I have the following question:
Say
$u_j^{n+1}=(1-2\alpha-2\beta)u^n_j+\alpha(u_{j+1}^n+u^n_{j-1})+\beta(u^n_{j+2}+u^n_{j-2})$
is a scheme for $u_t=u_{xx}$.
When $\Delta t/(\Delta x)^2$ and $\Delta t, \Delta x$ both tend to zero, how would you show that the scheme is only consistent when
$\alpha+4\beta=\Delta t/(\Delta x)^2?$
And, related, how do you show that it's fourth-order accurate in $x$ when
$\beta=\alpha/16?$
My "ideas" of solutions:
I know you could possibly use Von Neumann Analysis to determine the consistency, but I'm a bit confused about how this would work for this problem. Then, I'm guessing the fourth-order accuracy will follow from the analysis, but if it doesn't I'm not sure how I would show that part.
Any help would be greatly appreciated!
Thanks in advance.