Giesselmann, Jan ; LeFloch, Philippe G. (2016)
Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary.
doi: 10.48550/arXiv.1607.03944
Report, Bibliographie
Kurzbeschreibung (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.
| Typ des Eintrags: | Report |
|---|---|
| Erschienen: | 2016 |
| Autor(en): | Giesselmann, Jan ; LeFloch, Philippe G. |
| Art des Eintrags: | Bibliographie |
| Titel: | Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary |
| Sprache: | Englisch |
| Publikationsjahr: | 13 Juli 2016 |
| Verlag: | arXiV |
| Reihe: | Analysis of PDEs |
| Auflage: | 1. Version |
| DOI: | 10.48550/arXiv.1607.03944 |
| URL / URN: | http://arxiv.org/abs/1607.03944 |
| Kurzbeschreibung (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. |
| Zusätzliche Informationen: | Preprint; Erscheint auch in: Numerische Mathematik 144 (4), S. 751-785, Springer Verlag |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen |
| Hinterlegungsdatum: | 16 Okt 2018 11:14 |
| Letzte Änderung: | 29 Mai 2024 10:13 |
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