December  2011, 4(4): 955-989. doi: 10.3934/krm.2011.4.955

Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

1. 

Centre de Recerca Matemática, Campus de Bellaterra, Edifici C, E-08193 Bellaterra, Barcelona, Spain

2. 

ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain

3. 

Division of Applied Mathematics, Brown University, Providence, RI 02912

Received  December 2010 Revised  July 2011 Published  November 2011

We construct a new family of semi-discrete numerical schemes for the approximation of the one-dimensional periodic Vlasov-Poisson system. The methods are based on the coupling of discontinuous Galerkin approximation to the Vlasov equation and several finite element (conforming, non-conforming and mixed) approximations for the Poisson problem. We show optimal error estimates for all the proposed methods in the case of smooth compactly supported initial data. The issue of energy conservation is also analyzed for some of the methods.
Citation: Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
References:
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show all references

References:
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[2]

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[6]

B. Ayuso, J. A. Carrillo and C.-W. Shu, Discontinuous Galerkin methods for the one-dimensional vlasov-poisson system, Technical Report 2009-41, Brown University, 2009. Available from: http://www.dam.brown.edu/scicomp/reports/2009-41/. Google Scholar

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B. Ayuso de Dios, J. A. Carrillo and C.-W. Shu, Discontinuous Galerkin methods for the multi-dimensional Vlasov-Poisson problem,, preprint CRM-UAB., ().   Google Scholar

[8]

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[9]

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[10]

N. Besse, Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 42 (2004), 350-382 (electronic). doi: 10.1137/S0036142902410775.  Google Scholar

[11]

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[12]

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[13]

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[14]

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[15]

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[16]

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[17]

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[18]

F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods," Springer Series in Computational Mathematics, 15, Springer-Verlag, New York, 1991.  Google Scholar

[19]

M. Campos Pinto and M. Mehrenberger, Convergence of an adaptive semi-Lagrangian scheme for the Vlasov-Poisson system, Numer. Math., 108 (2008), 407-444. doi: 10.1007/s00211-007-0120-z.  Google Scholar

[20]

J. A. Carrillo and F. Vecil, Nonoscillatory interpolation methods applied to Vlasov-based models, SIAM J. Sci. Comput., 29 (2007), 1179-1206 (electronic). doi: 10.1137/050644549.  Google Scholar

[21]

P. Castillo, B. Cockburn, I. Perugia and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38 (2000), 1676-1706 (electronic). doi: 10.1137/S0036142900371003.  Google Scholar

[22]

F. Celiker and B. Cockburn, Superconvergence of the numerical traces of discontinuous Galerkin and hybridized methods for convection-diffusion problems in one space dimension, Math. Comp., 76 (2007), 67-96 (electronic). doi: 10.1090/S0025-5718-06-01895-3.  Google Scholar

[23]

C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space, J. Computational Phys., 22 (1976), 330-351. doi: 10.1016/0021-9991(76)90053-X.  Google Scholar

[24]

Y. Cheng, I. Gamba, A. Majorana and C.-W. Shu, A discontinuous Galerkin solver for Boltzmann-Poisson systems in nano devices, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3130-3150. doi: 10.1016/j.cma.2009.05.015.  Google Scholar

[25]

P. G. Ciarlet, Basic error estimates for elliptic problems, in "Handbook of Numerical Analysis, Vol. II," Handb. Numer. Anal., II, 17-351, North-Holland, Amsterdam, 1991.  Google Scholar

[26]

B. Cockburn, J. Guzmán and H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comp., 78 (2009), 1-24. doi: 10.1090/S0025-5718-08-02146-7.  Google Scholar

[27]

B. Cockburn, G. Kanschat, I. Perugia and D. Schötzau, Superconvergence of the local discontinuous Galerkin method for elliptic problems on Cartesian grids, SIAM J. Numer. Anal., 39 (2001), 264-285 (electronic). doi: 10.1137/S0036142900371544.  Google Scholar

[28]

B. Cockburn, G. E. Karniadakis and C.-W. Shu, The development of discontinuous Galerkin methods, in "Discontinuous Galerkin Methods" (Newport, RI, 1999), 3-50, Lect. Notes Comput. Sci. Eng., 11, Springer, Berlin, 2000.  Google Scholar

[29]

B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comp., 52 (1989), 411-435.  Google Scholar

[30]

B. Cockburn and C.-W. Shu, The Runge-Kutta local projection $P_1$-discontinuous-Galerkin finite element method for scalar conservation laws, RAIRO Modél. Math. Anal. Numér., 25 (1991), 337-361.  Google Scholar

[31]

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463 (electronic). doi: 10.1137/S0036142997316712.  Google Scholar

[32]

B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems, J. Comput. Phys., 141 (1998), 199-224. doi: 10.1006/jcph.1998.5892.  Google Scholar

[33]

J. Cooper and A. Klimas, Boundary value problems for the Vlasov-Maxwell equation in one dimension, J. Math. Anal. Appl., 75 (1980), 306-329. doi: 10.1016/0022-247X(80)90082-7.  Google Scholar

[34]

G.-H. Cottet and P.-A. Raviart, Particle methods for the one-dimensional Vlasov-Poisson equations, SIAM J. Numer. Anal., 21 (1984), 52-76. doi: 10.1137/0721003.  Google Scholar

[35]

N. Crouseilles and F. Filbet, A conservative and entropic method for the Vlasov-Fokker-Planck-Landau equation, in "Numerical Methods for Hyperbolic and Kinetic Problems," 59-70, IRMA Lect. Math. Theor. Phys., 7, Eur. Math. Soc., Zürich, 2005.  Google Scholar

[36]

N. Crouseilles, G. Latu and E. Sonnendrücker, A parallel Vlasov solver based on local cubic spline interpolation on patches, J. Comput. Phys., 228 (2009), 1429-1446. doi: 10.1016/j.jcp.2008.10.041.  Google Scholar

[37]

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