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.