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Centered Difference Schemes and Grid Decimation
Suppose we are interested in developing a finite difference scheme to calculate the solution to (4.17) numerically. We first define grid functions
, and
which, for convenience, will run over half-integer values of
and
, i.e.,
They are intended to approximate
and
at the points
, where
is the spatial grid step, and
the time step. We note that we have used the same variable,
, to stand for both the continuous-time current which solves (4.17), as well as the discrete-valued variable representing the spatial coordinate on the grid.
We have the centered difference approximations
![$\displaystyle \begin{eqnarray}\left.\frac{\partial w}{\partial t}\right\vert _{...
...2})\Delta,nT)-w((i-\frac{1}{2})\Delta,nT)}{\Delta}+O(\Delta^{2}) \end{eqnarray}$](img1240.png) |
(4.19a) |
where
stands for either of
or
.
Employing these differences in (4.17), and replacing the continuous time/space variables
and
by their respective grid functions yields the difference scheme
Here, we have chosen
for half-integer
. Because the centered difference approximations (4.19) are second-order accurate,
and
may be approximated to the same order without any decrease in accuracy. We leave the exact form of these approximations,
and
unspecified for the moment, but will return to various settings in §4.3.6. Also, in order to remain consistent with the notation in the MDWD schemes of the last chapter, we have set
Thus difference equations (4.20) are consistent with (4.17), and accurate to
.
In a difference scheme for a general system of PDEs, it would be necessary to update all the grid functions every time step, and at every grid point--that is to say, at every increment in
and
of one-half, new values of the grid functions would have to be calculated, and indeed, we can proceed in this manner in with the scheme (4.20) as well. In this case, however, it is easy to see that updating
, for
and
even requires access only to
at the previous time step, and at neighboring grid locations (thus for
odd and
odd), as well as
at the same location, two time steps previously (
and
again even) [131,184]. Similarly, updating
for
odd and
odd involves only values of
for
even and
even, and
for
odd and
odd. It is then obvious that only values of
for which
is even and
even (and values of
with
odd and
odd) need enter into our scheme. We can thus decimate the grid in the manner shown in Figure 4.7.
Figure 4.7:
Interleaved sampling grid for the (1+1)D transmission line.
![\begin{figure}\begin{center}
\begin{picture}(350,60)
% graphpaper(0,0)(350,60) ...
...tiny {$(i-1)\Delta$}}
\put(370,37){$x$}
\end{picture} \end{center} \end{figure}](img1251.png) |
We calculate the values of
at the grey dots in Figure 4.7, and
at the white dots. The difference scheme on the decimated grid can be written as
![$\displaystyle \begin{eqnarray}I_{i+\frac{1}{2}}(n+{\textstyle \frac{1}{2}}) - I...
...{2}})-I_{i-\frac{1}{2}}(n-{\textstyle \frac{1}{2}})\right) &=& 0 \end{eqnarray}$](img1253.png) |
(4.22a) |
for
,
integer. We perform the calculation on the decimated grid with no decrease in accuracy, although we are of course approximating the solution at fewer grid points. In analogy with the continuous case, when
and
are constant it is possible to combine the difference equations (4.22) into a single equation for the voltage grid function
, which is
![$\displaystyle U_{i}(n+1) -2U_{i}(n)+U_{i}(n-1) = \frac{\gamma^{2}}{v_{0}^{2}}\left(U_{i+1}(n)-2U_{i}(n)+U_{i-1}(n)\right)$](img1254.png) |
(4.23) |
and which solves the (1+1)D wave equation (4.18). For the so-called magic time step [184],
the difference scheme (4.23) reduces to
![$\displaystyle U_{i}(n+1) + U_{i}(n-1) = U_{i+1}(n)+U_{i-1}(n)$](img1256.png) |
(4.24) |
a form which has great relevance to the discussion to follow on the waveguide implementation. It is interesting that in this case, the grid may be further decimated; we need only calculate
for
even (or odd), for
,
integer. We will examine this point in further detail in higher dimensions in Appendix A.
Next: A (1+1)D Waveguide Network
Up: The (1+1)D Transmission Line
Previous: First-order System and the
Stefan Bilbao
2002-01-22