The difficulty with the given program is that it doesn't work for all possible numerical values of .You can see that when is too large (when is too small) the solution in the interior region of the data table contains growing oscillations. What is happening is that the low-frequency part of the solution is OK (for a while), but the high-frequency part is diverging. The precise reason the divergence occurs is the subject of some mathematical analysis that will be postponed till page . At wavelengths long compared to or , we expect the difference approximation to agree with the true heat-flow equation, smoothing out irregularities in temperature. At short wavelengths the wild oscillation shows that the difference equation can behave in a way almost opposite to the way the differential equation behaves. The short wavelength discrepancy arises because difference operators become equal to differential operators only at long wavelengths. The divergence of the solution is a fatal problem because the subsequent round-off error will eventually destroy the low frequencies too.

By supposing that the instability arises because the time derivative
is centered at a slightly different time *t*+1/2 than
the second *x*-derivative at time *t*,
we are led to the so-called
*leapfrog*
method, in which the time derivative is taken as a difference
between *t*-1 and *t*+1:

(33) |

3rx |
||||

3c | ||||

1c| | 1|c | |||

1c| | -1 | 1|c | ||

1c| | 1|c | |||

1|c| | ||||

1|c| | 4 | |||

1|c| | ||||

1c| | 1|c | |||

1c| | +1 | 1|c | ||

1c| | 1|c | |||

t |
4c |

To avoid all these problems (and get more accurate answers as well),
we now turn to some slightly more complicated solution methods known as
*implicit*
methods.

10/31/1997