Solving Poisson's Equation Using Finite Difference

[aaaa202]

I am using the Finite Difference Method to solve Poisson's equation

$$\frac{\partial \phi}{\partial z^2} = \frac{\rho}{\epsilon}$$

To do it is discretized according to the Finite Difference Approximation of the second order derivative yielding the following set of equations for each grid point

$$\frac{1}{2\Delta z^2}(\phi_{i+1}+\phi_{i-1}-2\phi_{i}) = \frac{\rho_{i}}{\epsilon}$$

My question is the following: Is there any simple way to see how many points and thus what spacing one should choose to produce reliable results? Obviously you should choose more than 5 but should I choose 500, 5000 or 50000? Asked another way: Over what length scale will the electrostatic potential vary significantly? I suppose if this is known one should make the spacing smaller than this length scale.
I once did a similar thing for the Schrödinger Equation. In this case it was straightforward to see what the characteristic wavelength of the solutions were and then to choose the spacing much smaller than this.

 

[Fightfish]

Well, there is clearly no concept of a characteristic wavelength here, but what we know for sure is that the spatial variation of $\rho$ is going to be the most important factor, so the first step is of course to examine the charge density distribution in the region of interest.

Most of the time in numerical calculations like these, there is no simple way to determine to correct grid spacing to use etc. from the onset. Rather what is often employed is to look for convergence: if increasing the number of points doesn't change the result significantly (to the level of tolerance or error that you desire) then we are done.