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 (Δt2 + hm/Δt), when the exact solution belongs to Hm.
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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {573--595},
year = {2010},
publisher = {EDP Sciences},
volume = {44},
number = {3},
doi = {10.1051/m2an/2010012},
mrnumber = {2666655},
zbl = {1188.83010},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2010012/}
}
TY - JOUR AU - Bechouche, Philippe AU - Besse, Nicolas TI - Analysis of a semi-lagrangian method for the spherically symmetric Vlasov-Einstein system JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2010 SP - 573 EP - 595 VL - 44 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2010012/ DO - 10.1051/m2an/2010012 LA - en ID - M2AN_2010__44_3_573_0 ER -
%0 Journal Article %A Bechouche, Philippe %A Besse, Nicolas %T Analysis of a semi-lagrangian method for the spherically symmetric Vlasov-Einstein system %J ESAIM: Modélisation mathématique et analyse numérique %D 2010 %P 573-595 %V 44 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2010012/ %R 10.1051/m2an/2010012 %G en %F M2AN_2010__44_3_573_0
Bechouche, Philippe; Besse, Nicolas. Analysis of a semi-lagrangian method for the spherically symmetric Vlasov-Einstein system. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 3, pp. 573-595. doi: 10.1051/m2an/2010012
[1] , Difference equations and inequalities, Monographs and Textbooks in pure and applied mathematics. Marcel Dekker, New York, USA (1992). | Zbl
[2] and , A numerical investigation of stability states and critical phenomena for the spherically symmetric Einstein-Vlasov system. Class. Quant. Grav. 23 (2006) 3659-3677. | Zbl
[3] and , Regular compactly supported wavelets in Sobolev spaces. Duke Math. J. 87 (1996) 481-508. | Zbl
[4] , , , and , Two dimensional semi-Lagrangian Vlasov simulations of laser-plasma interaction in the relativistic regime. J. Plasma Phys. 62 (1999) 367-388.
[5] , Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system. SIAM J. Numer. Anal. 42 (2004) 350-382. | Zbl
[6] , 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
[7] and , Gyro-water-bag approch in nonlinear gyrokinetic turbulence. J. Comput. Phys. 228 (2009) 3973-3995.
[8] and , Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system. Math. Comp. 77 (2008) 93-123. | Zbl
[9] and , Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space. J. Comput. Phys. 191 (2003) 341-376. | Zbl
[10] , , , and , A Wavelet-MRA-based adaptive semi-Lagrangian method for the relativistic Vlasov-Maxwell system. J. Comput. Phys. 227 (2008) 7889-7916. | Zbl
[11] and , Plasmas physics via computer simulation. McGraw-Hill, USA (1985).
[12] and , The integration of the Vlasov equation in configuration space. J. Comput Phys. 22 (1976) 330-351.
[13] , Universality and scaling in gravitational collapse of a scalar field. Phys. Rev. Lett. 70 (1993) 9-12.
[14] and , Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry. Phys. Rev. D 65 (2001) 024007.
[15] , and , Critical behaviour in gravitational collapse of a Yang-Mills field. Phys. Rev. Lett. 77 (1996) 424-427. | Zbl
[16] , Problème de Cauchy pour le système intégro-différentiel d'Einstein-Liouville. Ann. Inst. Fourier 21 (1971) 181-201. | Zbl | Numdam
[17] , Numerical analysis of wavelet methods, Studies in mathematics and its applications 32. Elsevier, North-Holland (2003). | Zbl
[18] , Particle simulation of plasmas. Rev. Modern Phys. 55 (1983) 403-447.
[19] and , On the convergence for particle methods for multidimensional Vlasov-Poisson systems. SIAM J. Numer. Anal. 26 (1989) 249-288. | Zbl
[20] and , Convergence of a particle method for the relativistic Vlasov-Maxwell system. SIAM J. Numer. Anal. 28 (1991) 1-25. | Zbl
[21] and , 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
[22] and , Convergence of a Particle-In-Cell scheme for the spherically symmetric Vlasov-Einstein system. Ind. Un. Math. J. 52 (2003) 821-861. | Zbl
[23] , and , A regularity theorem for solutions of the spherical symmetric Vlasov-Einstein system. Commun. Math. Phys. 168 (1995) 467-478. | Zbl
[24] , and , Critical collapse of collisionless matter-a numerical investigation. Phys. Rev. D 58 (1998) 044007.
[25] , 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
[26] , Discrete approximation of the Poisson-Vlasov system. Quart. Appl. Math. 45 (1987) 59-73. | Zbl
[27] and , Relativistic stellar dynamics on computer I, Motivation and numerical methods. Astrophys. J. 298 (1985) 34-57.
[28] and , Relativistic stellar dynamics on computer II, Physical applications. Astrophys. J. 298 (1985) 58-79.
[29] and , Relativistic stellar dynamics on computer IV, Collapse of a stellar cluster to a black hole. Astrophys. J. 307 (1986) 575-592.
[30] and , Semi-Lagrangian integration schemes for atmospheric models-a review. Mon. Weather Rev. 119 (1991) 2206-2223.
[31] and , The convergence theory of particle-in-cell methods for multi-dimensional Vlasov-Poisson systems. SIAM J. Numer. Anal. 28 (1991) 1207-1241. | Zbl
[32] , and , The convergence analysis of fully discretized particle methods for solving Vlasov-Poisson systems. SIAM J. Numer. Anal. 28 (1991) 955-989. | Zbl
Cité par Sources :
