  We study nonlinear hyperbolic conservation laws posed on a differential
(n+1)-manifold with boundary referred to as a spacetime, and defined from a
prescribed flux field of n-forms depending on a parameter (the unknown
variable), a class of equations proposed by LeFloch and Okutmustur in 2008. Our
main result is a proof of the convergence of the finite volume method for weak
solutions satisfying suitable entropy inequalities. A main difference with
previous work is that we allow for slices with a boundary and, in addition,
introduce a new formulation of the finite volume method involving the notion of
total flux functions. Under a natural global hyperbolicity condition on the
flux field and the spacetime and by assuming that the spacetime admits a
foliation by compact slices with boundary, we establish an existence and
uniqueness theory for the initial and boundary value problem, and we prove a
contraction property in a geometrically natural L1-type distance.
