Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary

We study nonlinear hyperbolic conservation laws posed on a differential $$(n+1)$$ ( 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 (Far East J. Math. Sci. 31:49–83, 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 $$L^1$$ L 1 -type distance.

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