| 1. |
- Bernhoff, Niclas, 1971-, et al.
(författare)
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Weak Shock Wave Solutions for the Discrete Boltzmann Equation
- 2007
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Ingår i: Rarefied Gas Dynamics: 25th International Symposium on Rarefied Gas Dynamics, Saint-Petersburg, Russia, July 21-28, 2006 (M.S. Ivanov and A.K. Rebrov, eds). - Novosibirsk : Publishing House of the Siberian Branch of the Russian Academy of Sciences. - 9785769209246 ; , s. 173-178
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Konferensbidrag (refereegranskat)abstract
- The analytically difficult problem of existence of shock wave solutions is studied for the general discrete velocity model (DVM) with an arbitrary finite number of velocities (the discrete Boltzmann equation in terminology of H. Cabannes). For the shock wave problem the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this work we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed , corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is then shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998 [1]. In their technical proof Bose et al. are following the lines of the pioneering work for the continuous Boltzmann equation by Caflisch and Nicolaenko [2]. In this work, we follow a more straightforward way, suiting the discrete case. Our approach is based on results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points [3] to general dynamical systems of the same type as in the shock wave problem for DVMs. Our proof is constructive, and it is also shown (at least implicitly) how close to the typical speed , the shock speed must be for our results to be valid. All results are mathematically rigorous. Our results are also applicable for DVMs for mixtures. ACKNOWLEDGEMENTS. The support by the Swedish Research Council grant 20035357 are gratefully acknowledged by both of the authors.REFERENCES[1] C. Bose, R. Illner, S. Ukai, Transp. Th. Stat. Phys., 27, 35-66 (1998) [2] R.E. Caflisch, B. Nicolaenko, Comm. Math. Phys., 86, 161-194 (1982)[3] A.V. Bobylev, N. Bernhoff, Lecture Notes on the Discretization of the Boltzmann Equation, World Scientific, 2003, pp. 203-222
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| 2. |
- Bernhoff, Niclas, 1971-, et al.
(författare)
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Weak shock waves for the general discrete velocity model of the Boltzmann equation
- 2007
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Ingår i: Communications in Mathematical Sciences. - Somerville, MA : International Press of Boston. - 1539-6746 .- 1945-0796. ; 5:4, s. 815-832
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Tidskriftsartikel (refereegranskat)abstract
- We study the shock wave problem for the general discrete velocity model (DVM), with an arbitrary finite number of velocities. In this case the discrete Boltzmann equation becomes a system of ordinary differential equations (dynamical system). Then the shock waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians). In this paper we give a constructive proof for the existence of solutions in the case of weak shocks. We assume that a given Maxwellian is approached at infinity, and consider shock speeds close to a typical speed c, corresponding to the sound speed in the continuous case. The existence of a non-negative locally unique (up to a shift in the independent variable) bounded solution is proved by using contraction mapping arguments (after a suitable decomposition of the system). This solution is shown to tend to a Maxwellian at minus infinity. Existence of weak shock wave solutions for DVMs was proved by Bose, Illner and Ukai in 1998. In this paper, we give a constructive proof following a more straightforward way, suiting the discrete case. Our approach is based on earlier results by the authors on the main characteristics (dimensions of corresponding stable, unstable and center manifolds) for singular points to general dynamical systems of the same type as in the shock wave problem for DVMs. The same approach can also be applied for DVMs for mixtures
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| 3. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Boltzmann equation and hydrodynamics at the Burnett level
- 2012
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Ingår i: Kinetic and Related Models. - : American Institute of Mathematical Sciences. - 1937-5093 .- 1937-5077. ; 5:2, s. 237-260
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Tidskriftsartikel (refereegranskat)abstract
- The hydrodynamics at the Burnett level is discussed in detail. First we explain the shortest way to derive the classical Burnett equations from the Boltzmann equation. Then we sketch all the computations needed for details of these equations. It is well known that the classical Burnett equations are ill-posed. We therefore explain how to make a regularization of these equations and derive the well-posed generalized Burnett equations (GBEs). We discuss briefly an optimal choice of free parameters in GBEs and consider a specific version of these equations. It is remarkable that this version of GBEs is even simpler than the original Burnett equations, it contains only third derivatives of density. Finally we prove a linear stability for GBEs. We also present some numerical results on the sound propagation based on GBEs and compare them with the Navier-Stokes results and experimental data.
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| 4. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Discrete Velocity Models and Dynamical Systems
- 2003
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Ingår i: Lecture Notes on the Discretization of the Boltzmann Equation. - Singapore : World Scientific. - 9789812382252 - 9789812796905 ; , s. 203-222
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Bokkapitel (övrigt vetenskapligt/konstnärligt)
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| 5. |
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| 6. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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DSMC Methods for Multicomponent Plasmas
- 2012. - 1
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Ingår i: DSMC and Related Simulations. - New York : American Institute of Physics (AIP). - 9780735411159 ; , s. 541-548
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Konferensbidrag (refereegranskat)abstract
- A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of the Landau-Fokker-Planck equations by the Boltzmann equations of a quasi-Maxwellian kind. This means that the total collision frequency for the corresponding Boltzmann equation does not depend on velocities. This allows one to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes (as particular cases) the well-known methods of Takizuka & Abe(1977) and Nanbu(1997) and generalizes the approach of Bobylev & Nanbu(2000). The numerical scheme of this paper is simpler than the schemes by Takizuka & Abe and by Nanbu. We derive it for the general case of multicomponent plasmas
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| 7. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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DSMC Modeling of a Single Hot Spot Evolution Using the Landau-Fokker-Planck Equation
- 2014
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Ingår i: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications. - : Springer Science and Business Media LLC. - 0167-8019 .- 1572-9036. ; 132:1, s. 107-116
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Tidskriftsartikel (refereegranskat)abstract
- Numerical solution of a fully nonlinear one dimensional in space and three dimensional in velocity space electron kinetic equation is presented. Direct Simulation Monte Carlo (DSMC) method used for the nonlinear Landau-Fokker-Planck (LFP) collision operator is combined with Particle-in-Cell (PiC) simulations. An assumption of a self-consistent ambipolar electric field is used. The illustrative simulation results for the relaxation of the initial temperature perturbation are compared with the antecedent analytical and numerical results.
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| 8. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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From Particle Systems to the Landau Equations : A Consistency Result
- 2013
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Ingår i: Communications in Mathematical Physics. - : Springer. - 0010-3616 .- 1432-0916. ; 319:3, s. 693-702
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Tidskriftsartikel (refereegranskat)abstract
- We consider a system of N classical particles, interacting via a smooth, short-range potential, in a weak-coupling regime. This means that N tends to infinity when the interaction is suitably rescaled. The j-particle marginals, which obey to the usual BBGKY hierarchy, are decomposed into two contributions: one small but strongly oscillating, the other hopefully smooth. Eliminating the first, we arrive to establish the dynamical problem in term of a new hierarchy (for the smooth part) involving a memory term. We show that the first order correction to the free flow converges, as N →∞, to the corresponding term associated to the Landau equation. We also show the related propagation of chaos.
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| 9. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Group analysis of the generalized Burnett equations
- 2020
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Ingår i: Journal of Nonlinear Mathematical Physics. - : Taylor & Francis. - 1402-9251 .- 1776-0852. ; 27:3, s. 494-494
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Tidskriftsartikel (refereegranskat)abstract
- In this paper group properties of the so-called Generalized Burnett equations are studied. In contrast to the clas-sical Burnett equations these equations are well-posed and therefore can be used in applications. We considerthe one-dimensional version of the generalized Burnett equations for Maxwell molecules in both Eulerian andLagrangian coordinates and perform the complete group analysis of these equations. In particular, this includesfinding and analyzing admitted Lie groups. Our classifications of the Lie symmetries of the Navier-Stokes equa-tions of compressible gas and generalized Burnett equations provide a basis for finding invariant solutions ofthese equations. We also consider representations of all invariant solutions. Some particular classes of invariantsolutions are studied in more detail by both analytical and numerical methods
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| 10. |
- Bobylev, Alexander, 1947-, et al.
(författare)
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Kinetic modeling of economic games with large number of participants
- 2011
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Ingår i: Kinetic and Related Models. - : American Institute of Mathematical Sciences. - 1937-5093 .- 1937-5077. ; 4:1, s. 169-185
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Tidskriftsartikel (refereegranskat)abstract
- We study a Maxwell kinetic model of socio-economic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys., 291 (2009), 599-644. The model depends on three non-negative parameters where is the control parameter. Two other parameters are fixed by market conditions. Self-similar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed.
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