Splitting can turn out to be much more accurate than might be imagined.
In many cases there is
*
no
*
loss of accuracy.
Then the method can be taken to an extreme limit.
Think about a radical approach to
equations (2) and (3) in which,
instead of alternating back and forth between them at alternate
time steps, what is done is to march
(2) through all time steps.
Then this
intermediate result is used as an
initial condition for (3), which is marched
through all time steps to produce a final
result.
It might seem surprising that this
radical method can produce the correct solution to equation (1).
But if is a constant
function of *x* and *y*, it does.
The process is depicted in Figure 2
for an impulsive initial disturbance.

Figure 2

A differential equation like (1) is said to be
*
fully separable
*
when the correct solution is obtainable
by the radical method.
It should not be
too surprising that full separation works
when is a constant, because then
Fourier transformation may be used,
and the two-dimensional
solution equals the
succession of one-dimensional solutions .It turns out, and will later be shown,
that the condition required for
applicability of full separation is
that should commute
with ,that is, the order of differentiation should be irrelevant.
Technically there is also a boundary-condition requirement,
but it creates no difficulty when the
disturbance dies out before reaching
a boundary.

Surprisingly, no notice is made of
full separability in many textbooks on numerical
solutions.
Perhaps this is because the
total number of additions and
multiplications is the same whether a
solution is found by splitting or by
full separation.
But as a practical matter,
the cost of solving large problems does not mount up
simply according to the number of multiplications.
When
the data base does not fit entirely into
the random-access memory, as is almost the
definition of a
*
large problem,
*
then each step of the splitting method demands
that the data base be transposed, say, from (*x*,*y*) storage order
to (*y*,*x*) storage order.
Transposing requires no multiplications, but
in many environments transposing would be by far
the most costly part of the whole computation.
So if transposing cannot be avoided, at least
it should be reduced to a practical minimum.

There are circumstances which
dictate a middle road between splitting and separation--for
example, if were a
slowly variable function of *x* or *y*.
Then you might find that although
does not
strictly commute with ,it comes close enough that a number of time steps
may be made with (2)
before you transpose the data and switch over to (3).
Circumstances like this one but with
more geophysical interest arise with
the wave-extrapolation equation that is considered next.
The significance in seismology
of the splitting and full separation concepts
was first recognized by Brown [1983].

10/31/1997