I want to show that for the Crank-Nicolson method, the following holds:

$|T_i^{n+\frac{1}{2}}| \leq \frac{\tau^2}{12} M_{ttt}+\frac{h^2}{12}M_{xxxx}$,

where $M_{ttt}=||u_{ttt}||_{\infty}, M_{xxxx}=||u_{xxxx}||_{\infty}$

and $T_{i}^{n+\frac{1}{2}}=\frac{u(t_{n},x_i)-u(t_{n-1},x_i)}{\tau}-\frac{1}{2} \frac{u(t_{n+1},x_{i+1})-2u(t_{n+1},x_i)+u(t_{n+1},x_{i-1})}{h^2}-\frac{1}{2} \frac{u(t_n,x_{i+1})-2u(t_n,x_i)+u(t_n,x_{i-1})}{h^2}$

I found the following using Taylor expansions:


- $\frac{u(t_n,x_i)-u(t_n-\tau,x_i)}{\tau}=u_t(t_n,x_i)-\frac{\tau}{2}u_{tt}(t_n,x_i)+\frac{\tau^2}{6}u_{ttt}(\…