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| \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
| \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).