Scheme for $\displaystyle u_t = u_{xx}$:
$\displaystyle u_j^{n+1}= (1-2\alpha - 2\beta)u_j^n + \alpha(u_{j+1}^n + u_{j-1}^n) + \beta(u_{j+2}^n + u_{j-2}^n)$
Denote $\displaystyle \mu = \Delta t/(\Delta x)^2$. Show that when $\displaystyle \mu$ is a constant, that the scheme is inconsistent unless $\displaystyle \alpha + 4\beta = \mu$
Show that the scheme is four-order accurate in x if $\displaystyle \beta = -\alpha/16$