CFD Online Discussion Forums - How to compute WENO-reconstructed flux in local characteristic field?

I'm trying to apply finite-difference characteristic-wise WENO method to 1D Euler equations:
$U_t + F(U)_x = 0, \; \;\text{where} \;\; U = (\rho, \rho u, e) \;\; \text{and} \;\;F(U) = (\rho u, \rho u^2+p, (e+p)u)$
Physical values are set at grid points

, so we need to approximate fluxes at mid-points $x_{j\pm \frac{1}{2}}$ :
$U_t \approx \frac{1}{h} \left[ \widehat{f}_{j+\frac{1}{2}} - \widehat{f}_{j-\frac{1}{2}} \right]$
where $\widehat{f}_{j\pm\frac{1}{2}}$ are sought-for approximation of physical fluxes $f(U(x_{j\pm\frac{1}{2}}))$ .

My algorithm is the following:

1) For each $x_{j+\frac{1}{2}}$ , we calculate simple average state $U_{j+\frac{1}{2}} = \frac{1}{2} (U_j + U_{j+1})$ and, using it, local eigenvalues $(u-a, u, u+a)_{j+\frac{1}{2}}$ , right eigenvector matrix $R_{j+\frac{1}{2}}$ and it's left counterpart $R^{-1}_{j+\frac{1}{2}}$ . (Subscript ${j+\frac{1}{2}}$ will be omitted below)

2) Now we transform

, its differences $\Delta U$ and flux differences $\Delta F(U)$ to local characteristic field:
$W=R^{-1}U, \;\; \Delta W=R^{-1}\Delta U, \;\; \Delta F(W)=R^{-1}\Delta F(U)$
It is done only for relevant grid points which in my case (

) are $x_{j-1}, x_j, x_{j+1}, x_{j+2}$ .

3) Then we reconstruct characteristic variable values by WENO method using

, $\Delta W$ and $\Delta F(W)$ . We get two "candidates":
$\widehat{W}^- = R_j(x_{j+\frac{1}{2}}) \; \text{and} \; \widehat{W}^+ = R_{j+1}(x_{j+\frac{1}{2}})$
where $R_j(x), R_{j+1}(x)$ are a convex combinations of corresponding polynomial functions obtained by WENO method.

The next step would be to compute 2-point flux function $h(\widehat{W}^-,\widehat{W}^+)$ using these reconstructed values. I'm using Lax-Friedrich's function:
$h(a,b) = \frac{1}{2} \left[ F(a) + F(b) + \alpha(b-a) \right]$
where $\alpha$ is maximum value of eigenvalue (different for each of equations, since they are decoupled now).

But how to compute these

and

using $\widehat{W}^\pm$ ? In component-wise approach it is simple: we reconstruct base values - $(\widehat{\rho}^\pm, \widehat{u}^\pm, \widehat{p}^\pm)$ , for example - and the use explicit formulas to compute $F(a) = (\widehat{\rho}^- \widehat{u}^-, \; \widehat{\rho}^- (\widehat{u}^-)^2 + \widehat{p}^-,\; ...)$ and $F(b) = (\widehat{\rho}^+ \widehat{u}^+, \; \widehat{\rho}^+ (\widehat{u}^+)^2 + \widehat{p}^+,\; ...)$ , but we don't have such formulas for $F(\widehat{W})$ because $\widehat{W}$ is expressed in local characteristic field...