I have to solve a diffusion probleme, radial diffusion into or out of a sphere. So far so good, I have the explicit solution via difference quotient. But I have trouble evaluation the stability analysis. Compared to linear diffusion, where I an summorize the exponential terms in to a cosine, thats not the for the radial diffusion in a sphere since there is an addition first derrivative included.
So my solution so far is:
- The time dependend concentration for a sphere is given by:
$C_{r}^{t+\delta t}=C_{r}^{t}+D\delta t*\left(\frac{ C_{r+\delta r}^t -2C_{r}^t+C_{r-\delta r}^t}{\delta r^2}+\frac{2}{r} \frac{C_{r+\delta r}^t-C_r^t}{\delta r}\right)$
Putting von Neumann in there results in:
$\xi=1+\frac{2D\delta t}{r \delta r^2}\left[r\left(cos(k \delta r)-1 \right)+\delta r e^{ik\delta r}-\frac{1}{2} \right]$
I new of problems with the interpretation of the second term $\delta r e^{ik\delta r}$. Is there a generall solution like $D\frac{\delta t}{\delta x^2}\leq\frac{1}{2}$ for the linear problem? Or do I have to solve it numerically in the case I want to solve for $\delta r$??
Many thanks in advance