I am solving the Riemann problem for 1-D shallow water equations. I used the generalized Riemann invariants to obtain the values of the water depth and velocity in the star region. Do you think this is the right method? When I plotted the result I see 2 shock waves moving( with time) in opposite directions for both depth and velocity .
But as I kept reading the book(Shock capturing methods by E.F. Toro), it seems like one has to understand the Nature of the eigenvalues(wave types) of the system, before one would know the condition(s) to apply.
Do I need to know if each wave is a shock, rarefraction or shear wave before employing the relevant conditions?
Please kindly advise me.
Yes. Riemann invariants are so in case GNL waves are of rarefaction type. If one (or both) is of shock type, the entropy consistency requires that you solve with shock curves instead.
Read the first LeVeque's book (the one at Birkhauser) for learning more on these topics.
Thanks Prof Laurent. I know that for a scalar conservation law, I can use the concavity of the flux function to identify if a shock or rarefraction is present. Though am still reading, It seems more complicated to know the type of wave for a system of conservation laws. Please kindly give me a brief summary on how to identify the wave type for a system? Thanks.
You work with Lax entropy conditions. For instance let's say you want to know the nature of the 1-wave between Ul and Ur: you have a shock in case
\lambda(Ul)>\lambda(Ur) [it pushes stronger on the left than on the right].
In the opposite case, you have a rarefaction and it's ok working with riemann invariants. Look for LeVeque's book and reads chapters 7,8,9.
Dear Prof, thanks for your guide in this problem. I was able to obtain the results.
But I have this current problem.
I have just started studying the shallow water equations. I have solved and implemented the Riemann Problem for the 1-D case. I now want to implement a finite volume scheme for the same 1-D SWEs but the result I get seem totally different from the exact solution. I evaluated numerical fluxes F at boundary interfaces using F(U_) where U_ is the variable value at boundary cells and evaluated inter-cell fluxes using F(Ustar) where Ustar is a solution of the Riemann Problem with initial data from the average values of U in the bounding cells.
What could be the cause of the disparity?
I could paste the exact functions here, if you don't mind.
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