Explicit Scheme, stability condition
However, although the explicit scheme is easy to implement, it has a very weak
stability condition, given by
$$
\begin{equation*}
\Delta t/\Delta x^2 \le 1/2.
\end{equation*}
$$
This means that if \( \Delta x = 0.01 \) (a rather frequent choice), then \( \Delta t= 5\times 10^{-5} \). This has obviously
bad consequences if our time interval is large.
In order to derive this relation we need some results from studies of iterative schemes.
If we require that our solution approaches a definite value after
a certain amount of time steps we need to require that the so-called
spectral radius \( \rho(\mathbf{A}) \) of our matrix \( \mathbf{A} \) satisfies the condition
$$
\begin{equation}
\tag{8}
\rho(\mathbf{A}) < 1.
\end{equation}
$$