Lax-Wendroff method for nonlinear hyperbolic systems of conservation laws

Start with the Taylor expansion $$ u(x,t+k)=u(x,t)+u_t(x,t)k+\frac12u_{tt}(x,t)k^2+... $$ Use the PDE to convert time into space derivatives, $u_t=-[f(u)]_x$, then \begin{align} u_{tt}&=-[f(u)]_{xt}=-[f'(u)u_t]_x\\ &=[f'(u)[f(u)]_x]_x\\ \end{align} Now realize this derivative structure using symmetric divided differences and a half-step scheme. Let the index denote the offset in $x$ direction in units of $h$ $$ [f'(u)[f(u)]_x]_x=\frac{f'(u_{+1/2})[f(u_{+1/2})]_x-f'(u_{-1/2})[f(u_{-1/2})]_x}h $$ The $x$ derivatives at the half-steps are again approximated using central divided differences with step size $h/2$, $$ [f'(u)[f(u)]_x]_x=\frac{f'(u_{+1/2})\frac{f(u_{+1})-f(u)}h-f'(u_{-1/2})\frac{f(u)-f(u_{-1})}h}h $$ and finally the midpoint arguments of the derivative of $f$ by $u$ can be approximated by the mean of the neighboring integer grid values, $u_{\pm 1/2}=\frac12(u+u_{\pm 1})$.

If one inserts this in backwards, using the full-step central derivative for the degree-1 term, $[f(u)]_x=\frac{f(u_{+1})-f(u_{-1})}{2h}$ one gets back to the given formula of the method.