Lax-Wendroff and Godunov schemes for $u_t + (u^4)_x = 0$ 
Consider the nonlinear conservation law
  $$ u_t + (u^4)_x = 0 $$
  Write the following schemes for the equation in the form $u_j^{n+1} = F(u_{j-1}^{n},u_{j}^{n},u_{j+1}^{n})$.

Im trying to do for Lax-Wendroff and Godunov method.
Since this is advection, we know $u(x-a \Delta t, t^n) = u(x,t^{n+1})$. So I was thinking to interpolate the points $(x_{i}, u_i)$ at n-level for $i=j-1,j,j+1$ so as to obtain a quadratic and then using the above we may write a three-point stencil as asked. But, I don't really quite understand how to for Godunov. Any advice would be greatly appreciated.
 A: The present conservation law $u_t + f(u)_x = 0$ is nonlinear, with a convex flux $f(u)=u^4$. The derivative of the flux is $f'(u)=4u^3$, and the only solution of $f'(u_s) = 0$ is $u_s=0$. The Lax-Wendroff method is well-described in the Wikipedia article. The method can be written in conservation form $$u_{j}^{n+1} = u_{j}^n - \frac{\Delta t}{\Delta x} (F_{j+1/2}^n - F_{j-1/2}^n) $$ with the  numerical flux
$$
F_{j+1/2}^n = \frac{1}{2} \left({f(u_j^n) + f(u_{j+1}^n)}\right) - \frac12 \frac{\Delta t}{\Delta x} A_{j+1/2} \left(f(u_{j+1}^n)-f(u_{j}^n)\right) 
$$
where
$ 
A_{j+ 1/2}=f'\big(\tfrac12(u_{j}^n + u_{j+ 1}^n)\big)
$,
which is of the desired form.
Godunov's method is usually written in conservation form too, with the numerical flux (see (1) p. 228)
$$
F_{j+1/2}^n = \left\lbrace
\begin{aligned}
&f(u_j^n) & &\text{if}\quad u_j^n > u_s \;\text{and}\; s_{j+1/2} > 0 ,\\
&f(u_{j+1}^n) & &\text{if}\quad u_{j+1}^n < u_s\;\text{and}\; s_{j+1/2} < 0 ,\\
&f(u_s) & &\text{if}\quad u_{j}^n < u_s < u_{j+1}^n ,
\end{aligned}\right.
$$
where
$
s_{j+1/2} = [{f(u_{j+1}^n) - f(u_j^n)}]/[{u_{j+1}^n - u_{j}^n}]
$,
which is also of the desired form.

(1) R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge university press, 2002. doi:10.1017/CBO9780511791253
