Skip to main content Accessibility help

A study on conserving invariants of the Vlasov equation in semi-Lagrangian computer simulations

  • L. Einkemmer (a1)


The semi-Lagrangian discontinuous Galerkin method, coupled with a splitting approach in time, has recently been introduced for the Vlasov–Poisson equation. Since these methods are conservative, local in space and able to limit numerical diffusion, they are considered a promising alternative to more traditional semi-Lagrangian schemes. In this paper we study the conservation of important physical invariants and the long-time behaviour of the semi-Lagrangian discontinuous Galerkin method. To that end we conduct a theoretical analysis and perform a number of numerical simulations. In particular, we find that the entropy is non-decreasing for the discontinuous Galerkin scheme, while unphysical oscillations in the entropy are observed for the traditional method based on cubic spline interpolation.


Corresponding author

Email address for correspondence:


Hide All
Brunetti, M., Califano, F. & Pegoraro, F. 2000 Asymptotic evolution of nonlinear Landau damping. Phys. Rev. E 62 (3), 4109.
Califano, F., Galeotti, L. & Mangeney, A. 2006 The Vlasov–Poisson model and the validity of a numerical approach. Phys. Plasmas 13 (8), 082102.
Casas, F., Crouseilles, N., Faou, E. & Mehrenberger, M. 2017 High-order Hamiltonian splitting for Vlasov–Poisson equations. Numer. Math. 135 (3), 769801.
Cheng, C. & Knorr, G. 1976 The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22 (3), 330351.
Crouseilles, N., Faou, E. & Mehrenberger, M.2011a High order Runge–Kutta–Nyström splitting methods for the Vlasov–Poisson equation.
Crouseilles, N., Latu, G. & Sonnendrücker, E. 2009a A parallel Vlasov solver based on local cubic spline interpolation on patches. J. Comput. Phys. 228 (5), 14291446.
Crouseilles, N., Mehrenberger, M. & Sonnendrücker, E. 2010 Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229 (6), 19271953.
Crouseilles, N., Mehrenberger, M. & Vecil, F. 2011b Discontinuous Galerkin semi-Lagrangian method for Vlasov–Poisson. In ESAIM: Proceedings, vol. 32, pp. 211230. EDP Sciences.
Crouseilles, N., Respaud, T. & Sonnendrücker, E. 2009b A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Comput. Phys. Commun. 180 (10), 17301745.
Einkemmer, L. 2016a A mixed precision semi-Lagrangian algorithm and its performance on accelerators. In International Conference on High Performance Computing and Simulation (HPCS). IEEE.
Einkemmer, L. 2016b High performance computing aspects of a dimension independent semi-Lagrangian discontinuous Galerkin code. Comput. Phys. Commun. 202, 326336.
Einkemmer, L. & Ostermann, A. 2014a A strategy to suppress recurrence in grid-based Vlasov solvers. Eur. Phys. J. B 68 (7), 17.
Einkemmer, L. & Ostermann, A. 2014b Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov–Poisson equations. SIAM J. Numer. Anal. 52 (2), 757778.
Einkemmer, L. & Ostermann, A. 2014c Convergence analysis of Strang splitting for Vlasov-type equations. SIAM J. Numer. Anal. 52 (1), 140155.
Eliasson, B. 2002 Outflow boundary conditions for the Fourier transformed two-dimensional Vlasov equation. J. Comput. Phys. 181 (1), 98125.
Fijalkow, E. 1999 Numerical solution to the Vlasov equation: the 1D code. Comput. Phys. Commun. 116 (2), 329335.
Galeotti, L. & Califano, F. 2005 Asymptotic evolution of weakly collisional Vlasov–Poisson plasmas. Phys. Rev. Lett. 95, 015002.
Galeotti, L., Califano, F. & Pegoraro, F. 2006 Echography of Vlasov codes. Phys. Lett. A 355 (4–5), 381385.
Hairer, E., Lubich, C. & Wanner, G. 2006 Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer.
Hou, Y. W., Ma, Z. W. & Yu, M. Y. 2011 The plasma wave echo revisited. Phys. Plasmas 18 (1), 012108.
Klimas, A. J. & Farrell, W. M. 1994 A splitting algorithm for Vlasov simulation with filamentation filtration. J. Comput. Phys. 110 (1), 150163.
Manfredi, G. 1997 Long-time behavior of nonlinear Landau damping. Phys. Rev. Lett. 79 (15), 28152818.
Mangeney, A., Califano, F., Cavazzoni, C. & Travnicek, P. 2002 A numerical scheme for the integration of the Vlasov–Maxwell system of equations. J. Comput. Phys. 179 (2), 495538.
Pollard, H. 1972 The convergence almost everywhere of Legendre series. Proc. Am. Math. Soc. 35 (2), 442444.
Qiu, J. M. & Shu, C. W. 2011 Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the Vlasov–Poisson system. J. Comput. Phys. 230 (23), 83868409.
Rossmanith, J. A. & Seal, D. C. 2011 A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations. J. Comput. Phys. 230 (16), 62036232.
Valentini, F., Veltri, P. & Mangeney, A. 2005 A numerical scheme for the integration of the Vlasov–Poisson system of equations, in the magnetized case. J. Comput. Phys. 210 (2), 730751.
Zerroukat, M., Wood, N. & Staniforth, A. 2005 A monotonic and positive-definite filter for a semi-Lagrangian inherently conserving and efficient (SLICE) scheme. Q. J. R. Meteorol. Soc. 131 (611), 29232936.
Zerroukat, M., Wood, N. & Staniforth, A. 2006 The parabolic spline method (PSM) for conservative transport problems. Intl J. Numer. Meth. Fluids 51 (11), 12971318.
MathJax is a JavaScript display engine for mathematics. For more information see


Related content

Powered by UNSILO

A study on conserving invariants of the Vlasov equation in semi-Lagrangian computer simulations

  • L. Einkemmer (a1)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.