Finite volume method, "philosophical" problem

Ciao,

I have a doubt about finite volume method "philosophy". Here the
unknowns are the field variables averaged over control volumes, not
their nodal values. Most people, however, forget about this when
writing the numerical fluxes across control volumes boundaries, and
treat the unkwons as point values.
In order to avoid this, in 1D we can write the fluxes across boundaries
in terms of the antiderivative of the field variable with respect to x.
For example, if dealing with inviscid Burgers equation:

u_t+(u^2/2)_x = 0

we can write the fluxes across the boundaries of a control volume in
terms of nodal values of H = \int_0^x u \, dx. This is simple because
nodal values of H are simply sums, weighted with the measure of control
volumes, of the values of u averaged of control volumes. i.e. our
unknowns.
Do you know of papers where this method is used? Is it possible to
extend it to 2D and 3D problems and unstructured grids? I ask this
because multidimensional integrals are very different from 1D
integrals. For example, there is no fundamental theorem of calculus for
3D integrals, so it is not clear to me if it is possible to talk about
some sort of "antiderivative" in a well defined sense, or not.
Thanks in advance for your help,

greetings,

Andrea

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