Roe-averaged Density

Nishikawa

Hello,

Roe's approximate Riemann solver is a famous scheme in FVM fof Euler equations. But it "seems" that many people do not understand the importance of Roe-averaged density which is \sqrt{rho_L*rho_R}. I saw an article in a Japanese journal which claims that density average cannot be determined, and is arbitrary , so arithmetic average can also be used. Also, I found in the Dr. Laney's book (Computational Gasdynamics) an ambiguous statement; "It is convenient and traditional ...".

In my opinion, the square root average is CONSISTENT with the Roe linearization. In Roe's method, you first linearize the Euler equations using Roe-averages of u, and H (density is not necessary here). Then the linear hyperbolic system is solved for a Riemann problem "exactly" by using the fact that any jump in U can be represented as a combination of the right eigenvectors, When we project the jump into the space of eigenvectors, we find the amplitudes in each basis, which thus will be determined UNIQUELY. If you actually compute them, you'll find that the square root average will appear naturally (I actually did this).

What do you say?

Nishikawa