Numerical solution of the Poisson equation

[ivantozavr]

TL;DR

How to solve the Poisson equation inside an electrostatic field source?

In my case, there is proton radiation acting on the material. Consequently, the protons get stuck in the sample and create an electrostatic field. I would like to solve the Poisson equation inside the sample. I assume that the medium is infinite and homogeneous, that is, the potential at infinity is zero (one-dimensional case):

Δφ(x) = - ρ(x) / ε_r ε₀
φ(±∞) = 0.​

The solution of such an equation is well known. That is (integration by range X' containing the source)

φ(x) = 1 / (4πε_r ε₀) ∫ dx' ρ(x') / |x-x'|.​

But then if I try to calculate the field inside the source itself, then singularities will occur due to the denominator in the integrand. I made small δx indents when calculating such an integral, i. e. sources with coordinates ζ: |ζ-x'| < δx were not taken into account when calculating an integral. This is hardly the right approach. Could you tell me how to solve such a problem correctly?

 

[Orodruin]

It is a perfectly valid approach if you insist on computing the result through the Green’s function. The contribution ignored goes as
$$\int_{0}^{\delta x} \frac{r^2 dr}r \propto \delta x^2$$
so for small $\delta x$ this should give negligible errors.

 

[ivantozavr]

It is a perfectly valid approach if you insist on computing the result through the Green's function. The contribution ignored goes as
$$\int_{0}^{\delta x} \frac{r^2 dr}r \propto \delta x^2$$
so for small $\delta x$ this should give negligible errors.

Thank you for your response!

Suppose that $\rho(x')\approx const$ in $|x' - x| < \delta x.$ Redefine the variable $x' - x = \xi.$ Wouldn't the error in a one-dimensional Cartesian system then be the same as below:
$$\delta\varphi \propto \int_{-\delta x}^{\delta x} \frac{d\xi}{|\xi|} = 2 \int_{0}^{\delta x} \frac{d\xi}{\xi}?$$
So the error tends to infinity?

In fact, I chose the solution via the green's function because I do not know how to construct a difference scheme for the Poisson equation with such boundary conditions.

 

[Orodruin]

Thank you for your response!

Suppose that $\rho(x')\approx const$ in $|x' - x| < \delta x.$ Redefine the variable $x' - x = \xi.$ Wouldn't the error in a one-dimensional Cartesian system then be the same as below:
$$\delta\varphi \propto \int_{-\delta x}^{\delta x} \frac{d\xi}{|\xi|} = 2 \int_{0}^{\delta x} \frac{d\xi}{\xi}?$$
So the error tends to infinity?

In fact, I chose the solution via the green's function because I do not know how to construct a difference scheme for the Poisson equation with such boundary conditions.

In a one-dimensional system the Green’s function is not divergent. It is a piecewise linear function.