Convergence of the finite volume method on a Schwarzschild background
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 5, pp. 1459-1476.

We introduce a class of nonlinear hyperbolic conservation laws on a Schwarzschild black hole background and derive several properties satisfied by (possibly weak) solutions. Next, we formulate a numerical approximation scheme which is based on the finite volume methodology and takes the curved geometry into account. An interesting feature of our model is that no boundary conditions is required at the black hole horizon boundary. We establish that this scheme converges to an entropy weak solution to the initial value problem and, in turn, our analysis also provides us with a theory of existence and stability for a new class of conservation laws.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2019037
Classification : 35L60, 65M05, 76L05
Mots-clés : Hyperbolic conservation law, Schwarzschild black hole, weak solution, finite volume scheme, convergence analysis
Dong, Shijie 1 ; LeFloch, Philippe G. 1

1 
@article{M2AN_2019__53_5_1459_0,
     author = {Dong, Shijie and LeFloch, Philippe G.},
     title = {Convergence of the finite volume method on a {Schwarzschild} background},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1459--1476},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {5},
     year = {2019},
     doi = {10.1051/m2an/2019037},
     mrnumber = {3982969},
     zbl = {1435.35381},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2019037/}
}
TY  - JOUR
AU  - Dong, Shijie
AU  - LeFloch, Philippe G.
TI  - Convergence of the finite volume method on a Schwarzschild background
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2019
SP  - 1459
EP  - 1476
VL  - 53
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2019037/
DO  - 10.1051/m2an/2019037
LA  - en
ID  - M2AN_2019__53_5_1459_0
ER  - 
%0 Journal Article
%A Dong, Shijie
%A LeFloch, Philippe G.
%T Convergence of the finite volume method on a Schwarzschild background
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2019
%P 1459-1476
%V 53
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2019037/
%R 10.1051/m2an/2019037
%G en
%F M2AN_2019__53_5_1459_0
Dong, Shijie; LeFloch, Philippe G. Convergence of the finite volume method on a Schwarzschild background. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 5, pp. 1459-1476. doi : 10.1051/m2an/2019037. http://www.numdam.org/articles/10.1051/m2an/2019037/

P. Amorim, P.G. Lefloch and B. Okutmustur, Finite volume schemes on Lorentzian manifolds. Comm. Math. Sc. 6 (2008) 1059–1086. | DOI | MR | Zbl

Y. Bakhtin and P.G. Lefloch, Ergodicity of spherically symmetric fluid flows outside of a Schwarzschild black hole with random boundary forcing. Stoch PDE: Anal. Comp. 6 (2018) 746–785. | Zbl

T. Ceylan, P.G. Lefloch and B. Okutmustur, A finite volume method for the relativistic Burgers equation on a FLRW background spacetime. Commun. Comput. Phys. 23 (2018) 500–519. | DOI | MR | Zbl

B. Cockburn, F. Coquel and P.G. Lefloch, Convergence of finite volume methods for multi-dimensional conservation laws. SIAM J. Numer. Anal. 32 (1995) 687–705. | DOI | MR | Zbl

K.A. Dennison and T.W. Baumgarte, A simple family of analytical trumpet slices of the Schwarzschild spacetime, Class. Quant. Grav. 31 (2014) 117001. | DOI | Zbl

R.J. Diperna, Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal. 88 (1985) 223–270. | DOI | MR | Zbl

J. Giesselman and P.G. Lefloch, Formulation and convergence of the finite volume method for conservation laws on spacetimes with boundary. Preprint arXiv:1607.03944 (2016). | MR

D. Kröner, T. Müller and L.M. Strehlau, Traces for functions of bounded variation on manifolds with applications to conservation laws on manifolds with boundary. SIAM J. Math. Anal. 47 (2015) 3944–3962. | DOI | MR | Zbl

S. Kruzkov, First-order quasilinear equations with several space variables. Math. USSR Sb. 10 (1970) 217–243. | DOI | MR | Zbl

P.G. Lefloch and H. Makhlof, A geometry-preserving finite volume scheme for compressible fluids on Schwarzschild spacetime. Commun. Comput. Phys. 15 (2014) 827–852. | DOI | MR | Zbl

P.G. Lefloch, H. Makhlof and B. Okutmustur, Relativistic Burgers equations on curved spacetimes. Derivation and finite volume approximation. SIAM J. Numer. Anal. 50 (2012) 2136–2158. | DOI | MR | Zbl

P.G. Lefloch and B. Okutmustur, Hyperbolic conservation laws on spacetimes. A finite volume scheme based on differential forms. Far East J. Math. Sci. 31 (2008) 49–83. | MR | Zbl

P.G. Lefloch and S. Xiang, Weakly regular fluid flows with bounded variation on the domain of outer communication of a Schwarzschild black hole spacetime. J. Math. Pures Appl. 106 (2016) 1038–1090. | DOI | MR | Zbl

P.G. Lefloch and S. Xiang, Weakly regular fluid flows with bounded variation on the domain of outer communication of a Schwarzschild black hole spacetime. II. J. Math. Pure Appl. 122 (2019) 272–317. | DOI | MR | Zbl

Cité par Sources :