I would like to apply Dirichlet conditions to the advection-diffusion equation using the finite-volume method. This answer, "How should boundary conditions be applied when using finite-volume method?" emphases the benefit of staying with integral form of the equations for as long as possible. This works fantastically for Robin boundary conditions because ghost cells nor interpolation is required.
Can Dirichlet conditions be applied to the integral form, or only the discretised equation?
For example, the advection-diffusion equation in flux form,
$$ u_t = -(\cal{F})_x + s $$
where,
$$ \mathcal{F}=au - du_x $$
After applying the finite volume method this becomes (in semi-discrete form),
$$ w_1^{\prime} = -\frac{\mathcal{F_{3/2}}}{h_1} + \frac{\mathcal{F_{1/2}}}{h_1} + \bar{s}_1 $$
Can Dirchlet conditions be applied to the flux term at the left hand side interface, $\mathcal{F_{1/2}}$, this would imply that we only know the advection part of the flux, i.e. the $au$ part where $u$ is known at the boundary.