Transient term

$\displaystyle \frac{\partial }{\partial t} \int_{V}^{} \rho v_i dv \approx \rho_P^{(m-1)} V_P \frac{ [v_i^{(m)}-v_i^{n-1}] }{\Delta t},$ (596)

where

$\displaystyle \rho_P^{(m-1)} = \frac{p_P^{(m-1)}}{r T_P^{(m-1)}}.$ (597)

$ p$ is the static pressure, $ T$ is the static temperature and $ r$ is the gas constant. The derivative was replaced by a backward Euler scheme. Notice that quantities such as pressure, temperature and density are taken from iteration $ (m-1)$. Therefore, the scheme will not be unconditionally stable (although this is a property of the backward Euler scheme).