Scheme for solving Laplace's (Poisson's) equation
We rewrite Eq.
(18)
$$
\begin{equation}
\tag{19}
4u_{i,j}= \left[u_{i,j+1}+u_{i,j-1}+u_{i+1,j}+u_{i-1,j}\right]
-h^2\rho_{i,j}=\Delta_{ij}-\tilde{\rho}_{ij},
\end{equation}
$$
where we have defined
$$
\begin{equation*}
\Delta_{ij}= \left[u_{i,j+1}+u_{i,j-1}+u_{i+1,j}+u_{i-1,j}\right],
\end{equation*}
$$
and
$$
\begin{equation*}
\tilde{\rho}_{ij}=h^2\rho_{i,j}.
\end{equation*}
$$