Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov--Poisson Equations


A rigorous convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov--Poisson equations is provided. It is shown that under suitable assumptions the error is of order $\mathcal{O}(\tau^2+h^q +h^q/\tau)$, where $\tau$ is the size of a time step, $h$ is the cell size, and $q$ the order of the discontinuous Galerkin approximation. In order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed.


  1. Strang splitting
  2. discontinuous Galerkin approximation
  3. convergence analysis
  4. Vlasov--Poisson equations
  5. recurrence

MSC codes

  1. 65M12
  2. 82D10
  3. 65L05
  4. 65M60

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T. D. Arber and R. Vann, A critical comparison of Eulerian-grid-based Vlasov solvers, J. Comput. Phys., 180 (2002), pp. 339--357.
E. A. Belli, Studies of Numerical Algorithms for Gyrokinetics and the Effects of Shaping on Plasma Turbulence, Ph.D. thesis, Princeton University, Princeton, NJ, 2006.
N. Besse and M. Mehrenberger, Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system, Math. Comp., 77 (2008), pp. 93--123.
N. Besse, Convergence of a semi-Lagrangian scheme for the one-dimensional Vlasov-Poisson system, SIAM J. Numer. Anal., 42 (2005), pp. 350--382.
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), pp. 639--670.
M. Bostan and N. Crouseilles, Convergence of a semi-Lagrangian scheme for the reduced Vlasov-Maxwell system for laser-plasma interaction, Numer. Math., 112 (2009), pp. 169--195.
F. Charles, B. Després, and M. Mehrenberger, Enhanced convergence estimates for semi-Lagrangian schemes. Application to the Vlasov--Poisson equation, SIAM J. Numer. Anal., 51 (2013), pp. 840--863.
C. Z. Cheng and G. Knorr, The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), pp. 330--351.
N. Crouseilles, E. Faou, and M. Mehrenberger, High Order Runge--Kutta--Nyström Splitting Methods for the Vlasov--Poisson Equation. PDF/cfm.pdf.
N. Crouseilles, M. Mehrenberger, and F. Vecil, Discontinuous Galerkin semi-Lagrangian method for Vlasov--Poisson, in CEMRACS 2010, ESAIM Proc. 32, EDP Sciences, Les Ulis, France, 2011, pp. 211--230.
L. Einkemmer and A. Ostermann, Convergence analysis of Strang splitting for Vlasov--type equations. SIAM J. Numer. Anal., 52 (2014), pp. 140--155.
M. R. Fahey and J. Candy, GYRO: A 5-d gyrokinetic-Maxwell solver, in Proceedings of the ACM/IEEE SC2004 Conference, 2008, p. 26.
F. Filbet and E. Sonnendrücker, Comparison of Eulerian Vlasov solvers, Comput. Phys. Comm., 150 (2003), pp. 247--266.
R. T. Glassey, The Cauchy Problem in Kinetic Theory, SIAM, Philadelphia, 1996.
R. E. Heath, I. M. Gamba, P. J. Morrison, and C. Michler, A discontinuous Galerkin method for the Vlasov-Poisson system, J. Comput. Phys., 231 (2011), pp. 1140--1174.
R. E. Heath, Analysis of the Discontinuous Galerkin Method Applied to Collisionless Plasma Physics, Ph.D. thesis, University of Texas at Austin, 2007.
T. Jahnke and C. Lubich, Error bounds for exponential operator splittings, BIT, 40 (2000), pp. 735--744.
S. M. H. Jenab, I. Kourakis, and H. Abbasi, Fully kinetic simulation of ion acoustic and dust-ion acoustic waves, Phys. Plasmas, 18 (2011), 073703.
A. Mangeney, F. Califano, C. Cavazzoni, and P. Travnicek, A numerical scheme for the integration of the Vlasov-Maxwell system of equations, J. Comput. Phys., 179 (2002), pp. 495--538.
M. Mehrenberger, C. Steiner, L. Marradi, N. Crouseilles, E. Sonnendrücker, and B. Afeyan, Vlasov on GPU (VOG project), preprint, arXiv:1301.5892, 2013.
K. W. Morton, A. Priestley, and E. Süli, Stability of the Lagrange-Galerkin method with non-exact integration, Modél. Math. Anal. Numér., 22 (1988), pp. 625--653.
G. M. Phillips, Interpolation and Approximation by Polynomials, Springer, New York, 2003.
J. M. Qiu and C. W. Shu, Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov--Poisson system, J. Comput. Phys., 230 (2011), pp. 8386--8409.
T. Respaud and E. Sonnendrücker, Analysis of a new class of forward semi-Lagrangian schemes for the 1D Vlasov Poisson equations, Numer. Math., 118 (2011), pp. 329--366.
J. A. Rossmanith and D. C. Seal, A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov--Poisson equations, J. Comput. Phys., 230 (2011), pp. 6203--6232.
A. Shadrin, Twelve proofs of the Markov inequality, in Approximation Theory: A Volume Dedicated to Borislav Bojanov, D. K. Dimitrov et al., eds., Marin Drinov, Sofia, 2004, pp. 233--298.
H. Yang and F. Li, Error Estimates of Runge--Kutta Discontinuous Galerkin Methods for the Vlasov--Maxwell System, preprint, arXiv:1306.0636, 2013.
T. Zhou, Y. Guo, and C. W. Shu, Numerical study on Landau damping, Phys. D, 157 (2001), pp. 322--333.

Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 757 - 778
ISSN (online): 1095-7170


Submitted: 12 November 2012
Accepted: 28 January 2014
Published online: 1 April 2014


  1. Strang splitting
  2. discontinuous Galerkin approximation
  3. convergence analysis
  4. Vlasov--Poisson equations
  5. recurrence

MSC codes

  1. 65M12
  2. 82D10
  3. 65L05
  4. 65M60



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