have for the two cells encountering the boundary :

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

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

$\bullet \qquad \,\,\,\,\, $ The question is now to adapt the relation (1.2.14) in order to determine the two [**boundary fluxes**]{} $\,\, f_{0}^{n+1/2} \,\,$ at the left of the domain and $\,\,
f_{J}^{n+1/2} \,\,$ at the right. For the advection equation with celerity $\, a>0
,\,$ we have observed in the first section that some boundary condition $\, v_0(t)
\,$ has to be assigned at $\, x=0 \,$ and it is not the case for $\, x=L .\,$ It is therefore natural to take into account this information at the input of the domain and to set :

(2.2.5) $\qquad \displaystyle
f_{0}^{n+1/2} \,\, =\,\, {{1}\over{\Delta t}} \, 
\int_{\displaystyle t^n}^{\displaystyle t^{n\!+\!1}}  \!\!\!\! 
\!\! a \, v_0(t) \, {\rm d} t \,$

or simply

(2.2.6) $\qquad 
f_{0}^{n+1/2} \,\, =\,\,  \, a \, v_0\bigl( (n+{{1}\over{2}})  \Delta t \bigr)
\,,\qquad a > 0 \,, \,$

if function $\, \, 