Riemann solvers and Numerical Methods for Fluid Dynamics
 

Hi,
I have been struggling to understand a specific part of the derivation of the exact Riemann solver for the Euler equations presented in Toro's book (Riemann Solvers and Numerical Methods for Fluid Dynamics).
The particular issue starts on page 119, where the author presents the generic equation connecting two states (L and R), the iterative solution of which provides the value of p* (the pressure in the star region). The functions that appear in this equation depend on whether the wave is a shock or a rerafaction wave, but this is intriguingly defined by the condition that p*>pL or p*<pL (for example for the left wave, equation 4.6). In my understanding what defines a shock (or a rarefaction) is the convergance (divergence) of the eigenvectors, rather than the pressure.
I would very much appreciate your help with this.
Thanks,
Ricardo

you are correct if consider an initial smooth solution that, depending on the coalescence or not of the charactereistic curves (of the same family) generates shock or expansion, but the Riemann problem is defined already as a discontinuous initial state.

Thanks. Yes, I agree, but my point was: what determines whether a given wave (e.g. a 1-wave) is a shock should be lambda1L<lambda1*, and not pL<p* (where lambdas are the eigenvalues and 'p' the pressure). It is unclear (not demostrated in the book) if the former implies the latter.

Ricardo

If you consider only the initial state provided by the Riemann problem, the u velocity is zero, therefore you have only dx/dt=+/-a for the characteristic equations. For t=0+, depending on the initial state, you have some possible framework as reported in the file. Ideally, two shock waves are possible.

PS: of course the sound velocity a is different for the state 1 and 4 due to the difference in the pressure.

Thanks again for your kind follow up. I still cannot fully appreciate this, as for a generic Riemann problem, u is not necessarily zero. In fact the left and right states can have any values of u.
I appreciate comments.

Yes, of course, you can have a non-vanishing u but the reasoning does not change... Depending on the initial state, among the four cases, you can have also two expansion as shown in
http://oai.cwi.nl/oai/asset/10964/10964D.pdf

The Riemann problem in such case has no shock even for the discontinuous initial state

Thanks again. I think there might have been a bit of misinterpretation since my original question. The question was about a specific definition presented in the mentioned book by E. Toro. It states that whether a particular wave is a shock or rarefaction is defined by (for example for a left shock or rarefaction): p*<pL or p*>p where the star denotes the intermediate state within the Riemann fan and 'L' the left (initial) state. Why
p*<pL and not u*-a*<uL-aL?
The convergence or divergence of the eigenvalues is determined by the three variables [as a=f(rho, p)], rather than p alone.

I don'have this book here to check this issue...are you sure is discussed for the general case with non-vanishing velocity? Maybe some hypothesis on the ratio p/rho for the omoentropic case, I should check carefully..
I have in mind that the waves are totally defined by the initial state (no p* is analysed), I never go deeper in the idea of your question.... Maybe some infos can be found in the book of LeVeque..

Thanks. Yes, this is given when defining the exact Riemann solver, so it has to be generic enough to include any initial state. I read LeVeque's book, but the exact solver is not included.
This is also given in the paper.
A FAST RIEMANN SOLVER WITH CONSTANT COVOLUME APPLIED TO THE RANDOM CHOICE METHOD
E.F. TOR0

http://onlinelibrary.wiley.com/doi/1...650090908/epdf

Thanks for taking the time to discuss the issue.