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I'm trying to derive the LTE for CN applied to the linear heat equation; $u_t = u_{xx}$.

The problem is that I end up with terms of the form $\frac{{\Delta t}^k}{{\Delta x}^2}$ when using a two dimensional Taylor expansion around $(x,t)$ for the term:

${\Delta x}^2 {\delta^2_x} = (u_{i+1}^{n+1} - 2 u_{i}^{n+1} + u_{i-1}^{n+1})$

What am I doing wrong?

(trying to prove LTE for the Crandall-Douglas scheme)

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  • $\begingroup$ I have the same problem! I did the expansion around x_i, t_n+1/2 and those terms cancel out and we have LTE h^2+k^2 but doing the expansion around any other point doesn't give the same result. What is wrong? Then when there is any method how do we say what is LTE if no point is specified? should not method have the same LTE around any point inside of the domain? thanks! $\endgroup$ – user9104 Apr 4 '11 at 19:04
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I don't know what is the $k$ in your $\frac{\Delta t^k}{\Delta x^2}$. And I cannot tell what's wrong in your result since you didn't provide enough details.

For the LTE (Local Truncation Error) of the C-N (Crank-Nicolson scheme) for the 1-$D$ heat equation, there is a complete discussion in Chapter 2.10 in the following book

Numerical Solution of partial Differential Equations (2nd Edition) by Morton and Mayers

where Crank-Nicolson scheme is called $\theta$ method where $\theta=\frac{1}{2}$.

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your taylor expansions should be around $\left(x_i,t_{n+1}\right)$

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