# How to compute WENO-reconstructed flux in local characteristic field?

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 August 20, 2017, 04:02 How to compute WENO-reconstructed flux in local characteristic field? #1 Member   Oleg Sutyrin Join Date: Feb 2016 Location: Russia Posts: 41 Rep Power: 9 I'm trying to apply finite-difference characteristic-wise WENO method to 1D Euler equations: Physical values are set at grid points , so we need to approximate fluxes at mid-points : where are sought-for approximation of physical fluxes . My algorithm is the following: 1) For each , we calculate simple average state and, using it, local eigenvalues , right eigenvector matrix and it's left counterpart . (Subscript will be omitted below) 2) Now we transform , its differences and flux differences to local characteristic field: It is done only for relevant grid points which in my case () are . 3) Then we reconstruct characteristic variable values by WENO method using , and . We get two "candidates": where are a convex combinations of corresponding polynomial functions obtained by WENO method. The next step would be to compute 2-point flux function using these reconstructed values. I'm using Lax-Friedrich's function: where is maximum value of eigenvalue (different for each of equations, since they are decoupled now). But how to compute these and using ? In component-wise approach it is simple: we reconstruct base values - , for example - and the use explicit formulas to compute and , but we don't have such formulas for because is expressed in local characteristic field...

 August 22, 2017, 06:03 #2 Member   Oleg Sutyrin Join Date: Feb 2016 Location: Russia Posts: 41 Rep Power: 9 It seems that I was misunderstanding how the flux is reconstructed: My idea was to reconstruct base values - (or if we are performing characteristic decomposition) - and then substitute them to formulas of (or ) to obtain reconstructed fluxes. It looks like the correct way is to apply reconstruction procedure to (or which is obtained by transformation ) itself. Some quick tests I performed with this approach show very plausible results.

 September 4, 2017, 14:24 #3 Senior Member   Join Date: Sep 2015 Location: Singapore Posts: 102 Rep Power: 10 Hi there, Here's what I am doing: after reconstructing the characteristic variables at the face, I use the left eigenvector matrix to convert them back to primitive variables. Then, I use these to compute the flux using the flux function, e.g. Lax-Friedrich method. It has performed well in several 1D test cases. USV

 Tags charateristic-wise, finite-difference, weno