I don't quite understand Sweeby's flux limited scheme to solve conservation laws.
\begin{equation} \frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x} = 0 \end{equation}
In particular, I don't really get how the expression of the Lax-Wendroff flux is derived and why it depends on the sign on the velocity.
Looking at the C.B. Laney book page 473, we assume the velocity $a(u)$ is positive and then we have the following lax-wendroff flux :
\begin{equation} \hat{f}^{LW}_{i+1/2} = f(u_i)+ \frac{1}{2}a_{i+1/2}^n\left(1-\lambda a_{i+1/2}^n\right)\left(u_{i+1}^n - u_{i}^n\right) \end{equation}
while it says page 474 that if $a(u) <0$ one has :
\begin{equation} \hat{f}^{LW}_{i+1/2} = f(u_i+1)- \frac{1}{2}a_{i+1/2}^n\left(1+\lambda a_{i+1/2}^n\right)\left(u_{i+1}^n - u_{i}^n\right) \end{equation}
how are those formulae determined ?
I do understand that the Lax-Wendroff scheme can be written as :
\begin{equation} u_{i}^{n+1} = u_i^n - \lambda\left(f(u_i)-f(u_{i-1)}\right) - \frac{\lambda}{2}\left(1-a_{i+1/2}^n\right)\left(f(u_{i+1})-f(u_i)\right) + \frac{\lambda}{2}\left(1-a_{i-1/2}^n\right)\left(f(u_{i})-f(u_{i-1})\right)\end{equation}
and that this equation can be written
\begin{equation} u_{i}^{n+1} = u_i^n + C^+_{i+1/2}\left(f(u_{i+1})-f(u_i)\right) - C^-_{i-1/2}\left(f(u_{i})-f(u_{i-1})\right) \end{equation}
applying wave splitting, with
\begin{eqnarray} C^+_{i+1/2} & =& - \frac{\lambda}{2}a_{i+1/2}\left(1-a_{i+1/2}^n\right)\\ C^-_{i+1/2} & =& \frac{\lambda}{2}a_{i+1/2}\left(1+a_{i+1/2}^n\right) \end{eqnarray}
This looks like the formulae for the fluxes are the above equation with either $C^+_{i+1/2}=0$ or $C^-_{i+1/2}=0$, but I don't see why $a(u)>0$ would be equivalent to $C^+_{i+1/2}=0$