Exercise 4: The Crank-Nicolson scheme for $u_t + a u_x = 0$ is given by $$ \frac{U_{j,n+1}-U_{j,n}}{\Delta t} + \frac{a}{2}\frac{D_xU_{j,n}}{2\Delta x} + \frac{a}{2}\frac{D_xU_{j,n+1}}{2\Delta x} = 0 .$$ Show that the LTE is given by $$ \mathcal{L}_\Delta u = au_{xxx} \left(\frac{1}{6} + \frac{p^2}{12}\right) {\Delta x}^2 + O({\Delta x}^3,{\Delta t}^3) , $$ where $p = a{\Delta t}/{\Delta x}$. Find the amplification factor and find the conditions for stability.
I have expanded this about a dozen times, I do not get the correct answer. Can someone please show me.
My working
$$\frac{ u(x , t + \delta ) - u( x,t)} { \delta t} + \frac{a}{2} \frac{u( x+ \delta x,t)-u(x-\delta x,t)}{ 2 \delta x} + \frac{a}{2} \frac{u ( x + \delta , t+ \delta t ) - u ( x- \delta x, t+ \delta t)} {2 \delta x}$$
expanding using taylor series I get
$$\frac{1}{\delta t} [ u + u_{t} \delta t + \frac{u_{tt}}{2} \delta t^{2} + O ( \Delta t^{3})] $$
$$ \frac{1}{\delta x^{2}} [ u_xx \delta x^{2} + \frac{1}{12} u_{xxxx} + O(\Delta x^{6})] $$
$$ u_{xx} + u_{xxt} \delta t + O(\Delta^{2}) +\frac{1}{12}u_{xxxx} \delta x^{2} + \frac{1}{12} u_{xxxxt} \delta x^{2} \delta t $$
these are the three terms I get after expanding for each one, however after rearranging, and using $u_t = -au_x$, I still am not able to get the correct answer.
I am hoping someone can show me as I really need to know this for my exam.
Thank you,
