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arXiv · 1607.03944

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

Jan Giesselmann·Philippe G. LeFloch
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Abstract

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.

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Jan Giesselmann, Philippe G. LeFloch. 2016-07-13. Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary. https://arxiv.org/abs/1607.03944

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