Editing Euler equations (fluid dynamics) (section)

===Finite volume form===
{{see also|Finite volume method}}
On the other hand, by integrating a generic conservation equation:

<math display="block"> \frac {\partial \mathbf y}{\partial t} + \nabla \cdot \mathbf F = \mathbf s,</math>

on a fixed volume ''V<sub>m</sub>'', and then basing on the [[divergence theorem]], it becomes:

<math display="block"> \frac {d}{dt} \int_{V_m} \mathbf y dV + \oint_{\partial V_m} \mathbf F \cdot \hat n ds = \mathbf S .</math>

By integrating this equation also over a time interval:

<math display="block"> \int_{V_m} \mathbf y(\mathbf r, t_{n+1}) \, dV - \int_{V_m} \mathbf y(\mathbf r, t_n) \, dV+ \int_{t_n}^{t_{n+1}} \oint_{\partial V_m} \mathbf F \cdot \hat n \, ds \, dt = \mathbf 0 .</math>

Now by defining the node conserved quantity:

<math display="block">\mathbf y_{m,n} \equiv \frac 1 {V_m} \int_{V_m} \mathbf y(\mathbf r, t_n) \, dV ,</math>

we deduce the finite volume form:

<math display="block">\mathbf{y}_{m,n+1}=\mathbf{y}_{m,n} - \frac{1}{V_m} \int_{t_n}^{t_{n+1}} \oint_{\partial V_m} \mathbf{F} \cdot \hat{n}\, ds \, dt .</math>

In particular, for Euler equations, once the conserved quantities have been determined, the convective variables are deduced by back substitution:

<math display="block">\begin{align} 
\displaystyle  \mathbf u_{m,n} &= \frac{\mathbf j_{m,n}}{\rho_{m,n}}, \\[1.2ex]
\displaystyle          e_{m,n} &= \frac{E^t_{m,n}}{\rho_{m,n}} - \frac{1}{2} u^2_{m,n}.
\end{align}</math>

Then the explicit finite volume expressions of the original convective variables are:{{sfn|Quartapelle|Auteri|2013|p=161|loc=par. 11.10: Forma differenziale: metodo dei volumi finiti}}

{{Equation box 1
|indent=:
|title='''Euler equations'''<br/>(''Finite volume form'')
|equation=<math>\begin{align} 
       \rho_{m,n+1} &= \rho_{m,n} - \frac{1}{V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m}\rho\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex]
  \mathbf u_{m,n+1} &= \mathbf u_{m,n} - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} (\rho\mathbf{u} \otimes \mathbf{u} - p\mathbf{I}) \cdot \hat{n}\, ds\, dt \\[1.2ex]
  \mathbf e_{m,n+1} &= \mathbf e_{m,n} - \frac{1}{2}\left(u^2_{m,n+1} - u^2_{m,n}\right) - \frac{1}{\rho_{m,n} V_m}\int_{t_n}^{t_{n+1}}\oint_{\partial V_m} \left(\rho e + \frac{1}{2}\rho u^2 + p\right)\mathbf{u} \cdot \hat{n}\, ds\, dt \\[1.2ex]
\end{align}</math>
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