scholarly journals A criterion for asymptotic preserving schemes of kinetic equations to be uniformly stationary preserving

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Casimir Emako ◽  
Farah Kanbar ◽  
Christian Klingenberg ◽  
Min Tang
Keyword(s):  
Distribution Function ◽  
Kinetic Models ◽  
Kinetic Equations ◽  
Asymptotic Preserving ◽  
Asymptotic Preserving Schemes

<p style='text-indent:20px;'>In this work we are interested in the stationary preserving property of asymptotic preserving (AP) schemes for kinetic models. We introduce a criterion for AP schemes for kinetic equations to be uniformly stationary preserving (SP). Our key observation is that as long as the Maxwellian of the distribution function can be updated explicitly, such AP schemes are also SP. To illustrate our observation, three different AP schemes for three different kinetic models are considered. Their SP property is proved analytically and tested numerically, which confirms our observations.</p>

scholarly journals Numerical methods for kinetic equations

Acta Numerica ◽  
2014 ◽  
Vol 23 ◽  
pp. 369-520 ◽  
Author(s):  
G. Dimarco ◽  
L. Pareschi
Keyword(s):  
Computational Complexity ◽  
Numerical Methods ◽  
State Of The Art ◽  
Kinetic Equations ◽  
Numerical Techniques ◽  
Asymptotic Preserving ◽  
Hybrid Schemes ◽  
Velocity Models ◽  
Current State ◽  
Number Of Particles

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

scholarly journals Boltzmann Equation Theory of Charged Particle Transport in Neutral Gases: Perturbation Treatment

1999 ◽  
Vol 52 (6) ◽  
pp. 999 ◽  
Author(s):  
Slobodan B. Vrhovac ◽  
Zoran Lj. Petrovic
Keyword(s):  
Distribution Function ◽  
Perturbation Theory ◽  
Charged Particle ◽  
Boltzmann Equation ◽  
Particle Transport ◽  
Kinetic Equations ◽  
Initial Distribution ◽  
Formal Structure ◽  
Time Dependent ◽  
Charged Particle Transport

This paper examines the formal structure of the Boltzmann equation (BE) theory of charged particle transport in neutral gases. The initial value problem of the BE is studied by using perturbation theory generalised to non-Hermitian operators. The method developed by R�sibois was generalised in order to be applied for the derivation of the transport coecients of swarms of charged particles in gases. We reveal which intrinsic properties of the operators occurring in the kinetic equation are sucient for the generalised diffusion equation (GDE) and the density gradient expansion to be valid. Explicit expressions for transport coecients from the (asymmetric) eigenvalue problem are also deduced. We demonstrate the equivalence between these microscopic expressions and the hierarchy of kinetic equations. The establishment of the hydrodynamic regime is further analysed by using the time-dependent perturbation theory. We prove that for times t ? τ0 (τ0 is the relaxation time), the one-particle distribution function of swarm particles can be transformed into hydrodynamic form. Introducing time-dependent transport coecients ? *(p) (?q,t), which can be related to various Fourier components of the initial distribution function, we also show that for the long-time limit all ? *(p) (?q,t) become time and ?q independent in the same characteristic time and achieve their hydrodynamic values.

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