5.3 - Stability analysis in the unsteady problem

As explained in the previous challenge, the stability of the transient methods depends on the factor k=alpha*dt/(dx^2). This factor is also known as CFL number.

The unstable solutions can look differently depending on the scheme (homogeneous vs alterning) but they will always show temperatures way too high or way too low.

1) Explicit Method:

This is the common representation of an unstable solution. We can see temperatures too high and other way too low (even negative, which doesnt exist in Kelvin scale). This is beacuse the scheme is not able to converge to a solution and alternates trying to aproximate to it from both sides.

In this method, the solution is unstable when k>0.25.

2) Implicit SOR Method:

This is the other example of the divergent solution. In this case, temperatures reaches values e+93, obviously impossible.

In this implicit method, the solution is unnstable when k>0.5.

Note that the implicit method allows for higher time steps without still being unstable. It is also important to notice that the time step cannot be too small, cause nummerical difussion is inverserly proportional to this:

Therefore, we need to chose a time step that is not too small, nor too high (unstable)