Analysis of the Crank-Nicolson scheme

The scheme $$[D_t u = \dfc D_xD_x \overline{u}^x]^{n+\half}_q$$ leads to $$A = \frac{ 1 - 2F\sin^2p}{1 + 2F\sin^2p}$$ $$u^n_q = \left(\frac{ 1 - 2F\sin^2p}{1 + 2F\sin^2p}\right)^ne^{ikp\Delta x}$$

Stability: The criteria $A>-1$ and $A < 1$ are fulfilled for any $\Delta t >0$