In deriving the Roe Jacobian matrix, I can see that the system is always under-determined, what then is the right way of finding the approx (Roe) Jacobian matrix?
To be specific, In chapter 11 of the book ""Riemann solvers and numerical methods for fluid dynamics: A practical introduction, 2nd edition"" by E. F. Toro, equations (11.38) or equation (11.39) needs to be solved to obtain matrices A and B . In equation (11.38), I can only see 4 equations but 8 unknowns(4 for matrix A and 4 for B). If we consider equation (11.39), we see 2 equations but 4 unknowns, so how did he obtain A and B as in equation (11.41)?
Even the papers will only quote results for Roe matrix, how are these matrices derived? I want to know the trick!!!
If I can get an explanation to equation(11.41) of Toro's book, then I can fix the rest.
Finally, I have figured it out. Just mare manipulation. But Toro's book didn't do any better in explaining while Leveque's book gives a true picture at a glance.
Thanks to his(Leveque's) book on Numerical methods for conservation laws.
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