Implicit Scheme
In deriving the equations for the explicit scheme we started with the so-called
forward formula for the first derivative, i.e., we used the discrete approximation
$$
\begin{equation*}
u_t\approx \frac{u(x_i,t_j+\Delta t)-u(x_i,t_j)}{\Delta t}.
\end{equation*}
$$
However, there is nothing which hinders us from using the backward formula
$$
\begin{equation*}
u_t\approx \frac{u(x_i,t_j)-u(x_i,t_j-\Delta t)}{\Delta t},
\end{equation*}
$$
still with a truncation error which goes like \( O(\Delta t) \).