# Analysis of a semi-Lagrangian method for the spherically symmetric Vlasov-Einstein system

Philippe Bechouche; Nicolas Besse

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 3, page 573-595
- ISSN: 0764-583X

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topBechouche, Philippe, and Besse, Nicolas. "Analysis of a semi-Lagrangian method for the spherically symmetric Vlasov-Einstein system." ESAIM: Mathematical Modelling and Numerical Analysis 44.3 (2010): 573-595. <http://eudml.org/doc/250818>.

@article{Bechouche2010,

abstract = {
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
$\mathcal\{O\}$ (Δt2 + hm/Δt), when the exact solution belongs to Hm.
},

author = {Bechouche, Philippe, Besse, Nicolas},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Vlasov-Einstein system; semi-Lagrangian methods; convergence analysis; general relativity},

language = {eng},

month = {4},

number = {3},

pages = {573-595},

publisher = {EDP Sciences},

title = {Analysis of a semi-Lagrangian method for the spherically symmetric Vlasov-Einstein system},

url = {http://eudml.org/doc/250818},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Bechouche, Philippe

AU - Besse, Nicolas

TI - Analysis of a semi-Lagrangian method for the spherically symmetric Vlasov-Einstein system

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/4//

PB - EDP Sciences

VL - 44

IS - 3

SP - 573

EP - 595

AB -
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
$\mathcal{O}$ (Δt2 + hm/Δt), when the exact solution belongs to Hm.

LA - eng

KW - Vlasov-Einstein system; semi-Lagrangian methods; convergence analysis; general relativity

UR - http://eudml.org/doc/250818

ER -

## References

top- R.P. Agarwal, Difference equations and inequalities, Monographs and Textbooks in pure and applied mathematics. Marcel Dekker, New York, USA (1992).
- 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. Zbl1096.83035
- F. Bastin and P. Laubin, Regular compactly supported wavelets in Sobolev spaces. Duke Math. J.87 (1996) 481–508. Zbl0883.42026
- 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.
- N. Besse, Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system. SIAM J. Numer. Anal.42 (2004) 350–382. Zbl1071.82037
- 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. Zbl1168.82025
- N. Besse and P. Bertrand, Gyro-water-bag approch in nonlinear gyrokinetic turbulence. J. Comput. Phys.228 (2009) 3973–3995. Zbl1273.82071
- 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. Zbl1131.65080
- 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. Zbl1030.82011
- 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. Zbl1194.83013
- C.K. Birdsall and A.B. Langdon, Plasmas physics via computer simulation. McGraw-Hill, USA (1985).
- C.Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space. J. Comput Phys.22 (1976) 330–351.
- M.W. Choptuik, Universality and scaling in gravitational collapse of a scalar field. Phys. Rev. Lett.70 (1993) 9–12.
- M.W. Choptuik and I. Obarrieta, Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry. Phys. Rev. D65 (2001) 024007.
- 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.
- Y. Choquet-Bruhat, Problème de Cauchy pour le système intégro-différentiel d'Einstein–Liouville. Ann. Inst. Fourier21 (1971) 181–201. Zbl0208.14303
- A. Cohen, Numerical analysis of wavelet methods, Studies in mathematics and its applications32. Elsevier, North-Holland (2003).
- J.M. Dawson, Particle simulation of plasmas. Rev. Modern Phys.55 (1983) 403–447.
- K. Ganguly and H. Victory, On the convergence for particle methods for multidimensional Vlasov-Poisson systems. SIAM J. Numer. Anal.26 (1989) 249-288. Zbl0669.76146
- 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. Zbl0725.65124
- 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.] Zbl0774.53056
- 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. Zbl1080.83003
- 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. Zbl0830.35141
- G. Rein, A.D. Rendall and J. Schaeffer, Critical collapse of collisionless matter-a numerical investigation. Phys. Rev. D58 (1998) 044007.
- 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). Zbl0972.35168
- J. Schaeffer, Discrete approximation of the Poisson-Vlasov system. Quart. Appl. Math.45 (1987) 59–73. Zbl0646.65097
- S.L. Shapiro and S.A. Teukolsky, Relativistic stellar dynamics on computer I, Motivation and numerical methods. Astrophys. J.298 (1985) 34–57.
- S.L. Shapiro and S.A. Teukolsky, Relativistic stellar dynamics on computer II, Physical applications. Astrophys. J.298 (1985) 58–79.
- 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.
- A. Staniforth and J. Cote, Semi-Lagrangian integration schemes for atmospheric models-a review. Mon. Weather Rev.119 (1991) 2206–2223.
- 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. Zbl0741.65072
- 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. Zbl0777.65058

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