Analysis of a semi-lagrangian method for the spherically symmetric Vlasov-Einstein system
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 3, p. 573-595

We consider the spherically symmetric Vlasov-Einstein system in the case of asymptotically flat spacetimes. From the physical point of view this system of equations can model the formation of a spherical black hole by gravitational collapse or describe the evolution of galaxies and globular clusters. We present high-order numerical schemes based on semi-lagrangian techniques. The convergence of the solution of the discretized problem to the exact solution is proven and high-order error estimates are supplied. More precisely the metric coefficients converge in L∞ and the statistical distribution function of the matter and its moments converge in L2 with a rate of 𝒪t2 + hmt), when the exact solution belongs to Hm.

DOI : https://doi.org/10.1051/m2an/2010012
Classification:  65M15,  65P40,  83C05
Keywords: Vlasov-Einstein system, semi-lagrangian methods, convergence analysis, general relativity
@article{M2AN_2010__44_3_573_0,
     author = {Bechouche, Philippe and Besse, Nicolas},
     title = {Analysis of a semi-lagrangian method for the spherically symmetric Vlasov-Einstein system},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {3},
     year = {2010},
     pages = {573-595},
     doi = {10.1051/m2an/2010012},
     zbl = {1188.83010},
     mrnumber = {2666655},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_3_573_0}
}
Bechouche, Philippe; Besse, Nicolas. Analysis of a semi-lagrangian method for the spherically symmetric Vlasov-Einstein system. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 3, pp. 573-595. doi : 10.1051/m2an/2010012. http://www.numdam.org/item/M2AN_2010__44_3_573_0/

[1] R.P. Agarwal, Difference equations and inequalities, Monographs and Textbooks in pure and applied mathematics. Marcel Dekker, New York, USA (1992). | Zbl 0925.39001

[2] H. Andréasson and G. Rein, A numerical investigation of stability states and critical phenomena for the spherically symmetric Einstein-Vlasov system. Class. Quant. Grav. 23 (2006) 3659-3677. | Zbl 1096.83035

[3] F. Bastin and P. Laubin, Regular compactly supported wavelets in Sobolev spaces. Duke Math. J. 87 (1996) 481-508. | Zbl 0883.42026

[4] M.L. Bégué, A. Ghizzo, P. Bertrand, E. Sonnendrücker and O. Coulaud, Two dimensional semi-Lagrangian Vlasov simulations of laser-plasma interaction in the relativistic regime. J. Plasma Phys. 62 (1999) 367-388.

[5] N. Besse, Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system. SIAM J. Numer. Anal. 42 (2004) 350-382. | Zbl 1071.82037

[6] N. Besse, Convergence of a high-order semi-Lagrangian scheme with propagation of gradients for the Vlasov-Poisson system. SIAM J. Numer. Anal. 46 (2008) 639-670. | Zbl 1168.82025

[7] N. Besse and P. Bertrand, Gyro-water-bag approch in nonlinear gyrokinetic turbulence. J. Comput. Phys. 228 (2009) 3973-3995. | Zbl pre05566332

[8] N. Besse and M. Mehrenberger, Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system. Math. Comp. 77 (2008) 93-123. | Zbl 1131.65080

[9] N. Besse and E. Sonnendrücker, Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space. J. Comput. Phys. 191 (2003) 341-376. | Zbl 1030.82011

[10] N. Besse, G. Latu, A. Ghizzo, E. Sonnendrücker and P. Bertrand, A Wavelet-MRA-based adaptive semi-Lagrangian method for the relativistic Vlasov-Maxwell system. J. Comput. Phys. 227 (2008) 7889-7916. | Zbl 1194.83013

[11] C.K. Birdsall and A.B. Langdon, Plasmas physics via computer simulation. McGraw-Hill, USA (1985).

[12] C.Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space. J. Comput Phys. 22 (1976) 330-351.

[13] M.W. Choptuik, Universality and scaling in gravitational collapse of a scalar field. Phys. Rev. Lett. 70 (1993) 9-12.

[14] M.W. Choptuik and I. Obarrieta, Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry. Phys. Rev. D 65 (2001) 024007.

[15] M.W. Choptuik, T. Chmaj and P. Bizoń, Critical behaviour in gravitational collapse of a Yang-Mills field. Phys. Rev. Lett. 77 (1996) 424-427. | Zbl 0986.83030

[16] Y. Choquet-Bruhat, Problème de Cauchy pour le système intégro-différentiel d'Einstein-Liouville. Ann. Inst. Fourier 21 (1971) 181-201. | Numdam | Zbl 0208.14303

[17] A. Cohen, Numerical analysis of wavelet methods, Studies in mathematics and its applications 32. Elsevier, North-Holland (2003). | Zbl 1038.65151

[18] J.M. Dawson, Particle simulation of plasmas. Rev. Modern Phys. 55 (1983) 403-447.

[19] K. Ganguly and H. Victory, On the convergence for particle methods for multidimensional Vlasov-Poisson systems. SIAM J. Numer. Anal. 26 (1989) 249-288. | Zbl 0669.76146

[20] R.T. Glassey and J. Schaeffer, Convergence of a particle method for the relativistic Vlasov-Maxwell system. SIAM J. Numer. Anal. 28 (1991) 1-25. | Zbl 0725.65124

[21] G. Rein and A.D. Rendall, Global existence of solutions of the spherically symmetric Vlasov-Einstein with small initial data. Commun. Math. Phys. 150 (1992) 561-583. [Erratum. Comm. Math. Phys. 176 (1996) 475-478.] | Zbl 0847.53062

[22] G. Rein and T. Rodewis, Convergence of a Particle-In-Cell scheme for the spherically symmetric Vlasov-Einstein system. Ind. Un. Math. J. 52 (2003) 821-861. | Zbl 1080.83003

[23] G. Rein, A.D. Rendall and J. Schaeffer, A regularity theorem for solutions of the spherical symmetric Vlasov-Einstein system. Commun. Math. Phys. 168 (1995) 467-478. | Zbl 0830.35141

[24] G. Rein, A.D. Rendall and J. Schaeffer, Critical collapse of collisionless matter-a numerical investigation. Phys. Rev. D 58 (1998) 044007.

[25] T. Rodewis, Numerical treatment of the symmetric Vlasov-Poisson and Vlasov-Einstein system by particle methods. Ph.D. Thesis, Mathematisches Institut der Ludwig-Maximilians-Universität München, Munich, Germany (1999). | Zbl 0972.35168

[26] J. Schaeffer, Discrete approximation of the Poisson-Vlasov system. Quart. Appl. Math. 45 (1987) 59-73. | Zbl 0646.65097

[27] S.L. Shapiro and S.A. Teukolsky, Relativistic stellar dynamics on computer I, Motivation and numerical methods. Astrophys. J. 298 (1985) 34-57.

[28] S.L. Shapiro and S.A. Teukolsky, Relativistic stellar dynamics on computer II, Physical applications. Astrophys. J. 298 (1985) 58-79.

[29] S.L. Shapiro and S.A. Teukolsky, Relativistic stellar dynamics on computer IV, Collapse of a stellar cluster to a black hole. Astrophys. J. 307 (1986) 575-592.

[30] A. Staniforth and J. Cote, Semi-Lagrangian integration schemes for atmospheric models-a review. Mon. Weather Rev. 119 (1991) 2206-2223.

[31] H.D. Victory and E.J. Allen, The convergence theory of particle-in-cell methods for multi-dimensional Vlasov-Poisson systems. SIAM J. Numer. Anal. 28 (1991) 1207-1241. | Zbl 0741.65072

[32] H.D. Victory, G. Tucker and K. Ganguly, The convergence analysis of fully discretized particle methods for solving Vlasov-Poisson systems. SIAM J. Numer. Anal. 28 (1991) 955-989. | Zbl 0777.65058