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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.

share|improve this question
    
Please typically tell us the PDE problem. –  doraemonpaul Nov 22 '12 at 22:15
    
The problem is linear, so you should be able to obtain an exact solution (albeit possibly difficultly), to compare with your numerical methods. –  Daryl Nov 23 '12 at 0:12
    
What exactly is $\Omega$? –  Pragabhava Nov 23 '12 at 0:13
    
$\Omega$ is the spatial domain. Probably my notation is poor, please correct it if you feel it should be corrected. –  Federico Nov 23 '12 at 0:31
    
@Federico I meant what domain is it? The unitary disk? The shape of the Everest? –  Pragabhava Nov 23 '12 at 0:40

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