The main idea behind the Finite Volume Method FVM is that what goes in must
come out and vice-versa.
If the domain is divided into a number of volumes, areas for 2D or lines for
1D, which are bounded by surfaces, lines in 2D and points in 1D, that are shared
between pairs of such sub volumes, then it is obvious that if something leaves
one volume through a shared surface then what just left will have gone into the
neighbor volume through that same shared surface.
The exception to this is the outer boundaries which are not shared between sub
volumes, but only belongs to one volume. At the boundaries, there need to be
boundary conditions as usual.
The full text is in the form of a PDF document that can be downloaded at the bottom of this page.