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  • 1. Acedo, L.
    et al.
    Santos, A.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow2002Inngår i: J. Statist. Phys, Vol. 109:5-6Artikkel i tidsskrift (Fagfellevurdert)
  • 2.
    Andriash, A. V.
    et al.
    All Russia Res Inst Automat VNIIA, Moscow, Russia.
    Bobylev, Alexander V.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Brantov, A. V.
    All Russia Res Inst Automat VNIIA, Moscow, Russia.;RAS, PN Lebedev Phys Inst, Moscow 117901, Russia.
    Bychenkov, V. Yu.
    All Russia Res Inst Automat VNIIA, Moscow, Russia.;RAS, PN Lebedev Phys Inst, Moscow 117901, Russia.
    Karpov, S. A.
    All Russia Res Inst Automat VNIIA, Moscow, Russia.
    Potapenko, I. F.
    RAS, MV Keldysh Appl Math Inst, Moscow 117901, Russia.
    Stochastic simulation of the nonlinear kinetic equation with high-frequency electromagnetic fields2013Inngår i: PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, ISSN 1562-6016, nr 4, s. 233-237Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of Landau-Fokker-Planck (LFP) equations by Boltzmann equations of quasi-Maxwellian kind. High-frequency fields are included into consideration and comparison with the well-known results are given.

  • 3.
    Bernhoff, Niclas
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Weak Shock Wave Solutions for the Discrete Boltzmann Equation2007Inngår 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 , 2007, s. 173-178Konferansepaper (Fagfellevurdert)
    Abstract [en]

    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

  • 4.
    Bernhoff, Niclas
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Weak shock waves for the general discrete velocity model of the Boltzmann equation2007Inngår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 5, nr 4, s. 815-832Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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

    Fulltekst (pdf)
    fulltext
  • 5.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Boltzmann Equation and Hydrodynamics Beyond the Navier-Stokes Level (Harold Grad Lecture)2007Konferansepaper (Fagfellevurdert)
  • 6.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Generalized Burnett Hydrodynamics2008Inngår i: J. Statist. Phys, Vol. 132:3Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Equations of hydrodynamics (derived from the Boltzmann equation) beyond the Navier-Stokes level are studied by a method proposed earlier by the author. The main question we consider is the following: What is the most natural replacement for classical (ill-posed) Burnett equations?

    It is shown that, in some sense, it is a two-parameter set of Generalized Burnett Equations (GBEs) derived in this paper. Some equations of this class are even simpler than original Burnett equations. The region of stability in the space of parameters and other properties of GBEs are discussed.

  • 7.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations2006Inngår i: J. Statist. PhysArtikkel i tidsskrift (Fagfellevurdert)
  • 8.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Discrete Velocity Models and Dynamical Systems2003Inngår i: Lecture Notes on the Discretization of the Boltzmann Equation / [ed] N. Bellomo, R. Gatignol, Singapore: World Scientific, 2003, s. 203-222Kapittel i bok, del av antologi (Annet vitenskapelig)
  • 9.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Bisi, M.
    Dipartimento di Matematica, Università di Parma.
    Cassinari, M.P.
    Dipartimento di Matematica “F. Enriques,” Università di Milano.
    Spiga, G.
    Dipartimento di Matematica, Università di Parma.
    Shock wave structure for generalized Burnett equations2011Inngår i: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 23, nr 3, s. 030607-030607-10Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Stationary shock wave solutions for the generalized Burnett equations (GBE) [ A. V. Bobylev, Generalized Burnett hydrodynamics, J. Stat. Phys. 132, 569 (2008) ] are studied. Based on the results of Bisi et al. [Qualitative analysis of the generalized Burnett equations and applications to half-space problems, Kinet. Relat. Models 1, 295 (2008) ], we choose a unique (optimal) form of GBE and solve numerically the shock wave problem for various Mach numbers. The results are compared with the numerical solutions of NavierStokes equations and with the MottSmith approximation for the Boltzmann equation (all calculations are done for Maxwell molecules) since it is believed that the MottSmith approximation yields better results for strong shocks. The comparison shows that GBE yield certain improvement of the NavierStokes results for moderate Mach numbers

  • 10.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Brantov, Andrei
    RAS, Lebedev Phys Inst, Moscow 117901, Russia.;Dukhov All Russia Res Inst Automat, Moscow, Russia..
    Bychenkov, Valerii
    RAS, Lebedev Phys Inst, Moscow 117901, Russia.;Dukhov All Russia Res Inst Automat, Moscow, Russia..
    Karpov, Stanislav
    Dukhov All Russia Res Inst Automat, Moscow, Russia..
    Potapenko, Irina
    Dukhov All Russia Res Inst Automat, Moscow, Russia.;RAS, Keldysh Inst Appl Math, Moscow 117901, Russia..
    DSMC Modeling of a Single Hot Spot Evolution Using the Landau-Fokker-Planck Equation2014Inngår i: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 132, nr 1, s. 107-116Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 11.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Exact eternal solutions of the Boltzmann equation2002Inngår i: J. Statist. Phys, Vol. 106:5-6Artikkel i tidsskrift (Fagfellevurdert)
  • 12.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Moment equations for a granular material in a thermal bath2002Inngår i: J. Statist. Phys, Vol. 106:3-4Artikkel i tidsskrift (Fagfellevurdert)
  • 13.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions2003Inngår i: J. Statist. Phys, Vol. 110:1-2Artikkel i tidsskrift (Fagfellevurdert)
  • 14.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Self-Similar Asymptotics for the Boltzmann equation with Inelastic Interactions, in Granular Gas Dynamics2004Inngår i: Lecture Notes in Physics, 2004Kapittel i bok, del av antologi (Annet vitenskapelig)
  • 15.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Self-similar solutions of the Boltzmann equation and their applications2002Inngår i: J. Statist. Phys, Vol. 106:5-6Artikkel i tidsskrift (Fagfellevurdert)
  • 16.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Self-similar solutions of the Boltzmann equation for non-Maxwell molecules2002Inngår i: J. Statist. Phys, Vol. 108:3-4Artikkel i tidsskrift (Fagfellevurdert)
  • 17.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation2002Inngår i: Appl. Math. Lett, Vol. 15:7Artikkel i tidsskrift (Fagfellevurdert)
  • 18.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Weak eternal solutions of the Boltzmann equation2003Konferansepaper (Fagfellevurdert)
  • 19.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Gamba, I. M.
    Generalized kinetic Maxwell type models of granular gases2008Inngår i: Mathematical Models of Granular Matter, Berlin: Springer , 2008Kapittel i bok, del av antologi (Annet vitenskapelig)
  • 20.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Politecnico di Milano.
    Gamba, I.M.
    Department of Mathematics, The University of Texas at Austin.
    On the Self-Similar Asymptotics for Generalized Nonlinear Kinetic Maxwell Models2009Inngår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 291, nr 3, s. 599-644Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Maxwell models for nonlinear kinetic equations have many applications in physics, dynamics of granular gases, economics, etc. In the present manuscript we consider such models from a very general point of view, including those with arbitrary polynomial non-linearities and in any dimension space. It is shown that the whole class of generalized Maxwell models satisfies properties one of which can be interpreted as an operator generalization of usual Lipschitz conditions. This property allows to describe in detail a behavior of solutions to the corresponding initial value problem. In particular, we prove in the most general case an existence of self similar solutions and study the convergence, in the sense of probability measures, of dynamically scaled solutions to the Cauchy problem to those self-similar solutions, as time goes to infinity. A new application of multi-linear models to economics and social dynamics is discussed

  • 21.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Toscani, G.
    Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials2003Inngår i: J. Statist. Phys, Vol. 111:1-2Artikkel i tidsskrift (Fagfellevurdert)
  • 22.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Dorodnitsyn, Vladimir
    Keldysh Institute of Applied Mathematics, Russian Academy of Science.
    Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation2009Inngår i: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 24, nr 1, s. 35-57Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

     In this paper we consider Lie group symmetries of evolutionequations with non-local operators in context of applications tononlinear kinetic equations. As an illustration we consider theBoltzmann equation and calculate the admitted group of pointtransformations.               

  • 23.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Esposito, Raffaele
    University Aquila, Italy .
    Transport Coefficients in the 2-dimensional Boltzmann Equation2013Inngår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 6, nr 4, s. 789-800Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We show that a rarefied system of hard disks in a plane, described in the Boltzmann-Grad limit by the 2-dimensional Boltzmann equation, has bounded transport coefficients. This is proved by showing opportune compactness properties of the gain part of the linearized Boltzmann operator.

  • 24.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Gamba, I. M
    Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails2006Inngår i: J. Statist. PhysArtikkel i tidsskrift (Fagfellevurdert)
  • 25.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Gamba, I. M.
    Panferov, V. A.
    Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions2004Inngår i: J. Statist. Phys, Vol. 116:5-6Artikkel i tidsskrift (Fagfellevurdert)
  • 26.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Gamba, Irene
    Department of Mathematics, The University of Texas at Austin.
    Solutions of the linear Boltzmann equation and some Dirichlet series2012Inngår i: Forum Mathematicum, ISSN 1435-5337, Vol. 24, nr 2, s. 239-251Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    It is shown that a broad class of generalized Dirichlet series (including the polylogarithm, related to the Riemann zeta-function) can be presented as a classof solutions of the Fourier transformed spatially homogeneous linear Boltzmannequation with a special Maxwell-type collision kernel. The result is based on anexplicit integral representationof solutions to the Cauchy problem for the Boltzmann equation. Possibleapplications to the theory of Dirichlet seriesare briefly discussed.

  • 27.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Groppi, Maria
    Spiga, Giampiero
    Approximate solutions to the problem of stationary shear flow of smooth granular materials2002Inngår i: Eur. J. Mech. B Fluids, Vol. 21:1Artikkel i tidsskrift (Fagfellevurdert)
  • 28.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Grzhibovskis, R.
    Heintz, A.
    Entropy inequalities for evaporation/condensation problem in rarefied gas dynamics2001Inngår i: J. Statist. Phys, Vol. 102:5-6Artikkel i tidsskrift (Fagfellevurdert)
  • 29.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Hansen, Alex
    Piasecki, J.
    Hauge, E. H.
    From the Liouville equation to the generalized Boltzmann equation for magnetotransport in the 2D Lorentz model2001Inngår i: J. Statist. Phys, Vol. 102:5-6Artikkel i tidsskrift (Fagfellevurdert)
  • 30.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Karpov, S.A.
    Keldysh Institute for Applied Mathematics.
    Potapenko, I.F.
    Keldysh Institute for Applied Mathematics.
    Monte-Carlo method for two component plasmas2012Inngår i: Matematicheskoe Modelirovanie, ISSN 0234-0879, Vol. 24, nr 9, s. 35-49Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The new direct simulation method of Monte-Carlo type (DSMC) for Coulomb collisions in the case of two component plasma is considered.  A brief literature review and preliminary information concerning the problem are given. Further the idea that lies in the basis of the method is discussed and its scheme is provided. The illustrative numerical simulation of the initial distribution relaxation for one and two sorts of particles in 3D case in the velocity space is performed. Simulation results are compared with the numerical results based on the completely conservative finite difference schemes for the Landau-Fokker-Planck equation. Estimation of calculation accuracy obtained from numerical results is given.

  • 31.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Meleshko, Sergey V.
    Suranaree University of Technology, Thailand.
    Group analysis of the generalized Burnett equations2020Inngår i: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 27, nr 3, s. 494-494Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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

    Fulltekst (pdf)
    fulltext
  • 32.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Mossberg, Eva
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Potapenko, I.
    DSMC methods for the Landau equation and for the Boltzmann equation with long range forces2007Konferansepaper (Fagfellevurdert)
  • 33.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Ohwada, T.
    The error of the splitting scheme for solving evolutionary equations2001Inngår i: Appl. Math. Lett, Vol. 14:1Artikkel i tidsskrift (Fagfellevurdert)
  • 34.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Potapenko, Irina
    Russia.
    Monte Carlo methods and their analysis for Coulomb collisions in multicomponent plasmas2013Inngår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 246, s. 123-144Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A general approach to Monte Carlo methods for Coulomb collisions is proposed. Its key idea is an approximation of Landau-Fokker-Planck equations by Boltzmann equations of quasi-Maxwellian kind. It means that the total collision frequency for the corresponding Boltzmann equation does not depend on the velocities. This allows to make the simulation process very simple since the collision pairs can be chosen arbitrarily, without restriction. It is shown that this approach includes the well-known methods of Takizuka and Abe (1977) [12] and Nanbu (1997) as particular cases, and generalizes the approach of Bobylev and Nanbu (2000). The numerical scheme of this paper is simpler than the schemes by Takizuka and Abe [12] and by Nanbu. We derive it for the general case of multicomponent plasmas and show some numerical tests for the two-component (electrons and ions) case. An optimal choice of parameters for speeding up the computations is also discussed. It is also proved that the order of approximation is not worse than O(root epsilon), where epsilon is a parameter of approximation being equivalent to the time step Delta t in earlier methods. A similar estimate is obtained for the methods of Takizuka and Abe and Nanbu. (C) 2013 Elsevier Inc. All rights reserved.

  • 35.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Potapenko,, Irina F.
    Keldysh Institute for Applied Mathematics.
    Karpov, Stanislav A.
    Keldysh Institute for Applied Mathematics.
    DSMC Methods for Multicomponent Plasmas2012Inngår i: DSMC and Related Simulations: 28th International Symposium on Rarefied Gas Dynamics  2012 / [ed] Michel Mareschal, Andrés Santos, New York: American Institute of Physics (AIP), 2012, 1, s. 541-548Konferansepaper (Fagfellevurdert)
    Abstract [en]

    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

  • 36.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Pulvirenti, Mario
    Dipartimento di Matematica Guido Castelnuovo, Università La Sapienza, Roma.
    Saffirio, Chiara
    Dipartimento di Matematica Guido Castelnuovo, Università La Sapienza, Roma.
    From Particle Systems to the Landau Equations: A Consistency Result2013Inngår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 319, nr 3, s. 693-702Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 37.
    Bobylev, Alexander V.
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Mossberg, Eva
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    On some properties of linear and linearized Boltzmann collision operators for hard spheres2008Inngår i: Kinetic and related models, ISSN 1937-5093, Vol. 1, nr 4, s. 521-555Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The linear and the linearized Boltzmann collision operators are studied in the case of hard spheres. The equations for the integral operators are reduced to ordinary differential equations. From these equations, we show that the collision operators have discrete eigenvalues, and we demonstrate how to compute them. We also use the differential equations to investigate asymptotics for the linearized collision operator.

  • 38.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Vinerean (Bernhoff), Mirela
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Symmetric extensions of normal discrete velocity models2012Inngår i: 28th International Symposium on Rarefied Gas Dynamics 2012 / [ed] Michel Mareschal, Andrés Santos, American Institute of Physics (AIP), 2012, 1, Vol. 1501, nr 1, s. 254-261Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In this paper we discuss a general problem related to spurious conservation laws for discrete velocity models (DVMs) of the classical (elastic) Boltzmann equation. Models with spurious conservation laws appeared already at the early stage of the development of discrete kinetic theory. The well-known theorem of uniqueness of collision invariants for the continuous velocity space very often does not hold for a set of discrete velocities. In our previous works we considered the general problem of the construction of normal DVMs, we found a general algorithm for the construction of all such models and presented a complete classification of normal DVMs with small number n of velocities (n<11). Even if we have a general method to classify all normal discrete kinetic models (and in particular DVMs), the existing method is relatively slow and the amount of possible cases to check increases rapidly with n. We remarked that many of our normal DVMs appear to be axially symmetric. In this paper we consider a connection between symmetric transformations and normal DVMs. We first develop a new inductive method that, starting with a given normal DVM, leads by symmetric extensions to a new normal DVM. This method can produce very fast many new normal DVMs with larger number of velocities, showing that the class of normal DVMs contains a large subclass of symmetric models. We finally apply the method to several normal DVMs and construct new models that are not only normal, but also symmetric relatively to more and more axes. We hope that such symmetric velocitysets can be used for DSMC methods of solving Boltzmann equation.

  • 39.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Vinerean (Bernhoff), Mirela
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Windfäll, Åsa
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Discrete velocity models of the Boltzmann equation and conservation laws2010Inngår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, nr 1, s. 35-58Artikkel i tidsskrift (Fagfellevurdert)
  • 40.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Vinerean-Bernhoff, Mirela
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Construction and Classification of Discrete Kinetic Models without Spurious Invariants2007Inngår i: Riv. Mat. Univ. Parma (7) 7 (2007), pp.1-80, Vol. 7Artikkel i tidsskrift (Fagfellevurdert)
  • 41.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Vinerean-Bernhoff, Mirela
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Construction of Discrete Kinetic Models with Given Invariants2008Inngår i: Journal of Statistical PhysicsArtikkel i tidsskrift (Fagfellevurdert)
  • 42.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Vinerean-Bernhoff, Mirela
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Discrete Kinetic Models and Conservation Laws2006Inngår i: Modelling and Numerics of Kinetic Dissipative Systems, L. Pareschi and G. Russo and G. Toscani, eds., Nova Science Publishers, 2006, pp. 147-162, Nova Science , 2006Kapittel i bok, del av antologi (Annet vitenskapelig)
  • 43.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Vinerean-Bernhoff, Mirela
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    General Methods of the Construction of Discrete Kinetic Models with Given Conservation Laws2007Konferansepaper (Fagfellevurdert)
    Abstract [en]

    In the present work we consider the general problem of the construction of discrete kinetic models (DKMs) with given conservation laws. This problem was first stated by R. Gatignol [1] in connection with discrete models of the Boltzmann equation (BE), when it became clear that the velocity discretization can lead to equations with spurious conservation laws (not linear combinations of physical invariants). The problem has been addressed in the last decade by several authors, in particular by Cercignani, Bobylev, Vedenyapin, and Cornille. Even though a practical criterion for the non-existence of spurious conservation laws has been devised [2], and a method for enlarging existing physical models by new velocity points without adding non-physical invariants has been proposed [3], a general algorithm for the construction of all normal (physical) discrete models with assigned conservation laws, in any dimension and for any number of points, is still lacking in the literature. We develop such a general algorithm in the present work.

    We introduce the most general class of discrete kinetic models and obtain a general method for the construction and classification of normal DKMs. In particular, it is proved that for any given dimension d 2 and for any sufficiently large number N of velocities (for example, N 6 for the planar case d = 2) there exists just a finite number of distinct (non-equivalent) classes of DKMs. We apply the general method in the particular cases of discrete velocity models (DVMs) of the inelastic BE and elastic BE. In the first case, we show that all normal models can be explicitly described. In the second case, we give a complete classification of normal models up to 9 velocities and show that the extension method [3], does not lead to all normal DVMs.

    Using our general approach to DKMs and our results on normal DVMs for a single gas, we develop a method for the construction of the most natural (from physical point of view) subclass of normal DVMs for binary gas mixtures. We call such models supernormal models (SNMs) (they have the property that by isolating the velocities of one-kind particles of the single gases involved in the mixture, the corresponding discrete models for a single gas are also normal models). We apply this method and obtain SNMs with up to 20 velocities and their spectrum of mass ratio.

    Finally, we develop a new method that can lead, by symmetric transformations, from a given normal DVM to extended normal DVMs.



    ACKNOWLEDGEMENTS. The support by the grant 2003-5357 from Swedish Research Council for both authors is gratefully acknowledged.



    REFERENCES

    [1] R. Gatignol, Théorie Cinétique des Gaz à Répartition Discrète de Vitesses, Springer-Verlag, New-York, 1975

    [2] V. V. Vedenyapin, Y. N. Orlov, Teoret. and Math. Phys., 121, 1516-1523 (1999)

    [3] A. V. Bobylev, C. Cercignani, J. Statist. Phys., 97, 677-686 (1999)

  • 44.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Windfäll, Åsa
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Boltzmann equation and hydrodynamics at the Burnett level2012Inngår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 5, nr 2, s. 237-260Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    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.

  • 45.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Windfäll, Åsa
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Kinetic modeling of economic games with large number of participants2011Inngår i: Kinetic and Related Models, ISSN 1937-5093, Vol. 4, nr 1, s. 169-185Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

                     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.

  • 46.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Windfäll, Åsa
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    On a special function related to the Riemann zeta-functionManuskript (preprint) (Annet vitenskapelig)
  • 47. Potapenko, I.
    et al.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Mossberg, Eva
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Deterministic and stochastic methods for nonlinear Landau-Fokker-Planck kinetic equations with applications to plasma physics2008Inngår i: Transport Theory and Statistical Physics, 37:113-170Artikkel i tidsskrift (Fagfellevurdert)
  • 48.
    Vinerean, Mirela
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Windfäll, Åsa
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Construction of Normal Discrete Velocity Models of the Boltzmann Equation2010Inngår i: Nuovo cimento della societa italiana de fisica. C, Geophysics and space physics, ISSN 1124-1896, E-ISSN 1826-9885, Vol. 33, nr 1, s. 257-264Artikkel i tidsskrift (Fagfellevurdert)
1 - 48 of 48
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