Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms

We consider nonlinear hyperbolic conservation laws, posed on a differential (n + 1)-manifold with boundary referred to as a spacetime, and in which the “flux” is defined as a flux field of n-forms depending on a parameter (the unknown variable). We introduce a formulation of the initial and boundary value problem which is geometric in nature and is more natural than the vector field approach recently developed for Riemannian manifolds. Our main assumption on the manifold and the flux field is a global hyperbolicity condition, which provides a global time-orientation as is standard in Lorentzian geometry and general relativity. Assuming that the manifold admits a foliation by compact slices, we establish the existence of a semi-group of entropy solutions. Moreover, given any two hypersurfaces with one lying in the future of the other, we establish a “contraction” property which compares two entropy solutions, in a (geometrically natural) distance equivalent to the L distance. To carry out the proofs, we rely on a new version of the finite volume method, which only requires the knowledge of the given n-volume form structure on the (n + 1)-manifold and involves the total flux across faces of the elements of the triangulations, only, rather than the product of a numerical flux times the measure of that face.