Dear alvesker,
you don't have to write the FEM approximated form of each parameter http://www.cfdonline.com/Forums/vbL...ab3d0f451.gif is more accurate, though, I would suggest writing the the mean value of those parameters to simplify the weak form. about the integration on each time step it depends on the grid form you are using to split the domain. 
Dear seyedashraf,
Thanks for the quick reply. Can you explain what do you understand by the mean value? is it like the following? and how write it to the terms like where u, h are the velocity component and height field So if apply the FEM we have what is the average in the case when in the multiplication one of unknows one contain gradient (we have these in the advection formulation) 
the grid form  I use triangular grid

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