# Amplification factor of Crank Nicolson scheme in cylindrical coordinates

When one considers the 1-D diffusion equation in cartesian coordinates

$$\frac{\partial u}{\partial t}=\chi\frac{\partial^2u}{\partial x^2},$$

one finds that the amplification factor for the Crank Nicolson scheme (with central differences in the spatial derivatives) is

$$A=\frac{1-2F\sin^2(k\Delta x/2)}{1+2F\sin^2(k\Delta x/2)},$$

where $$F=\chi\Delta t/(\Delta x)^2$$. Meaning that this method is unconditionally stable, i.e., the method is stable whatever $$F$$ is.

Now, let us consider the equation in cylindrical coordinates with a source term:

$$\frac{\partial u}{\partial t}=\chi\frac{1}{r}\frac{\partial}{\partial r}\left[r\frac{\partial u}{\partial r}\right]+S_{ext}$$

with boundary conditions $$\partial_rT(r=0)=0$$ and $$T(r=1)=T_a$$.

When $$S_{ext}=0$$, I get that the amplification factor is given by

$$A=\frac{1-\frac{F}{2}\left[4\sin^2\left(\frac{k\Delta r}{2}\right)-\frac{\sin(k\Delta r)}{q}\right]}{1+\frac{F}{2}\left[4\sin^2\left(\frac{k\Delta r}{2}\right)-\frac{\sin(k\Delta r)}{q}\right]}$$

where $$q$$ is the cell number. I am not sure if this is the correct result, namely it is strange to me the fact that it depends on $$q$$. Indeed, for small $$q$$ it is easy to see that $$A$$ may become larger than 1 for some of the wavenumbers $$k$$ and independently of $$F$$ .

Questions:

1. Does this mean that the method may become unstable?

2. When onde considers $$S_{ext}\neq0$$ the same result for $$A$$ applies?

3. And what about the accuracy in the $$S_{ext}\neq0$$ case? Because I have found that, with increasing $$S_{ext}$$, the deviation from the theoretical result increases a lot before the steady-state is reached.

• I am not sure what is your numerical scheme here. How do you deal with $r$ and $1/r$ in your scheme ? And what is $K_{ext}$ ? – Héhéhé Jul 19 '19 at 23:11
• I used the Crank Nicolson scheme. Regarding the $1/r$ at $r=0$, I make use of the boundary condition $\partial_ rT(r=0)=0$. Should I insert the discretized equation in my question? About the $K_{ext}$, that was a typo: I should have written $S_{ext}$. – AJHC Jul 20 '19 at 8:47
• Yeah I think you should insert it, because it is still not clear for me how you discretize $r$ and $1/r$. – Héhéhé Jul 20 '19 at 9:30

The Von Neumann stability analysis only works for linear PDEs with constant coefficients. It is not the case for your problem because of the $$r$$ and $$1/r$$ terms.