# Unphysical Behaviour Characteristic-Wise WENO5-Z

I am currently working on a scheme that uses finite differences WENO5-Z with 3rd Order Runge-Kutta time integration for solving the Euler equations. The code projects the conserved variables and fluxes onto the eigenspace using Roe-averages at i+1/2 and uses a local Lax-Friedrichs flux splitting before applying the WENO reconstruction to the interface fluxes.

When I perform several numerical test (Sod Problem, Modified Sod, and Lax Problem) I get a spurious behavior in the left-traveling rarefaction wave (looks a lot like an expansion shock) in two of them, specifically in the Sod and Lax Problem. The lax problem also displays an oscillatory behavior in across the contact discontinuity.

I attached below the plots for the density, pressure, and velocity. I was wondering if any of you have encountered this issue before. I am surprised to see that kind of behavior since I am implementing Lax-Friedrichs flux splitting. Thank you very much in advance for your help! The projection on the characteristic space was being done only on the point-values of the vector of conserved variables, that is: $$V_i = L(u_{i+1/2})U_i$$ $$G_i = L(u_{i+1/2})F(U_i)$$
where $U_i$ is the vector of conserved variables, $F(U_i)$ is the vector of fluxes, and $L(u_{i+1/2})$ represents the left eigenvectors evaluated at $x_{i+1/2}$ using Roe-averages. Note that this is incorrect since the eigenprojection must be done for ALL the numerical stencil associate with $x_{i+1/2}$. The right expressions would be: $$V_k = L(u_{i+1/2})U_k$$ $$G_k = L(u_{i+1/2})F(U_k)$$ where for WENO5 $k= i-2,i-1,i,i+1,i+2,i+3$. I did not find this obvious when reading most of the references so I am attaching a quick diagram below for completeness. It is not intended to be rigorous but simply informative: Cheers!