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For scalar conservation laws Roe scheme gives non entropic weak solutions when Riemann problem has transonic rarefaction. We replace the Roe flux with HLL when the left and right states admits a transonic rarefaction to make the scheme entropic. (For shock case Roe and Godunov fluxes are same and hence scheme is entropic)

However for strictly hyperbolic systems of conservation laws, when Riemann problem has single schock solution (i.e simple wave solution which is a shock) the Roe flux and Godunov fluxes are same.

How to fix the flux in the other cases so that the scheme becomes entropic? Please explain me how the Harten and Hymen entropy fix makes the scheme entropic, when the solution is not a simple shock wave solution.

Is there any relations between eigen values of Roe matrix and Jacobian matrix? As far as I understand the modification is based on the difference of these eigen values.

I am reading LeVeque's book and papers by Harten and Hymen but I am not understanding.

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