## Diffusion Limit Of The Lorentz Model: Asymptotic Preserving Schemes

Citations: | 10 - 2 self |

### BibTeX

@MISC{Buet_diffusionlimit,

author = {Christophe Buet and Stephane Cordier and B. Lucquin-Desreux and S. Mancini},

title = {Diffusion Limit Of The Lorentz Model: Asymptotic Preserving Schemes},

year = {}

}

### OpenURL

### Abstract

This paper deals with the diusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diusive limit, the right discrete diusion equation with the same value of the diusion coecient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization points, in order to reduce the cost of computation.

### Citations

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44 |
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A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes
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The discrete-ordinate method in diffusive regimes, Transport Theory Stat
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Citation Context ...hey determine the quadrature points in the integral with respect to velocity such that, in the diffusion limit, one recovers a heat equation with the correct diffusion coefficient. A first paper, see =-=[22]-=-, is devoted to the discretization in velocities, the distribution function depending continuously of the space and time variables. The authors construct quadrature sets corresponding to a small numbe... |

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15 |
A well balanced scheme designed for inhomogeneous scalar conservation laws
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Citation Context ...rature sets corresponding to a small number of discrete velocities such that the diffusion coefficient and also the boundary conditions are compatible with the diffusive limit. In a second paper, see =-=[23]-=-, they investigate the fully discrete case (both on the velocity and the space variables), the problem being stationary. This can be seen as the problem for one time step iteration using an implicit s... |

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Citation Context ...tiff source terms (see [4, 5, 28]) due, for example, to the modeling of rapid chemical reactions. These systems read: ∂tU + ∂xF (U) = 1 ε S(f) and are related to the so called relaxation methods (see =-=[24,28,40,41]-=-) that consists in replacing a nonlinear problem of the above form by a relaxed linear system of the following form: ∂tU + ∂xV =0, ∂tV + a∂xU = −1 (V − F (U)) ε hal-00076839, version 1 - 24 Oct 2012 w... |

13 |
The asymptotics of collision operators for two species of particles of disparate masses
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Citation Context ...first order term of the inter-species collision operator representing the collisions of the heavy particles on the light ones, when doing an asymptotic expansion in terms of the small mass ratio (see =-=[8,35]-=-). This operator does not depend on the energy variable, i.e. on the modulus of the velocity. It is defined in the Boltzmann case by (for simplicity, Keywords and phrases. Hilbert expansion, diffusion... |

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10 |
Souganidis, Existence of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates
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10 |
The asymptotic diffusion limit of discretized transport problems
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8 | Central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms
- Pareschi
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Citation Context ...tiff source terms (see [4, 5, 28]) due, for example, to the modeling of rapid chemical reactions. These systems read: ∂tU + ∂xF (U) = 1 ε S(f) and are related to the so called relaxation methods (see =-=[24,28,40,41]-=-) that consists in replacing a nonlinear problem of the above form by a relaxed linear system of the following form: ∂tU + ∂xV =0, ∂tV + a∂xU = −1 (V − F (U)) ε hal-00076839, version 1 - 24 Oct 2012 w... |

7 | The Convergence of numerical transfer schemes in diusive regimes I: the discrete-ordinate method - Golse, Jin, et al. - 1999 |

7 |
Numerical Navier-Stokes solutions from gas kinetic theory
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6 |
The grazing collision limit for the Boltzmann-Lorentz model. Asymp. Analysis 25(2): 93–107
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- 2001
(Show Context)
Citation Context ... to the Fokker–Planck–Lorentz one (1.3) when the cross-section concentrates, i.e. when the scattering angle during a collision, θ ′ − θ, is very small. This is the so-called “grazing collision limit” =-=[3, 6, 36]-=-, which is also valid in the non-linear case [7, 9]. It is well known in literature, that for ε ≪ 1 the solution of (1.1) converges to the solution of a diffusion problem, with respect to the space va... |

6 | Numerical integrations of systems of conservation laws of mixed type
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Diffusion of electrons by multicharged ions
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5 | The discrete-ordinate method in diusive regimes - Jin, Levermore - 1991 |

5 | The asymptotic diusion limit of discretized transport problems - Larsen - 1992 |

5 | Balancing source terms and gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm - LeVeque - 1998 |

5 | Numerical schemes for kinetic equations in diusive regimes - Naldi, Pareschi - 1998 |

4 |
A generalized Riemann problem for quasi one dimensional gas flows
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- 1984
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3 |
An introduction to kinetic schemes for gas dynamics. An introduction to recent developments in theory and numerics for conservation laws
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3 |
Two coupled particle-finite volume methods using Delaunay-Voronoï meshes for the approximation of VlasovPoisson and Vlasov-Maxwell equations
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2 | A nite volume method for the approximation of di usion operators on distorted meshes - Hermeline - 2000 |

2 | Central dierencing based numerical schemes for hyperbolic conservation laws with relaxation terms - Pareschi |

2 | A finite element approximation of grazing collisions (submitted
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(Show Context)
Citation Context ... to the Fokker–Planck–Lorentz one (1.3) when the cross-section concentrates, i.e. when the scattering angle during a collision, θ ′ − θ, is very small. This is the so-called “grazing collision limit” =-=[3, 6, 36]-=-, which is also valid in the non-linear case [7, 9]. It is well known in literature, that for ε ≪ 1 the solution of (1.1) converges to the solution of a diffusion problem, with respect to the space va... |

2 |
A linear-discontinuous spatial differencing scheme for Sn radiative transfer calculations
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