Full separation

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 $\sigma$ is a constant function of x and y, it does. The process is depicted in Figure 2 for an impulsive initial disturbance.

 

temperature
Figure 2 Temperature distribution in the (x,y)-plane beginning from a delta function (left). After heat is allowed to flow in the x-direction but not in the y-direction the heat is located in a ``wall'' (center). Finally allowing heat to flow for the same amount of time in the y-direction but not the x-direction gives the same symmetrical Gaussian result that would have been found if the heat had moved in x- and y-directions simultaneously (right).

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 $\sigma$ is a constant, because then Fourier transformation may be used, and the two-dimensional solution $\exp [ - \sigma\, ( k_x^2\ +\ k_y^2 ) t ]$ equals the succession of one-dimensional solutions $\exp ( - \sigma\, {k_x^2} t )$$\exp ( - \sigma\, {k_y^2} t )$.It turns out, and will later be shown, that the condition required for applicability of full separation is that $\sigma\, {\partial^2 / \partial x^2 }$ should commute with $\sigma\, {\partial^2 / \partial y^2 }$,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 $\sigma$ were a slowly variable function of x or y. Then you might find that although $\sigma\, {\partial^2 / \partial x^2 }$ does not strictly commute with $\sigma\, {\partial^2 / \partial y^2 }$,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].