Differences

The set of PDEs describing a lossless, source-free parallel-plate transmission line in (2+1)D is a direct generalization of system (4.17):

$$\begin{eqnarray}l\frac{\partial i_{x}}{\partial t} + \frac{\parti... ...tial i_{x}}{\partial x} + \frac{\partial i_{y}}{\partial y} &=& 0\end{eqnarray}$$ (4.60a)

Now $i_{x}(x,y,t)$ and $i_{y}(x,y,t)$ are the components of the current density vector in the $x$ and $y$ directions, respectively, and $u(x,y,t)$ is the voltage between the plates. $l(x,y)$ and $c(x,y)$, both assumed positive everywhere, are the inductance and capacitance per unit length.

If we assume that $l$ and $c$ are constant, then as in the (1+1)D case, the set of equations can be reduced to a single second order equation in the voltage alone:

$$\frac{\partial^{2} u}{\partial t^{2}}=\gamma^{2}\Big(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\Big)$$ (4.61)

and again, the wave speed $\gamma$ is given by

$$\gamma = \frac{1}{\sqrt{lc}}$$

The centered difference scheme for system (4.49) also generalizes simply. Define grid functions $I_{x.i,j}(n)$, $I_{y,i,j}(n)$ and $U_{i,j}(n)$ which run over half-integer values of $i$, $j$, and $n$, i.e.,

$$i,j,n = \hdots -1,-{\textstyle \frac{1}{2}},0,{\textstyle \frac{1}{2}},1\hdots$$

We will furthermore assume that the spatial step in the $x$ direction and the $y$ direction are the same and equal to $\Delta/2$. As before, the time step will be $T/2$. We can use the approximations (4.19), as well as an approximation to the derivative in the $y$ direction,

$$\left.\frac{\partial w}{\partial y}\right\vert _{i\Delta,j\Delta,nT} = \frac{w(i,j+\frac{1}{2},n)-w(i,j-\frac{1}{2},n)}{\Delta}+O(\Delta^{2})$$

where $w$ stands for either of $i_{y}$ or $u$. We obtain the difference scheme

where we have written

$$\bar{l}_{i,j} \triangleq l(i\Delta,j\Delta)+O(\Delta^{2})$$

$$\bar{c}_{i,j} \triangleq l(i\Delta,j\Delta)+O(\Delta^{2})$$

and

$$v_{0} \triangleq \frac{\Delta}{T}$$

As in the (1+1)D case, it is possible to subdivide the calculation scheme (4.51) into smaller, mutually exclusive subschemes. Using a decimated grid for the variable coefficient difference scheme amounts to rewriting scheme (4.51) as

$$\begin{eqnarray}\hspace{-0.6in}I_{x,i+\frac{1}{2},j}(n+{\textstyl... ...{y,i,j-\frac{1}{2}}(n-{\textstyle \frac{1}{2}})\right) =\,\,\, 0 \end{eqnarray}$$

where we now compute solutions for $i$, $j$ and $n$ integer. The interleaved grid is shown in Figure 4.18; a grey (white) dot at a grid location indicates that the adjacent named variable is to be calculated at times which are even (odd) multiples of $T/2$. This interleaved form was originally put forth by Yee [214] in the context of electromagnetic field problems, and forms the basis of the widely used finite-difference time domain (FDTD) family of difference methods [184], which were discussed briefly in §4.1. If system (4.52) is rewritten as a TE or TM system, the interleaved arrangement of the field components also has an interesting physical interpretation as a discrete counterpart to the integral form of Ampere's and Faraday's Laws [184]. This result also extends easily to the discretization of Maxwell's equations in (3+1)D [214]; see §4.10.6 for more details.

Figure 4.18: Interleaved computational grid for the (2+1)D parallel-plate system.

The (2+1)D analogue of (4.23), which holds in the case where $l$ and $c$ are constant, is

$$\frac{v_{0}^{2}}{\gamma^{2}}\Big(U_{i,j}(n\!+\!1)+U_{i,j}(n\!-\!1)\Big) = U_{i+1,j}(n)+U_{i-1,j}(n) + U_{i,j+1}(n)+U_{i,j-1}(n)\notag$$ (4.64)

$$+ 2\left(\frac{v_{0}^{2}}{\gamma^{2}}-2\right)U_{i,j}(n)$$ (4.65)

The magic time step will now be

$$v_{0} = \sqrt{2}\gamma$$

and (4.53) simplifies to

$$U_{i,j}(n\!+\!1) + U_{i,j}(n\!-\!1) = \frac{1}{2}\Big(U_{i+1,j}(n)+U_{i-1,j}(n)+ U_{i,j+1}(n)+U_{i,j-1}(n)\Big)$$ (4.66)

As in (1+1)D, when we are solving the wave equation by centered differences at the magic time step (or at CFL), the calculation further decomposes into two independent calculations; we need only update $U_{i,j}(n)$ for $i+j+n$ even (or odd). We will examine this interesting decomposition property in detail in Appendix A.