Density update

After calculating the update to the pressure (conservation of mass) and temperature (conservation of energy) the density is updated:

$ \rho_P ^{(m)}$ is obtained from $ p_P ^{(m)}$ and $ T_p ^{(m)} $:

$\displaystyle \rho_P ^{(m)} = \frac{p_P ^{(m)} }{r T_P ^{(m)} },$ (711)

whereas $ \overrightarrow{\rho }_f^{(m)}$ satisfies

$\displaystyle \overrightarrow{\rho }_f^{(m)} = \frac{\overrightarrow{p}_f ^{(m)} }{r \overline{T}_f ^{(m)} }.$ (712)