Crank-Nicolson scheme
It is possible to combine the implicit and explicit methods in a slightly more general
approach. Introducing a parameter \( \theta \) (the so-called \( \theta \)-rule) we can set up
an equation
$$
\begin{equation}
\tag{9}
\frac{\theta}{\Delta x^2}\left(u_{i-1,j}-2u_{i,j}+u_{i+1,j}\right)+
\frac{1-\theta}{\Delta x^2}\left(u_{i+1,j-1}-2u_{i,j-1}+u_{i-1,j-1}\right)=
\frac{1}{\Delta t}\left(u_{i,j}-u_{i,j-1}\right),
\end{equation}
$$
which for \( \theta=0 \) yields the forward formula for the first derivative and
the explicit scheme, while \( \theta=1 \) yields the backward formula and the implicit
scheme. These two schemes are called the backward and forward Euler schemes, respectively.
For \( \theta = 1/2 \) we obtain a new scheme after its inventors, Crank and Nicolson.
This scheme yields a truncation in time which goes like \( O(\Delta t^2) \) and it is stable
for all possible combinations of \( \Delta t \) and \( \Delta x \).