Splitting

The splitting method for numerically solving the heat-flow equation is to replace the two-dimensional heat-flow equation by two one-dimensional equations, each of which is used on alternate time steps:

$$\begin{eqnarray} {\partial T \over \partial t}\ \ \ & = & \ \ \ 2\, \sigma\ {\pa... ...{\partial^2 T \over \partial y^2}\ \ \ \ \ \ {\rm(all\ {\it x })}\end{eqnarray}$$ (2) (3)

In equation (2) the heat conductivity $\sigma$ has been doubled for flow in the x-direction and zeroed for flow in the y-direction. The reverse applies in equation (3). At odd moments in time heat flows according to (2) and at even moments in time it flows according to (3). This solution by alternation between (2) and (3) can be proved mathematically to converge to the solution to (1) with errors of the order of $\Delta t$.Hence the error goes to zero as $\Delta t$ goes to zero. The motivation for splitting is the infeasibility of higher-dimensional implicit methods.