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 (7) and (8) in which, instead of alternating back and forth between them at alternate time steps, what is done is to march (7) through all time steps. Then this intermediate result is used as an initial condition for (8), 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 (6). But if $\sigma$ is a constant function of x and y, it does. The process is depicted in Figure 1 for an impulsive initial disturbance.

 

temperature
Figure 1 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 (6) 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.

There are circumstances which dictate a middle road between splitting and full 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 (7) before you transpose the data and switch over to (8). Circumstances like this one but with more geophysical interest arise with the wave-extrapolation equation that is considered next.