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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

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