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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

 (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.

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
 (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

 (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

 (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