$\,\, W({\scriptstyle \bullet} ,\,
{\scriptstyle \bullet}) \,\,$ in the cell $\,\, K_{j+1/2} \, : \,$

(2.1.8) $\qquad \displaystyle
W_{j+1/2}^{n} \,\,= \,\, {{1}\over{\mid  K_{j+1/2} \mid}} \,
\int_{\displaystyle x_{j}}^{\displaystyle x_{j\!+\!1}} \, W(x,\,t^n) \, {\rm d}x \,$

and the time mean value $\,\, f_{j}^{n+1/2} \,\,$ of the so-called flux $\, a \, W(
{\scriptstyle \bullet} ,\, {\scriptstyle \bullet}) \,\,$ at the space position $\,
x_{j} \,$ and between discrete times $\, t^n \,$ and $\, t^{n+1} \,: \, $

(2.1.9) $\qquad \displaystyle
f_{j}^{n+1/2} \,\,= \,\, {{1}\over{\Delta t}} \,
\int_ { \displaystyle  t^n }^{ \displaystyle  t^{n\!+\!1}} \, 
(a \,  W) (x_{j},\,t) \, {\rm d}t \,. \,$

Then we have the following constitutive relation of finite volumes schemes :

(2.1.10) $\qquad \displaystyle
{{1}\over{\Delta t}} \bigl( W_{j+1/2}^{n+1} -  W_{j+1/2}^{n} \bigr) \,\,+\,\, 
{{1}\over{\Delta x}} \bigl( f_{j+1}^{n+1/2} - f_{j}^{n+1/2} \bigr) \,\, = \,\, 0
\,.\,$

This numerical modelling characterizes the so-called finite volume method which has been developed thanks to the work of S. Godunov \[Go59\], Godunov et al \[GZIKP79\], Patankar \[Pa80\], Harten, Lax and Van Leer \[HLV83\] or Faille, Gallouët and Herbin \[FGH91\] among others.

$\bullet \qquad \,\,\, $ The [**proof of Proposition 2.1**]{} consists in a precise evaluation of the left hand side of equality (2.1.7). 