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)