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I'm trying to find a transient temperature of a certain location of a 3D body, after a known perturbation. I'm solving the heat equation using finite differences. I've tried the explicit, implicit and Crank-Nicolson methods. I find that, even if all these methods converge equally, the implicit (and Crank-Nicolson) method calculates a different result for the same (transient state) point in time of the explicit method. Is there any error bound for this? I think that maybe I'm not stating this clearly enough, so I would really appreciate, besides answers, any comments requesting more information.

This is the exact problem: $ \frac{\partial u}{\partial t} = K \left(\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2}\right) $

being $\Omega \subset \mathbb{R}^3$ the spatial domain, $\ (x, y, z) \in \Omega$, $\ u(x, y, z, 0) = F(x, y, z)$ and $t \geq 0$.

Thanks in advance.

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    @Federico [Crank-Nicolson method](http://en.wikipedia.org/wiki/Crank%E2%80%93Nicolson_method) is numerically stable, and as long as the [Courant-Friedrichs number](http://en.wikipedia.org/wiki/Courant_number) is small (in this case $\mu = \frac{K\Delta t}{(\Delta x)^2}$), the scheme should work really well. If implicit, explicit and CN schemes are not giving the same answer for a sufficiently dense mesh, there is something wrong with the algorithms themselves. As Daryl suggested, try an explicitly solvable example to compare and figure out what's wrong.2012-11-23

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