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  • 1. Acedo, L.
    et al.
    Santos, A.
    Bobylev, Alexander
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On the derivation of a high-velocity tail from the Boltzmann-Fokker-Planck equation for shear flow2002In: J. Statist. Phys, Vol. 109:5-6Article in journal (Refereed)
  • 2.
    Andriash, A. V.
    et al.
    All Russia Res Inst Automat VNIIA, Moscow, Russia.
    Bobylev, Alexander V.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.
    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 fields2013In: PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, ISSN 1562-6016, no 4, p. 233-237Article in journal (Refereed)
    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.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Bobylev, Alexander
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Weak Shock Wave Solutions for the Discrete Boltzmann Equation2007In: 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, p. 173-178Conference paper (Refereed)
    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.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Bobylev, Alexander
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Weak shock waves for the general discrete velocity model of the Boltzmann equation2007In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 5, no 4, p. 815-832Article in journal (Refereed)
    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

  • 5.
    Bobylev, Alexander
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Boltzmann Equation and Hydrodynamics Beyond the Navier-Stokes Level (Harold Grad Lecture)2007Conference paper (Refereed)
  • 6.
    Bobylev, Alexander
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Generalized Burnett Hydrodynamics2008In: J. Statist. Phys, Vol. 132:3Article in journal (Refereed)
    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
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations2006In: J. Statist. PhysArticle in journal (Refereed)
  • 8.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Discrete Velocity Models and Dynamical Systems2003In: Lecture Notes on the Discretization of the Boltzmann Equation / [ed] N. Bellomo, R. Gatignol, Singapore: World Scientific, 2003, p. 203-222Chapter in book (Other academic)
  • 9.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    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 equations2011In: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 23, no 3, p. 030607-030607-10Article in journal (Refereed)
    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.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (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 Equation2014In: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications, ISSN 0167-8019, E-ISSN 1572-9036, Vol. 132, no 1, p. 107-116Article in journal (Refereed)
    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.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Exact eternal solutions of the Boltzmann equation2002In: J. Statist. Phys, Vol. 106:5-6Article in journal (Refereed)
  • 12.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Moment equations for a granular material in a thermal bath2002In: J. Statist. Phys, Vol. 106:3-4Article in journal (Refereed)
  • 13.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions2003In: J. Statist. Phys, Vol. 110:1-2Article in journal (Refereed)
  • 14.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Self-Similar Asymptotics for the Boltzmann equation with Inelastic Interactions, in Granular Gas Dynamics2004In: Lecture Notes in Physics, 2004Chapter in book (Other academic)
  • 15.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Self-similar solutions of the Boltzmann equation and their applications2002In: J. Statist. Phys, Vol. 106:5-6Article in journal (Refereed)
  • 16.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Self-similar solutions of the Boltzmann equation for non-Maxwell molecules2002In: J. Statist. Phys, Vol. 108:3-4Article in journal (Refereed)
  • 17.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    The inverse Laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation2002In: Appl. Math. Lett, Vol. 15:7Article in journal (Refereed)
  • 18.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Weak eternal solutions of the Boltzmann equation2003Conference paper (Refereed)
  • 19.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Gamba, I. M.
    Generalized kinetic Maxwell type models of granular gases2008In: Mathematical Models of Granular Matter, Berlin: Springer , 2008Chapter in book (Other academic)
  • 20.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    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 Models2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 291, no 3, p. 599-644Article in journal (Refereed)
    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.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Cercignani, C.
    Toscani, G.
    Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials2003In: J. Statist. Phys, Vol. 111:1-2Article in journal (Refereed)
  • 22.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Dorodnitsyn, Vladimir
    Keldysh Institute of Applied Mathematics, Russian Academy of Science.
    Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation2009In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 24, no 1, p. 35-57Article in journal (Refereed)
    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.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Esposito, Raffaele
    University Aquila, Italy .
    Transport Coefficients in the 2-dimensional Boltzmann Equation2013In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 6, no 4, p. 789-800Article in journal (Refereed)
    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.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Gamba, I. M
    Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails2006In: J. Statist. PhysArticle in journal (Refereed)
  • 25.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Gamba, I. M.
    Panferov, V. A.
    Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions2004In: J. Statist. Phys, Vol. 116:5-6Article in journal (Refereed)
  • 26.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Gamba, Irene
    Department of Mathematics, The University of Texas at Austin.
    Solutions of the linear Boltzmann equation and some Dirichlet series2012In: Forum Mathematicum, ISSN 1435-5337, Vol. 24, no 2, p. 239-251Article in journal (Refereed)
    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.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Groppi, Maria
    Spiga, Giampiero
    Approximate solutions to the problem of stationary shear flow of smooth granular materials2002In: Eur. J. Mech. B Fluids, Vol. 21:1Article in journal (Refereed)
  • 28.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Grzhibovskis, R.
    Heintz, A.
    Entropy inequalities for evaporation/condensation problem in rarefied gas dynamics2001In: J. Statist. Phys, Vol. 102:5-6Article in journal (Refereed)
  • 29.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Hansen, Alex
    Piasecki, J.
    Hauge, E. H.
    From the Liouville equation to the generalized Boltzmann equation for magnetotransport in the 2D Lorentz model2001In: J. Statist. Phys, Vol. 102:5-6Article in journal (Refereed)
  • 30.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Karpov, S.A.
    Keldysh Institute for Applied Mathematics.
    Potapenko, I.F.
    Keldysh Institute for Applied Mathematics.
    Monte-Carlo method for two component plasmas2012In: Matematicheskoe Modelirovanie, ISSN 0234-0879, Vol. 24, no 9, p. 35-49Article in journal (Refereed)
    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.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Mossberg, Eva
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Potapenko, I.
    DSMC methods for the Landau equation and for the Boltzmann equation with long range forces2007Conference paper (Refereed)
  • 32.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Ohwada, T.
    The error of the splitting scheme for solving evolutionary equations2001In: Appl. Math. Lett, Vol. 14:1Article in journal (Refereed)
  • 33.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Potapenko, Irina
    Russia.
    Monte Carlo methods and their analysis for Coulomb collisions in multicomponent plasmas2013In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 246, p. 123-144Article in journal (Refereed)
    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.

  • 34.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Potapenko,, Irina F.
    Keldysh Institute for Applied Mathematics.
    Karpov, Stanislav A.
    Keldysh Institute for Applied Mathematics.
    DSMC Methods for Multicomponent Plasmas2012In: 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, p. 541-548Conference paper (Refereed)
    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

  • 35.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    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 Result2013In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 319, no 3, p. 693-702Article in journal (Refereed)
    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.

  • 36.
    Bobylev, Alexander V.
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Mossberg, Eva
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On some properties of linear and linearized Boltzmann collision operators for hard spheres2008In: Kinetic and related models, ISSN 1937-5093, Vol. 1, no 4, p. 521-555Article in journal (Refereed)
    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.

  • 37.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Vinerean (Bernhoff), Mirela
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Symmetric extensions of normal discrete velocity models2012In: 28th International Symposium on Rarefied Gas Dynamics 2012 / [ed] Michel Mareschal, Andrés Santos, American Institute of Physics (AIP), 2012, 1, Vol. 1501, no 1, p. 254-261Conference paper (Refereed)
    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.

  • 38.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Vinerean (Bernhoff), Mirela
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Windfäll, Åsa
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Discrete velocity models of the Boltzmann equation and conservation laws2010In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, no 1, p. 35-58Article in journal (Refereed)
  • 39.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Vinerean-Bernhoff, Mirela
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Construction and Classification of Discrete Kinetic Models without Spurious Invariants2007In: Riv. Mat. Univ. Parma (7) 7 (2007), pp.1-80, Vol. 7Article in journal (Refereed)
  • 40.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Vinerean-Bernhoff, Mirela
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Construction of Discrete Kinetic Models with Given Invariants2008In: Journal of Statistical PhysicsArticle in journal (Refereed)
  • 41.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Vinerean-Bernhoff, Mirela
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Discrete Kinetic Models and Conservation Laws2006In: 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 , 2006Chapter in book (Other academic)
  • 42.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Vinerean-Bernhoff, Mirela
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    General Methods of the Construction of Discrete Kinetic Models with Given Conservation Laws2007Conference paper (Refereed)
    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)

  • 43.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Windfäll, Åsa
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Boltzmann equation and hydrodynamics at the Burnett level2012In: Kinetic and Related Models, ISSN 1937-5093, Vol. 5, no 2, p. 237-260Article in journal (Refereed)
    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.

  • 44.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Windfäll, Åsa
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Kinetic modeling of economic games with large number of participants2011In: Kinetic and Related Models, ISSN 1937-5093, Vol. 4, no 1, p. 169-185Article in journal (Refereed)
    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.

  • 45.
    Bobylev, Alexander
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Windfäll, Åsa
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On a special function related to the Riemann zeta-functionManuscript (preprint) (Other academic)
  • 46. Potapenko, I.
    et al.
    Bobylev, Alexander
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Mossberg, Eva
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Deterministic and stochastic methods for nonlinear Landau-Fokker-Planck kinetic equations with applications to plasma physics2008In: Transport Theory and Statistical Physics, 37:113-170Article in journal (Refereed)
  • 47.
    Vinerean, Mirela
    et al.
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Windfäll, Åsa
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Bobylev, Alexander
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Construction of Normal Discrete Velocity Models of the Boltzmann Equation2010In: Nuovo cimento della societa italiana de fisica. C, Geophysics and space physics, ISSN 1124-1896, E-ISSN 1826-9885, Vol. 33, no 1, p. 257-264Article in journal (Refereed)
1 - 47 of 47
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