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 $x_j$, 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 $U$, 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 ($r=2$) are $x_{j-1}, x_j, x_{j+1}, x_{j+2}$.
3) Then we reconstruct characteristic variable values by WENO method using $W$, $\Delta W$ 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 $F(a)$ and $F(b)$ 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...