What implementation details need to change if I use a cell average approach rather than a cell total approach for the finite-volume method?
For example, consider the conservation law,
$$ u_t + \mathcal{F}_x = s(x,t) $$
A cell average approach yields,
$$ \frac{\partial}{\partial t}\int_{x_{j-1/2}}^{x_{j+1/2}} u(x,t)~dx = -\int_{x_{j-1/2}}^{x_{j+1/2}} F_x~dx + \int_{x_{j-1/2}}^{x_{j+1/2}}s(x,t)~dx \\ \frac{\partial}{\partial t} \tilde{u}(x,t) = \frac{1}{x_{j+1/2} - x_{j-1/2}}\left(\mathcal{F}_{j-1/2} - \mathcal{F}_{j+1/2} \right) + \tilde{s}(x,t) $$
The cell total is the same but without dividing the flux term by the cell length $(x_{j+1/2} - x_{j-1/2})$ (I am considering 1D only).
When solving the equation numerically, do I need to define $\tilde{u}$ and $\tilde{s}$ differently in these two cases?
A little confused about this subtlety. My first impression what that nothing needs to change other than the dividing the flux term by $(x_{j+1/2} - x_{j-1/2})$ when using a cell averaged approach. Is that correct?