We present an original adaptive scheme using a dynamically refined grid for the simulation of the six-dimensional Vlasov--Poisson equations. The distribution function is represented in a hierarchical basis that retains only the most significant coefficients. This allows considerable savings in terms of computational time and memory usage. The proposed scheme involves the mathematical formalism of multiresolution analysis and computer implementation of adaptive mesh refinement. We apply a finite difference method to approximate the Vlasov--Poisson equations, although other numerical methods could be considered. Numerical experiments are presented for the $d$-dimensional Vlasov--Poisson equations in the full $2d$-dimensional phase space for $d=1,2$, or 3. The six-dimensional case is compared to a Gadget N-body simulation.


  1. adaptive mesh refinement
  2. hierarchical basis
  3. wavelet
  4. multiresolution analysis
  5. finite differences
  6. phase-space simulations
  7. Vlasov--Poisson equations
  8. astrophysics

MSC codes

  1. 65M05
  2. 65M50
  3. 85-08

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Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 583 - 614
ISSN (online): 1540-3467


Submitted: 20 December 2016
Accepted: 5 January 2018
Published online: 3 April 2018


  1. adaptive mesh refinement
  2. hierarchical basis
  3. wavelet
  4. multiresolution analysis
  5. finite differences
  6. phase-space simulations
  7. Vlasov--Poisson equations
  8. astrophysics

MSC codes

  1. 65M05
  2. 65M50
  3. 85-08



Funding Information

Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-13-MONU-0003
FP7 Fusion Energy Research https://doi.org/10.13039/100011270 : EURATOM-CfP-QP14-ER-01/IPP-03

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