I'm using the Explicit Euler method to find numerical approximations to the following laser heat equation in cylindrical coordinates. The laser is aimed at the point $r = 0$, the center of the cylinder. Due to symmetry, there is no dependence on angle:
$$ \frac{\partial}{\partial t} T(r,z,t)= \kappa \left ( \frac{\partial^2 T}{\partial r^2} + \frac{\partial T}{r \partial r} + \frac{\partial^2 T}{\partial z^2} \right ) + \frac{1}{\rho c}S(r,z,t) $$
My first question is, how do I deal with the singularity at the boundary, $r = 0$?
This paper claims $\frac{\partial^2 T}{\partial r^2} \approx \frac{1}{r}\frac{\partial T}{\partial r} \approx 0$ at $r=0$, But this doesn't make sense to me for two reasons:
$1)$ Intuitively, if there's a heat source at $r = 0$ the heat should dissipate outward, therefore $\frac{\partial T}{\partial r} < 0$ at $r=0$.
$2)$ If we take $\frac{\partial T}{\partial r} < 0$ at $r=0$ to be true and discretize accordingly, there will be no information shared between $T(0,z,t)$ and $T(1,z,t)$. We end up with the nonsensical situation where if $T(0,z,t) = 0^{\circ}K$ and $T(1,z,t)=100^{\circ}K$, then $T(0,z,t+1) = 0^{\circ}K$.
Am I missing something? Is the paper wrong? Am I wrong? How do I deal with this $r=0$ singularity?
My second question is how do you discretize the heat equation with Neumann Boundary conditions $\frac{\partial T}{\partial z} = 0$ at $z=0$? The original heat equation does not have a $\frac{\partial T}{\partial z}$ term, so how do I incorporate it into a Forward Euler discretization? I'm having a hard time finding a good resource.
Thanks for your help!