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The diffusion equation is

∂u/∂t =∂^2u/∂x^2.

Consider using an explicit finite difference method to solve it. What does it mean for that explicit finite difference method to be conditionally stable?

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It means that the method will be stable only if certain criteria are meet. For explicit methods this means that the time-step must be lower than a certain value set by the mesh spacing and diffusion constant. There's a fair bit of analysis that I'm stepping over but you'll end up with a CFL (Courant-Friedrichs-Lewy)-like condition: $\Delta t \le \frac{\Delta x ^2}{2\alpha}$

where $\alpha$ is your diffusion constant, $\Delta x$ is your mesh spacing.

This should be contrasted against implicit methods which can be unconditionally stable. This means that the solution will not 'blow up' or go to infinity even if you choose an excessively large timestep. Note that this does not mean it'll be accurate with large timestep.

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