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I am solving a 1d Shock tube problem of compressible fluid obeying Euler equations(Hyperbolic pde). I am trying to simulate it using Finite Volume Method using Roe's Scheme. Half of the tube contains highly compressed fluid and other half is not compressed with density ratio of ~1000.

Euler Equation: $$ ∂Q/∂t + ∂F/∂x = 0; A = ∂F/∂Q$$

In Roe's Scheme, the parameters of velocity 'u' and Enthalpy 'H', at the interface are need for determining the 'A' a 3x3 matrix(for 1D Euler equation) . They are obtained by satisfying the following condition.

$$A (Q_\text{Left} - Q_\text{Right}) = \text{F}_\text{Left} - \text{F}_\text{Right}\ .$$

But the density, $\rho$ remains undetermined. While reading through literature, it was mentioned that it is 'Natural' to consider

$$\rho = \sqrt{\rho_\text{Left}.\rho_\text{Right}}\ .$$

My question is why is it so and what happens if we consider density to be a arithmetic mean(or some other mean) of the two density values.

I am trying to obtain Matrix 'A' for a different Equation of State(EoS) other than ideal gas EoS. If I choose a particular value of the density(calculated from density on either side of the face), the Roe's average values are greatly simplified.

EoS considered: $$ Pressure, P(\rho, E) = (\gamma-1).\rho E+g(\rho)$$

I was obtaining the density from following condition for simplification of equations for 'u' and 'H'. $$ dg(\rho)/d\rho = (g(\rho_\text{Left})-g(\rho_\text{Right}))/(\rho_\text{Left}-\rho_\text{Right}) $$

The reason for my question is enquire about the mathematical implications in choosing a density, other than 'Natural' square root value.

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  • $\begingroup$ I assume that EoS means Equation of State, but it might not be obvious for everybody. $\endgroup$ – nicoguaro♦ Aug 4 '17 at 13:41
  • $\begingroup$ Thank you @ nicoguaro for point it out this. By EoS, I meant Equation of State. I made the necessary edit in the question $\endgroup$ – Rakesh Srr Aug 4 '17 at 14:37
  • $\begingroup$ Welcome to Scicomp.SE! I have to agree with nicoguaro, your question as it is currently formulated is near impossible to understand (I literally have no idea what you are asking). Can you add more details such as what state are you actually modeling, how the density enters into the equation, and what your goal here is? (Note: I'm not voting to close for now, because I believe we should give some time -- say, 24 hours -- for simple clarifications, as well as some guidance, before calling for a time-out.) $\endgroup$ – Christian Clason Aug 4 '17 at 14:39
  • $\begingroup$ I have edited the question to include more information $\endgroup$ – Rakesh Srr Aug 4 '17 at 15:33
  • $\begingroup$ As far as I know, any matrix is suitable as long as it fulfills the conditions to be a Roe matrix. The impact of the choice of an average state is minimal, so you should feel free chose the most convenient. I'll see if I can find some references to back this before I post it as an answer. $\endgroup$ – Matthieu Aug 8 '17 at 19:00
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A detailed derivation for roe averages is provided in one of the earlier articles of Roe and Pike. Which I am skipping as the link to article is provided below.

On a more vague note, roe average of density is not assumed but rather derived from the Jacobian matrix. Literature you have read are probably second or third account which states it is natural to assume such a geometric mean. By placing a geometric mean it is easy to show the derivation of other means, but the right way to derive the averages is rather complicated and is provided in this article.

Take away: You will violate flux conservation and rule of hyperbolicity by playing around with averages that are not physical.

Roe, P.L.; Pike, J., Efficient construction and utilisation of approximate Riemann solutions, Computing methods in applied sciences and engineering VI, Proc. 6th Int. Symp., Versailles 1983, 499-518 (1984). ZBL0558.76001.

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  • $\begingroup$ Thank you Ramanathan. I have revisited the paper you mention. My current understanding is apart from the following equation:$$A (Q_\text{Left} - Q_\text{Right}) = \text{F}_\text{Left} - \text{F}_\text{Right}\ .$$. It has to also satify: $\Delta$($\rho$u) = $\Delta$($\rho).$u+$\rho$ $\Delta.$(u).. Let me know if my understanding is right. $\endgroup$ – Rakesh Srr Aug 15 '17 at 16:47
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That Roe-averaged density guarantees the exact projection of the solution jump onto the right eigenvectors. See http://ossanworld.com/cfdnotes/cfdnotes_roe_averaged_density.html

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