# WENO methods: why the characteristic wise method resulting big errors?

I was doing my research/project using WENO as the limiter in finite volume methods to solve hyperbolic conservation law. I have no idea why the result in the characteristic wise method has a big error compared to component-wise. This is my current solution that I get from my solver, it shows that there is a numerical issue in characteristic wise reconstruction.

Velocity in $x$ direction: solution of sod shock at $t=0.1$:

My algorithm of characteristic wise reconstruction (I am using 5th order WENO). Let we evaluate cell $i$:

1. Using ROE averaging, get averaged value $\bar q_{i+\frac{1}{2}} = f(q_{i},q_{i+1})$ and compute left and right eigenvector
2. Transform variable $q$ to characteristic variable $v$ using left eigenvector, because 5 points is needed so we need $q_{j} = [q_{i-2},q_{i-1},q_{i},q_{i+1},q_{i+2}]$, and then $v_j = L(\bar q_{i+\frac{1}{2}})*q_{j}$
3. Perform WENO reconstruction
4. Transform back to physical space using $q^{\pm}_{i\mp\frac{1}{2}} = R(\bar q_{i+\frac{1}{2}})*v^{\pm}_{i\mp\frac{1}{2}}$, so we get $\ q^{+}_{i-\frac{1}{2}}$ and $q^{-}_{i+\frac{1}{2}}$
5. While at cell interface interface $i+\frac{1}{2}:\$ $q^{+}_{i+\frac{1}{2}}=q^{+}_{(i-\frac{1}{2})+ 1},\ q^{-}_{i+\frac{1}{2}}=q^{-}_{i+\frac{1}{2}}$.
6. Compute flux $F_{i+\frac{1}{2}} = f(q^{-}_{i+\frac{1}{2}},q^{+}_{i+\frac{1}{2}})$

Is there any inconsistency in my algorithm?

• Help please... Still don't have solutions for this problem – Mr. Robot Sep 30 '18 at 14:59