Some problems concerning the solvability of the nonlinear stationary Boltzmann equation in the framework of the BGK model

2016 ◽  
Vol 77 ◽  
pp. 87-106
Author(s):  
Agavard Kh. Khachatryan ◽  
Khachatur A. Khachatryan
Keyword(s):  
Boltzmann Equation ◽  
Bgk Model

THE CONTRIBUTION OF THE BHATNAGAR–GROSS–KROOK MODEL TO THE DEVELOPMENT OF RAREFIED GAS DYNAMICS IN THE EARLY YEARS OF THE SPACE AGE

2013 ◽  
Vol 25 (01) ◽  
pp. 1340025
Author(s):  
RODDAM NARASIMHA
Keyword(s):  
Boltzmann Equation ◽  
Gas Dynamics ◽  
Rarefied Gas ◽  
Rarefied Gas Dynamics ◽  
Early Years ◽  
Plane Shock ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Nonequilibrium Flows ◽  
Space Age

The advent of the space age in 1957 was accompanied by a sudden surge of interest in rarefied gas dynamics (RGD). The well-known difficulties associated with solving the Boltzmann equation that governs RGD made progress slow but the Bhatnagar–Gross–Krook (BGK) model, proposed three years before Sputnik, turned out to have been an uncannily timely, attractive and fruitful option, both for gaining insights into the Boltzmann equation and for estimating various technologically useful flow parameters. This paper gives a view of how BGK contributed to the growth of RGD during the first decade of the space age. Early efforts intended to probe the limits of the BGK model showed that, in and near both the continuum Euler limit and the collisionless Knudsen limit, BGK could provide useful answers. Attempts were therefore made to tackle more ambitious nonlinear nonequilibrium problems. The most challenging of these was the structure of a plane shock wave. The first exact numerical solutions of the BGK equation for the shock appeared during 1962 to 1964, and yielded deep insights into the character of transitional nonequilibrium flows that had resisted all attempts at solution through the Boltzmann equation. In particular, a BGK weak shock was found to be amenable to an asymptotic analysis. The results highlighted the importance of accounting separately for fast-molecule dynamics, most strikingly manifested as tails in the distribution function, both in velocity and in physical space — tails are strange versions or combinations of collisionless and collision-generated flows. However, by the mid-1960s Monte-Carlo methods of solving the full Boltzmann equation were getting to be mature and reliable and interest in the BGK waned in the following years. Interestingly, it has seen a minor revival in recent years as a tool for developing more effective algorithms in continuum computational fluid dynamics, but the insights derived from the BGK for strongly nonequilibrium flows should be of lasting value.

scholarly journals The Gaussian-BGK model of Boltzmann equation with small Prandtl number

2000 ◽  
Vol 19 (6) ◽  
pp. 813-830 ◽  
Author(s):  
Pierre Andries ◽  
Patrick Le Tallec ◽  
Jean-Philippe Perlat ◽  
Benoı̂t Perthame
Keyword(s):  
Boltzmann Equation ◽  
Prandtl Number ◽  
Bgk Model

scholarly journals Convergence of a Semi-Lagrangian Scheme for the BGK Model of the Boltzmann Equation

2012 ◽  
Vol 50 (3) ◽  
pp. 1111-1135 ◽  
Author(s):  
Giovanni Russo ◽  
Pietro Santagati ◽  
Seok-Bae Yun
Keyword(s):  
Boltzmann Equation ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Lagrangian Scheme

scholarly journals Boundary conditions for semi-Lagrangian methods for the BGK model

2016 ◽  
Vol 7 (3) ◽  
pp. 138-164 ◽  
Author(s):  
Maria Groppi ◽  
Giovanni Russo ◽  
Giuseppe Stracquadanio
Keyword(s):  
Boltzmann Equation ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
Numerical Simulations ◽  
Lagrangian Formulation ◽  
High Order Accuracy ◽  
Bgk Model ◽  
Order Accuracy ◽  
New Class ◽  
Iterative Procedures

Abstract A new class of high-order accuracy numerical methods based on a semi-Lagrangian formulation for the BGK model of the Boltzmann equation has been recently proposed in [1]. In this paper semi-Lagrangian schemes for the BGK equation have been extended to treat boundary conditions, in particular the diffusive ones. Two different techniques are proposed, using or avoiding iterative procedures. Numerical simulations illustrate the accuracy properties of these approaches and the agreement with the results available in literature.

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