Stability of the explicit heat-flow equation

Begin with the heat-flow equation and Fourier transform over space. Thus $\partial^2 / \partial x^2$ becomes simply - k², and  

$${\partial q \over \partial t } \eq - \ { \sigma \over c } \ k^2 \ q$$ (69)

Finite differencing explicitly over time gives an equation that is identical in form to the inflation-of-money equation:

$$\begin{eqnarray} { q_{t+1} \ -\ q_t \over \Delta t } \ \ \ &=&\ \ \ -\ { \sigma... ... \left( 1 \ - \ { \sigma \, \Delta t \over c } \ k^2 \right) \ q_t\end{eqnarray}$$ (70) (71)

For stability, the magnitude of q_t+1 should be less than or equal to the magnitude of q_t. This requires the factor in parentheses to have a magnitude less than or equal to unity. The dangerous case is when the factor is more negative than -1. There is instability when $k^2 \gt 2c / ( \sigma \, \Delta t )$.This means that the high frequencies are diverging with time. The explicit finite differencing on the time axis has caused disaster for short wavelengths on the space axis. Surprisingly, this disaster can be recouped by differencing the space axis coarsely enough! The second space derivative in the Fourier transform domain is $- \, k^2$.When the x-axis is discretized it becomes $- \hat k^2$.So, to discretize (69), (70) and (71), just replace k by $\hat k$.Equation (68) shows that $\hat k^2$ has an upper limit of $\hat k^2 \ =\ 4 / \Delta x^2$ at the Nyquist frequency $k \Delta x = \pi$.Finally, the factor in (71) will be less than unity and there will be stability if  

$$\hat k^2 \eq { 4 \over \Delta x^2 }\ \ \ \le\ \ \ { 2 \, c \over \sigma \, \Delta t }$$ (72)

Evidently instability can be averted by a sufficiently dense sampling of time compared to space. Such a solution becomes unbearably costly, however, when the heat conductivity $\sigma ( x )$ takes on a wide range of values. For problems in one space dimension, there is an easy escape in implicit methods. For problems in higher-dimensional spaces, explicit methods must be used.