I've the following explicit scheme in finite differences (for a one dimensional non uniform diffusion problem), being $k$ the time step, $h$ the space step, $A$ the thermal conductivity at position $i$ and $u_i^n$ the diffused quantity at position $i$ on time $n$:
$$u_i^{n+1} = \frac{k}{h^2}A_{i+1/2}u_{i+1} + \frac{k}{h^2}A_{i-1/2}u_{i+1} + [1 - \frac{k}{h^2}(A_{i+1/2} + A_{i-1/2})]u_i$$
We know that $k/h^2 > 0$ and if we assume that $1 - \frac{k}{h^2}(A_{i+1/2} + A_{i-1/2}) \geq 0$ then we can show that the stability condition is: $$ \frac{k}{h^2} \leq \frac{1}{\max(A_{i+1/2} + A_{i-1/2})}\ \forall i$$
I have a proof for this (I don't want to write it all here unless it is necessary) but I want to check it by asking if somebody has used this (or some similar) condition before.
Thanks in advance,
Federico
Looks very similar to the CFL condition somewhat generalized to a diffusion problem. You can rewrite your condition as $$ \frac{\max_i (A_{i+1/2} + A_{i - 1/2} )}{h} \frac{k }{h} \leq 1 $$ and if you define the first fraction as sort of a "diffusion speed", you have a very similar expression.
In the case of Finite Volume Schemes for hyperbolic conservation laws, you often find a CFL condition in the form of $$\max_i \bigg \vert f'\Big(u_i^{(n)}\Big) \bigg \vert \frac{\Delta t}{\Delta x} \leq \frac{1}{2} $$ which ensures that waves from neighboring cells do not interact. The same is actually ensured for the more precise/less restricted condition (similar to yours) like $$\max_i \Bigg \{ \bigg \vert f'\Big(u_i^{(n)}\Big) \bigg \vert + \bigg \vert f'\Big(u_{i-1}^{(n)}\Big) \bigg \vert \Bigg \} \frac{\Delta t}{\Delta x} \leq 1. $$