A conservative and entropic method for the Vlasov–Fokker–Planck–Landau equation

2009 ◽  
pp. 59-70
Author(s):  
N. Crouseilles ◽  
F. Filbet
Keyword(s):  
Landau Equation ◽  
Fokker Planck

A Numerical Scheme for the Quantum Fokker-Planck-Landau Equation Efficient in the Fluid Regime

2012 ◽  
Vol 12 (5) ◽  
pp. 1541-1561 ◽  
Author(s):  
Jingwei Hu ◽  
Shi Jin ◽  
Bokai Yan
Keyword(s):  
Numerical Scheme ◽  
Landau Equation ◽  
Collision Operator ◽  
Collision Term ◽  
Fluid Regime ◽  
Quantum Operator ◽  
Simple Quantum ◽  
Dirac Distribution ◽  
Bose Einstein ◽  
Fokker Planck

Abstract We construct an efficient numerical scheme for the quantum Fokker-Planck- Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such a scheme inevitably needs an implicit discretization of the nonlinear collision operator, which is difficult to invert. Inspired by work [9] we seek a linear operator to penalize the quantum FPL collision term QqFPL in order to remove the stiffness induced by the small Knudsen number. However, there is no suitable simple quantum operator serving the purpose and for this kind of operators one has to solve the complicated quantum Maxwellians (Bose-Einstein or Fermi-Dirac distribution). In this paper, we propose to penalize QqFPL by the "classical" linear Fokker-Planck operator. It is based on the observation that the classical Maxwellian, with the temperature replaced by the internal energy, has the same first five moments as the quantum Maxwellian. Numerical results for Bose and Fermi gases are presented to illustrate the efficiency of the scheme in both fluid and kinetic regimes.

scholarly journals An Entropic Scheme for an Angular Moment Model for the Classical Fokker-Planck-Landau Equation of Electrons

2014 ◽  
Vol 15 (2) ◽  
pp. 422-450 ◽  
Author(s):  
Jessy Mallet ◽  
Stéphane Brull ◽  
Bruno Dubroca
Keyword(s):  
Discrete Model ◽  
Landau Equation ◽  
Numerical Test ◽  
Test Cases ◽  
Minimum Entropy ◽  
Fundamental Properties ◽  
Direct Numerical Solution ◽  
Entropy Principle ◽  
The Moment ◽  
Fokker Planck

AbstractIn plasma physics domain, the electron transport is described with the Fokker-Planck-Landau equation. The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. That is why we propose in this paper a new model whose derivation is based on an angular closure in the phase space and retains only the energy of particles as kinetic dimension. To find a solution compatible with physics conditions, the closure of the moment system is obtained under a minimum entropy principle. This model is proved to satisfy the fundamental properties like a H theorem. Moreover an entropic discretization in the velocity variable is proposed on the semi-discrete model. Finally, we validate on numerical test cases the fundamental properties of the full discrete model.

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