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 and Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955
References:
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D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems,, SIAM J. Numer. Anal., 39 (): 1749.  doi: 10.1137/S0036142901384162.

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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/.

[7]

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

[8]

B. Ayuso and L. D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 47 (2009), 1391-1420. doi: 10.1137/080719583.

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I. Babuška and R. Narasimhan, The Babuška-Brezzi condition and the patch test: An example, Comput. Methods Appl. Mech. Engrg., 140 (1997), 183-199. doi: 10.1016/S0045-7825(96)01058-4.

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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.

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N. Besse, Convergence of a high-order semi-Lagrangian scheme with propagation of gradients for the one-dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 46 (2008), 639-670. doi: 10.1137/050635171.

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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.

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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.

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[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.

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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.

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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.

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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.

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show all references

References:
[1]

M. Adams, Discontinuous finite element transport solutions in thick diffusive problems, Nuclear Sci. Eng., 137 (2001), 298-333.

[2]

R. A. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[3]

S. Agmon, "Lectures on Elliptic Boundary Value Problems," Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr., Van Nostrand Mathematical Studies, No. 2., D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965.

[4]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760. doi: 10.1137/0719052.

[5]

D. N. Arnold, F. Brezzi, B. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems,, SIAM J. Numer. Anal., 39 (): 1749.  doi: 10.1137/S0036142901384162.

[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/.

[7]

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

[8]

B. Ayuso and L. D. Marini, Discontinuous Galerkin methods for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 47 (2009), 1391-1420. doi: 10.1137/080719583.

[9]

I. Babuška and R. Narasimhan, The Babuška-Brezzi condition and the patch test: An example, Comput. Methods Appl. Mech. Engrg., 140 (1997), 183-199. doi: 10.1016/S0045-7825(96)01058-4.

[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.

[11]

N. Besse, Convergence of a high-order semi-Lagrangian scheme with propagation of gradients for the one-dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 46 (2008), 639-670. doi: 10.1137/050635171.

[12]

N. Besse, F. Berthelin, Y. Brenier and P. Bertrand, The multi-water-bag equations for collisionless kinetic modeling, Kinet. Relat. Models, 2 (2009), 39-80. doi: 10.3934/krm.2009.2.39.

[13]

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 (electronic). doi: 10.1090/S0025-5718-07-01912-6.

[14]

C. K. Birdsall and A. B. Langdon, "Plasma Physics Via Computer Simulation," McGraw-Hill, New York, 1985.

[15]

R. Biswas, K. D. Devine and J. E. Flaherty, Parallel, adaptive finite element methods for conservation laws, in "Proceedings of the Third ARO Workshop on Adaptive Methods for Partial Differential Equations" (Troy, NY, 1992), Appl. Numer. Math., 14 (1994), 255-283. doi: 10.1016/0168-9274(94)90029-9.

[16]

F. Bouchut, F. Golse and M. Pulvirenti, "Kinetic Equations and Asymptotic Theory," Edited and with a foreword by Benoît Perthame and Laurent Desvillettes, Series in Applied Mathematics (Paris), 4, Gauthier-Villars, Éditions Scientifiques et Médicales Elsevier, Paris, 2000.

[17]

F. Brezzi, B. Cockburn, L. D. Marini and E. Süli, Stabilization mechanisms in discontinuous Galerkin finite element methods, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3293-3310. doi: 10.1016/j.cma.2005.06.015.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp., 48 (1987), 521-532. doi: 10.2307/2007825.

[38]

J. Dolbeault, An introduction to kinetic equations: The Vlasov-Poisson system and the Boltzmann equation, Current Developments in Partial Differential Equations (Temuco, 1999), Discrete Contin. Dyn. Syst., 8 (2002), 361-380. doi: 10.3934/dcds.2002.8.361.

[39]

J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods, in "Computing Methods in Applied Sciences" (Second Internat. Sympos., Versailles, 1975), 207-216, Lecture Notes in Phys., Vol. 58, Springer, Berlin, 1976.

[40]

J. Douglas, Jr., I. Martínez-Gamba and M. C. J. Squeff, Simulation of the transient behavior of a one-dimensional semiconductor device, Mat. Apl. Comput., 5 (1986), 103-122.

[41]

E. Fijalkow, A numerical solution to the Vlasov equation, Comput. Phys. Comm., 116 (1999), 319-328. doi: 10.1016/S0010-4655(98)00146-5.

[42]

F. Filbet, Convergence of a finite volume scheme for the Vlasov-Poisson system, SIAM J. Numer. Anal., 39 (2001), 1146-1169 (electronic). doi: 10.1137/S003614290037321X.

[43]

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