I am trying to implement the ADI scheme of Douglas and Rachford.
For $p(X,Z,t)$, there is:
$$ \begin{gathered} A=p^{n-1}+\Delta t_n\left[F_0\left(p^{n-1}, t_{n-1}\right)+F_1\left(p^{n-1}, t_{n-1}\right)+F_2\left(p^{n-1}, t_{n-1}\right)\right] \\ B-\alpha \Delta t_n F_1\left(B, t_n\right)=A-\alpha \Delta t_n F_1\left(p^{n-1}, t_{n-1}\right) \quad \quad(*) \\ C-\alpha \Delta t_n F_2\left(C, t_n\right)=B-\alpha \Delta t_n F_2\left(p^{n-1}, t_{n-1}\right) \quad \quad(**) \\ p^n=C, n=1, \ldots, N, \end{gathered} $$
with
$$ \begin{gathered} F_0(p, t)=\frac{\partial^2}{\partial X \partial Z}p \\ F_1(p, t)=-\frac{\partial}{\partial Z}p+\frac{1}{2} \frac{\partial^2}{\partial Z^2}p \\ F_2(p, t)=-\frac{\partial}{\partial X}p+\frac{1}{2} \frac{\partial^2}{\partial X^2}p \end{gathered} $$
I am confused with equations $(*)$ and $(**)$. How can I calculate the $B$ and $C$ if they are functions of themselves? I have checked other resources and the 2nd correction always depends on itself, as seen in the image below ($U^{m+1*}$ and $U^{m+1**}$ are functions of themselves). How do I go about calculating this?
