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