Csdrhrt

I am trying to solve Poisson's equation in an axisymmetric cylindrical domain using finite difference. So I start with my differential equation and boundary conditions and discretize them. However, I'm having trouble thinking of how to discretize the radial portion

begin{array}{lll}
frac{1}{r} frac{partial}{partial r} left(r frac{partial u}{partial r} right) + frac{partial^2 u}{partial z^2} = -frac{rho}{varepsilon_0} & longrightarrow & frac{r_{i+1/2} u_{i+1, j} - 2 r_{i} u_{i,j} + r_{i-1/2} u_{i-1, j}}{r_i Delta r^2} + frac{u_{i, j+1} - 2 u_{i, j} + u_{i, j - 1}}{Delta z^2} = -frac{rho_{i,j}}{varepsilon_0} \
left. frac{partial u}{partial r} right|_{r = 0} = 0 & longrightarrow & u_{0,j} = frac{1}{3} left(4 u_{1,j} - u_{2,j} right) ~ \
lim_{rtoinfty} u(r, z) = 0 & longrightarrow & ? ~ \
u(r, l(r)) = f_3(r) & longrightarrow & u_{i, l(r_i)/Delta z} = f_3(r_i) \
u(r, h) = f_4(r) & longrightarrow & u_{i, N} = f_4(r_i) \
end{array}

where

$$
l(r) = left{begin{aligned}
&h - frac{H}{R} r &&: r le R\
&0 &&: r > R
end{aligned}
right.$$
Looking at my old class notes, the professor mentioned a method called irregular singular points as a method to better approximate boundary conditions in (semi-)infinite domains but, I don't understand how I would apply this to 2D systems or to radial systems.

I would appreciate any suggestions.

The actual boundary condition should be

$$lim_{r rightarrow infty} frac{partial u}{partial r} = 0$$

which I approximated at some finite point using the backward second order finite difference.

The actual boundary condition can be seen to be correct by a simple thought experiment by setting $H = 0$ (a rectangular grid) with $rho = 0$ and we must recover the solution 1D cartesian solution to Laplace's equation.