Skip to main content
×
Home
    • Aa
    • Aa

A ‘metric’ semi-Lagrangian Vlasov–Poisson solver

  • Stéphane Colombi (a1) (a2) and Christophe Alard (a1)
Abstract

We propose a new semi-Lagrangian Vlasov–Poisson solver. It employs metric elements to follow locally the flow and its deformation, allowing one to find quickly and accurately the initial phase-space position $\boldsymbol{Q}(\boldsymbol{P})$ of any test particle $\boldsymbol{P}$ , by expanding at second order the geometry of the motion in the vicinity of the closest element. It is thus possible to reconstruct accurately the phase-space distribution function at any time $t$ and position $\boldsymbol{P}$ by proper interpolation of initial conditions, following Liouville theorem. When distortion of the elements of metric becomes too large, it is necessary to create new initial conditions along with isotropic elements and repeat the procedure again until next resampling. To speed up the process, interpolation of the phase-space distribution is performed at second order during the transport phase, while third-order splines are used at the moments of remapping. We also show how to compute accurately the region of influence of each element of metric with the proper percolation scheme. The algorithm is tested here in the framework of one-dimensional gravitational dynamics but is implemented in such a way that it can be extended easily to four- or six-dimensional phase space. It can also be trivially generalised to plasmas.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      A ‘metric’ semi-Lagrangian Vlasov–Poisson solver
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      A ‘metric’ semi-Lagrangian Vlasov–Poisson solver
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      A ‘metric’ semi-Lagrangian Vlasov–Poisson solver
      Available formats
      ×
Copyright
Corresponding author
Email address for correspondence: colombi@iap.fr
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

S. J. Aarseth , D. N. C. Lin  & J. C. B. Papaloizou 1988 On the collapse and violent relaxation of protoglobular clusters. Astrophys. J. 324, 288310.

C. Alard  & S. Colombi 2005 A cloudy Vlasov solution. Mon. Not. R. Astron. Soc. 359, 123163.

E. Bertschinger 1998 Simulations of structure formation in the universe. Annu. Rev. Astron. Astrophys. 36, 599654.

N. Besse , E. Deriaz  & É. Madaule 2017 Adaptive multiresolution semi-Lagrangian discontinuous Galerkin methods for the Vlasov equations. J. Comput. Phys. 332, 376417.

N. Besse , G. Latu , A. Ghizzo , E. Sonnendrücker  & P. Bertrand 2008 A wavelet-MRA-based adaptive semi-Lagrangian method for the relativistic Vlasov Maxwell system. J. Comput. Phys. 227, 78897916.

N. Besse  & E. Sonnendrücker 2003 Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space. J. Comput. Phys. 191, 341376.

J. Binney 2004 Discreteness effects in cosmological $N$ -body simulations. Mon. Not. R. Astron. Soc. 350, 939948.

C. M. Boily , E. Athanassoula  & P. Kroupa 2002 Scaling up tides in numerical models of galaxy and halo formation. Mon. Not. R. Astron. Soc. 332, 971984.

G. L. Camm 1950 Self-gravitating star systems. Mon. Not. R. Astron. Soc. 110, 305324.

M. Campos Pinto , E. Sonnendrücker , A. Friedman , D. P. Grote  & S. M. Lund 2014 Noiseless Vlasov–Poisson simulations with linearly transformed particles. J. Comput. Phys. 275, 236256.

C. Z. Cheng  & G. Knorr 1976 The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330351.

B. Cockburn , G. E. Karniadakis  & C. W. Shu 2000 The development of discontinuous Galerkin methods. In Discontinuous Galerkin Methods (ed. B. Cockburn , G. E. Karniadakis  & C. W. Shu ), Lecture Notes in Computational Science and Engineering, vol. 11, pp. 350. Springer.

S. Colombi 2001 Dynamics of the large-scale structure of the universe: $N$ -body techniques. New Astronomy Rev. 45, 373377.

S. Colombi , T. Sousbie , S. Peirani , G. Plum  & Y. Suto 2015 Vlasov versus $N$ -body: the Hénon sphere. Mon. Not. R. Astron. Soc. 450, 37243741.

S. Colombi  & J. Touma 2014 Vlasov–Poisson in 1D: waterbag. Mon. Not. R. Astron. Soc. 441, 24142432; (CT14).

N. Crouseilles , G. Latu  & E. Sonnendrücker 2009 A parallel Vlasov solver based on local cubic spline interpolation on patches. J. Comput. Phys. 228, 14291446.

N. Crouseilles , M. Mehrenberger  & E. Sonnendrücker 2010 Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys. 229, 19271953.

W. Dehnen  & J. I. Read 2011 $N$ -body simulations of gravitational dynamics. Eur. Phys. J. Plus 126, 55.

K. Dolag , S. Borgani , S. Schindler , A. Diaferio  & A. M. Bykov 2008 Simulation techniques for cosmological simulations. Space Sci. Rev. 134, 229268.

F. Filbet , E. Sonnendrücker  & P. Bertrand 2001 Conservative numerical schemes for the Vlasov equation. J. Comput. Phys. 172, 166187.

Y. Güçlü , A. J. Christlieb  & W. N. G. Hitchon 2014 Arbitrarily high order convected scheme solution of the Vlasov–Poisson system. J. Comput. Phys. 270, 711752.

M. Gutnic , M. Haefele , I. Paun  & E. Sonnendrücker 2004 Vlasov simulations on an adaptive phase-space grid. Comput. Phys. Commun. 164, 214219.

V. Grandgirard , J. Abiteboul , J. Bigot , T. Cartier-Michaud , N. Crouseilles , G. Dif-Pradalier , C. Ehrlacher , D. Esteve , X. Garbet , P. Ghendrih 2016 A 5D gyrokinetic full-f global semi-Lagrangian code for flux-driven ion turbulence simulations. Comput. Phys. Commun. 207, 3568.

O. Hahn  & R. E. Angulo 2016 An adaptively refined phase-space element method for cosmological simulations and collisionless dynamics. Mon. Not. R. Astron. Soc. 455, 11151133.

R. W. Hockney  & J. W. Eastwood 1988 Computer Simulation Using Particles. Hilger.

M. Joyce , B. Marcos  & F. Sylos Labini 2009 Energy ejection in the collapse of a cold spherical self-gravitating cloud. Mon. Not. R. Astron. Soc. 397, 775792.

H. E. Kandrup  & H. Smith Jr. 1991 On the sensitivity of the $N$ -body problem to small changes in initial conditions. Astrophys. J. 374, 255265.

D. J. Larson  & C. V. Young 2015 A finite mass based method for Vlasov–Poisson simulations. J. Comput. Phys. 284, 171185.

J. J. Monaghan 1992 Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 30, 543574.

J.-M. Qiu  & C.-W. Shu 2011 Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the Vlasov–Poisson system. J. Comput. Phys. 230, 83868409.

J. A. Rossmanith  & D. C. Seal 2011 A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations. J. Comput. Phys. 230, 62036232.

M. M. Shoucri  & R. R. J. Gagne 1978 Splitting schemes for the numerical solution of a two-dimensional Vlasov equation. J. Comput. Phys. 27, 315322.

L. Spitzer Jr. 1942 The dynamics of the interstellar medium. III. Galactic distribution. Astrophys. J. 95, 329.

G. B. Rybicki 1971 Exact statistical mechanics of a one-dimensional self-gravitating system. Astrophys. Space Sci. 14, 5672; (Papers appear in the Proceedings of IAU Colloquium No. 10 Gravitational $N$ -Body Problem (ed. by Myron Lecar), R. Reidel Publ. Co., Dordrecht-Holland).

E. Sonnendrücker , J. Roche , P. Bertrand  & A. Ghizzo 1999 The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys. 149, 201220.

T. Sousbie  & S. Colombi 2016 ColDICE: A parallel Vlasov–Poisson solver using moving adaptive simplicial tessellation. J. Comput. Phys. 321, 644697.

K. Yoshikawa , N. Yoshida  & M. Umemura 2013 Direct integration of the collisionless Boltzmann equation in six-dimensional phase space: self-gravitating systems. Astrophys. J. 762, 116.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Plasma Physics
  • ISSN: 0022-3778
  • EISSN: 1469-7807
  • URL: /core/journals/journal-of-plasma-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 6
Total number of PDF views: 50 *
Loading metrics...

Abstract views

Total abstract views: 76 *
Loading metrics...

* Views captured on Cambridge Core between 5th June 2017 - 21st September 2017. This data will be updated every 24 hours.