Now I meet with some difficulties in Roe-avarage flux difference splitting solving the fluxs.At sonic points there is wrong entropy pionts.so I have to fix the entropy in Roe scheme.but I don't know how to modify it in the vector systems .ie.for Euler Equations.please give me the method in details.thank you very much !!

If I have correctly understood your problem, a possible answer might be to add a perturbation to absolute values of the jacobian matrix eigenvalues.

Namely, lambda being such eigenvalue, instead of using |lambda|, replace it with sqrt[(1+epsilon)*lambda^2], epsilon being a small parameter you can tune accordingly to your needs.

I think you could also use the Harten and Hyman Entropy fix.

You use for |lambda| = max ( |lambda| , eps)

where eps = max (0 , (lambda(i+1/2)-lambda(i) , (lambda(i+1)- lambda(i+1/2) )

So you only use the correction on expansion shocks and not on compression shocks.

Sandro

Thank you very much I still can't undeerstand the relationship between roe characteristic speed and eiginvalues of Jacobian in the hypobolic conservative systems.for example 1) lamda = f'(u) ??? 2) lamda1<lamda2=lamda3<lamda4 ?? this sequence must be adopted? 3) Roe scheme is apllied to the conservative variables ENO scheme with FVM spatial discretion for me .HOW to modify the lamda without change the FVM? please give me a detailed interpretaion.i am eager for it in my master thesis .thank you

You are absolutely right. However I just intended to sketch a simple mean to fulfil the thread purpose. I hope I haven't confused anyone with my previous post.

The entropy fix problem is always annoying as, as far as I know, there's no "ultimate entropy fix user guide" around. If anyone has a nice reference, just let me know.