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:
[1]

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

[2]

R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975). Google Scholar

[3]

S. Agmon, "Lectures on Elliptic Boundary Value Problems,", Prepared for publication by B. Frank Jones, (1965). Google Scholar

[4]

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

[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. Google Scholar

[6]

B. Ayuso, J. A. Carrillo and C.-W. Shu, Discontinuous Galerkin methods for the one-dimensional vlasov-poisson system,, Technical Report 2009-41, (2009), 2009. Google Scholar

[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., (). Google Scholar

[8]

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

[9]

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

[10]

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

[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. doi: 10.1137/050635171. Google Scholar

[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. doi: 10.3934/krm.2009.2.39. Google Scholar

[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. doi: 10.1090/S0025-5718-07-01912-6. Google Scholar

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C. K. Birdsall and A. B. Langdon, "Plasma Physics Via Computer Simulation,", McGraw-Hill, (1985). Google Scholar

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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. doi: 10.1016/j.cma.2005.06.015. Google Scholar

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

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

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

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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. 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. doi: 10.1090/S0025-5718-06-01895-3. Google Scholar

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C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space,, J. Computational Phys., 22 (1976), 330. 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. doi: 10.1016/j.cma.2009.05.015. Google Scholar

[25]

P. G. Ciarlet, Basic error estimates for elliptic problems,, in, (1991), 17. Google Scholar

[26]

B. Cockburn, J. Guzmán and H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems,, Math. Comp., 78 (2009), 1. 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. doi: 10.1137/S0036142900371544. Google Scholar

[28]

B. Cockburn, G. E. Karniadakis and C.-W. Shu, The development of discontinuous Galerkin methods,, in, 11 (1999), 3. 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. 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. 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. 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. 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. 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. 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, 7 (2005), 59. 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. doi: 10.1016/j.jcp.2008.10.041. Google Scholar

[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. doi: 10.2307/2007825. Google Scholar

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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. doi: 10.3934/dcds.2002.8.361. Google Scholar

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

[42]

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

[43]

F. Filbet and E. Sonnendrücker, Comparison of Eulerian Vlasov solvers,, Comput. Phys. Comm., 150 (2003), 247. doi: 10.1016/S0010-4655(02)00694-X. Google Scholar

[44]

F. Filbet, E. Sonnendrücker and P. Bertrand, Conservative numerical schemes for the Vlasov equation,, J. Comput. Phys., 172 (2001), 166. doi: 10.1006/jcph.2001.6818. Google Scholar

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R. E. Heath, "Analysis of the Discontinuous Galerkin Method Applied to Collisionless Plasma Physics,", Ph.D thesis, (2007). Google Scholar

[48]

R. E. Heath, I. Gamba, P. Morrison and C. Michler, A discontinuous Galerkin method for the Vlasov-Poisson system,, to appear in J. Comput. Phys., (2011). Google Scholar

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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. Google Scholar

[2]

R. A. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975). Google Scholar

[3]

S. Agmon, "Lectures on Elliptic Boundary Value Problems,", Prepared for publication by B. Frank Jones, (1965). Google Scholar

[4]

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

[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. Google Scholar

[6]

B. Ayuso, J. A. Carrillo and C.-W. Shu, Discontinuous Galerkin methods for the one-dimensional vlasov-poisson system,, Technical Report 2009-41, (2009), 2009. Google Scholar

[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., (). Google Scholar

[8]

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

[9]

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

[10]

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

[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. doi: 10.1137/050635171. Google Scholar

[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. doi: 10.3934/krm.2009.2.39. Google Scholar

[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. doi: 10.1090/S0025-5718-07-01912-6. Google Scholar

[14]

C. K. Birdsall and A. B. Langdon, "Plasma Physics Via Computer Simulation,", McGraw-Hill, (1985). Google Scholar

[15]

R. Biswas, K. D. Devine and J. E. Flaherty, Parallel, adaptive finite element methods for conservation laws,, in, 14 (1994), 255. doi: 10.1016/0168-9274(94)90029-9. Google Scholar

[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, (2000). Google Scholar

[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. doi: 10.1016/j.cma.2005.06.015. Google Scholar

[18]

F. Brezzi and M. Fortin, "Mixed and Hybrid Finite Element Methods," Springer Series in Computational Mathematics, 15,, Springer-Verlag, (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. 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. 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. 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. 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. 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. doi: 10.1016/j.cma.2009.05.015. Google Scholar

[25]

P. G. Ciarlet, Basic error estimates for elliptic problems,, in, (1991), 17. Google Scholar

[26]

B. Cockburn, J. Guzmán and H. Wang, Superconvergent discontinuous Galerkin methods for second-order elliptic problems,, Math. Comp., 78 (2009), 1. 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. doi: 10.1137/S0036142900371544. Google Scholar

[28]

B. Cockburn, G. E. Karniadakis and C.-W. Shu, The development of discontinuous Galerkin methods,, in, 11 (1999), 3. 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. 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. 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. 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. 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. 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. 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, 7 (2005), 59. 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. doi: 10.1016/j.jcp.2008.10.041. Google Scholar

[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. doi: 10.2307/2007825. Google Scholar

[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. doi: 10.3934/dcds.2002.8.361. Google Scholar

[39]

J. Douglas, Jr. and T. Dupont, Interior penalty procedures for elliptic and parabolic Galerkin methods,, in, (1975), 207. Google Scholar

[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. Google Scholar

[41]

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

[42]

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

[43]

F. Filbet and E. Sonnendrücker, Comparison of Eulerian Vlasov solvers,, Comput. Phys. Comm., 150 (2003), 247. doi: 10.1016/S0010-4655(02)00694-X. Google Scholar

[44]

F. Filbet, E. Sonnendrücker and P. Bertrand, Conservative numerical schemes for the Vlasov equation,, J. Comput. Phys., 172 (2001), 166. doi: 10.1006/jcph.2001.6818. Google Scholar

[45]

K. Ganguly and H. D. Victory, Jr., On the convergence of particle methods for multidimensional Vlasov-Poisson systems,, SIAM J. Numer. Anal., 26 (1989), 249. doi: 10.1137/0726015. Google Scholar

[46]

R. T. Glassey, "The Cauchy Problem in Kinetic Theory,", Society for Industrial and Applied Mathematics (SIAM), (1996). Google Scholar

[47]

R. E. Heath, "Analysis of the Discontinuous Galerkin Method Applied to Collisionless Plasma Physics,", Ph.D thesis, (2007). Google Scholar

[48]

R. E. Heath, I. Gamba, P. Morrison and C. Michler, A discontinuous Galerkin method for the Vlasov-Poisson system,, to appear in J. Comput. Phys., (2011). Google Scholar

[49]

G. S. Jiang and C.-W. Shu, On a cell entropy inequality for discontinuous Galerkin methods,, Math. Comp., 62 (1994), 531. doi: 10.1090/S0025-5718-1994-1223232-7. Google Scholar

[50]

A. J. Klimas and W. M. Farrell, A splitting algorithm for Vlasov simulation with filamentation filtration,, J. Comput. Phys., 110 (1994), 150. Google Scholar

[51]

D. J. Knezevic and E. Süli, A heterogeneous alternating-direction method for a micro-macro dilute polymeric fluid model,, M2AN Math. Model. Numer. Anal., 43 (2009), 1117. doi: 10.1051/m2an/2009034. Google Scholar

[52]

D. J. Knezevic and E. Süli, Spectral Galerkin approximation of Fokker-Planck equations with unbounded drift,, M2AN Math. Model. Numer. Anal., 43 (2009), 445. doi: 10.1051/m2an:2008051. Google Scholar

[53]

P. Lasaint and P.-A. Raviart, On a finite element method for solving the neutron transport equation,, in, (1974), 89. Google Scholar

[54]

J.-G. Liu and C.-W. Shu, A high-order discontinuous Galerkin method for 2D incompressible flows,, J. Comput. Phys., 160 (2000), 577. doi: 10.1006/jcph.2000.6475. Google Scholar

[55]

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