Vtyjkyuk

I'm trying to figure out how the finite volume version of Lax-Wendroff scheme is derived.

Here is the PDE and Lax-Wendfroff scheme:
$$u=textfunction of x,t\hatu=frac1Delta xint_x_i-1/2^x_i+1/2uthinspace dx text (the average flux through volume)$$
$$fracpartial upartial t=-fracpartial f(u)partial xrightarrow \hatu_j^n+1=hatu_j^n - fracDelta tDelta x(F_j+1/2^n-F_j-1/2^n)\
F_j+1/2=frac12thinspace (f_j+1+f_j)-frac12thinspace a^2_j+1/2fracDelta tDelta xthinspace (hatu_j+1-hatu_j)\
a_j+1/2=begincases
fracf_j+1-f_jhatu_j+1-hatu_j & if enspace hatu_j+1neq hatu_j \
f'(u_j) & if enspace hatu_j+1=hatu_j
endcases$$

I know that in a finite difference Lax-Wendroff is derived like this :

$$u_t=-cu_x rightarrow u_tt=c^2u_xx\
text taylors expansion :thinspace u(t+Delta t, x)= u+Delta tthinspace u_t + fracDelta t^22u_tt rightarrow \u^m+1_n=u^m_n-cDelta t thinspace u_x+fracc^2Delta t^22u_xx$$

I know in finite volume we are measuring the average flux of $u$, so I attempt to get the equation into the right form using $hatu$ by dividing by $Delta x$ and taking an integral with respect to x.

$$hatu_n^m+1=hatu_n^m-fraccDelta tDelta x(u(x_i+1/2,t)-u(x_i-1/2,t)+fracc^2Delta t^22Delta x^2u_x|_x_i-1/2^x_i+1/2$$

Not sure what to do though from here.

I also tried just expanding $f$ like this :
$$
f(u(x+fracDelta x2))=f(u(x))+fracDelta x2f_u u_x+(fracDelta x2)^2(f_uuu_x^2+f_u u_xx)
$$

I don't see where this is going either. I just can't find it derived after a ton of google searching. Can anyone show me or link me to a derivation?