I am trying to discretize the heat equation in 1-D polar coordinates. I have:
$ \frac{\partial T}{\partial t}=\alpha\left( \frac{1}{r}\frac{\partial T}{\partial r} +\frac{\partial^2T}{\partial^2r} \right) $ with boundary condition as $-\frac{\partial T}{\partial r}\bigg|_{r=l}=\frac{h}{k}T$
Now there are a variety of methods to discretise this and try to handle the singularity at the center using axisymmetry etc. Discretising this only in the spatial domain I have a set of differential equations that give me the state space $\dot{x}=Ax$. Where $x_{i}$ is the temperature at the $i^{th} $discretized point that I have chosen.
Next, I am trying to measure the temperature at the left most point in this 1-D system. Hence I have the corresponding $C$ matrix in $y=Cx$
My problem is that if I increase the number of discretization points the system starts becoming unobservable. Physically the system should be observable and stays so for low values of discretization points.
What's going on here? Does some mathematical theorem exist that limits the number of discretization points?