Stability of the Explicit Method

When z takes on value more than 0.5, then we can observe that as time passes, we get an unstable system as the future value ($u^{k+1}_i$) will be dependent on the values neighboring index. (i.e. $u^{k+1}_i \simeq 2z u^k_{neighbors} + (1-2z) u^k_{i}$ ) so if $z > 0.5$ then $u^k_{neighbors}$ will weigh more. So eventually, at one time point, the values with odd induces will have similar values and even induces will have their own similarities as opposed to the immediate neighbors having similar values. This will give a factor that won't assure at all that the solution will be smooth, which we can easily envision as the desired result.

However, there are times when we want z value to be more than 0.5. Recall that $x = \frac{ks}{h^2}$ when k value within the z value is particularly large for fixed discretization (which fixes the $h^2$ factor of z), then we are forced to make s within z very very small. this means that we have to perform intensive amount of computation to get the value out of a certain time $t_F$ as opposed to the conductive material with relatively small k value with the number of iterations that are needed to compute enough time stages.

So whether we want the z value large or not would depend on whether we would employ effective methods to prevent the destabilization of the solution. If such method is supplied then larger z values would help us to see the future values of u much quicker and with much less computational effort. On the other hand, if we want more condensed snapshots, or simply cannot get to stabilize the system, then the z value has to stay small.

At the same time, it is interesting to note that no matter what the k value is, the way that heat flows within the discretized domain is exactly the same, just different in how fast it occurs.