INITIAL-VALUE PROBLEMS IN TWO AND THREE
DIMENSIONS
There are whole textbooks (for example, References
33
and
34)
devoted to solving
initial-value problems by difference approximations to differential equations. In
this section we will briefly cover the main ideas. The overall idea in two dimensions
is that one partitions a computer memory into one or a few two-dimensional grids
where field variables are represented as functions of two spatial dimensions. Then
you insert initial conditions, turn on the computer, and see what happens. There
have been numerous extensive studies devoted to the diffusion equation, but
far fewer studies have been devoted to the wave equation. The problem with
modeling the wave equation is that ten points per wavelength is probably not
enough, and even at that you cannot fit very many wavelengths onto a reasonable
grid. The energy then propagates rapidly to the edges of the grid where it bounces
back, whether you want it to or not. One way to ameliorate this kind of difficulty is
to develop coordinate systems which move with the waves. These coordinate
systems also facilitate projection of waves from the earth's surface, where they are
observed, back down into the earth. This kind of projection forms the basis for the
practical reflection seismic data processing techniques described in chapter
11.
