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  • 101.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles2016Ingår i: 30th International Symposium on Rarefied Gas Dynamics: RGD 30 / [ed] Andrew Ketsdever, Henning Struchtrup, American Institute of Physics (AIP), 2016, s. 040005-1-040005-8, artikel-id 040005Konferensbidrag (Refereegranskat)
    Abstract [en]

    An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that, in difference to in the continuous case, DVMs can have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species as well as for binary mixtures. For binary mixtures also the concept of supernormal DVMs has been introduced by Bobylevand Vinerean. Supernormal DVMs are defined as normal DVMs such that both restrictions to the different species are normal as DVMs for single species.

    In this presentation we extend the concept of supernormal DVMs to the case of multicomponent mixtures and introduce it for polyatomic molecules. By polyatomic molecules we mean here that each molecule has one of a finite number of different internal energies, which can change, or not, during a collision. We will present some general algorithms for constructing such models, but also give some concrete examples of such constructions.

    The two different approaches above can be combined to obtain multicomponent mixtures with a finite number of different internal energies, and then be extended in a natural way to chemical reactions.

    The DVMs are constructed in such a way that we for the shock-wave problem obtain similar structures as for the classical discrete Boltzmann equation (DBE) for one species, and therefore will be able to apply previously obtained results for the DBE. In fact the DBE becomes a system of ordinary dierential equations (dynamical system) and the shock profiles can be seen as heteroclinic orbits connecting two singular points (Maxwellians). The previous results for the DBE then give us the existence of shock profiles for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. For binary mixtures this extension has already been addressed before by the author.

  • 102.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Discrete Velocity Models for Polyatomic Molecules Without Nonphysical Collision Invariants2018Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 172, nr 3, s. 742-761Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. Unlike for the Boltzmann equation, for DVMs there can appear extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and hence, without spurious ones, is called normal. The construction of such normal DVMs has been studied a lot in the literature for single species, but also for binary mixtures and recently extensively for multicomponent mixtures. In this paper, we address ways of constructing normal DVMs for polyatomic molecules (here represented by that each molecule has an internal energy, to account for non-translational energies, which can change during collisions), under the assumption that the set of allowed internal energies are finite. We present general algorithms for constructing such models, but we also give concrete examples of such constructions. This approach can also be combined with similar constructions of multicomponent mixtures to obtain multicomponent mixtures with polyatomic molecules, which is also briefly outlined. Then also, chemical reactions can be added.

  • 103.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas2012Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 147, nr 6, s. 1156-1181Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider a non-linear half-space problem related to the condensation problem for the discrete Boltzmann equation and extend some known results for a single-component gas to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian at infinity, as the non-condensable gas tends to zero at infinity. We assume that the vapor is completely absorbed and that the non-condensable gas is diffusively reflected at the condensed phase and that the vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution, needed for the existence of a unique solution of the problem are investigated. We also find exact solvability conditions and solutions for a simplified six+four-velocity model, as the given distribution is a Maxwellian at rest, and study a simplified twelve+six-velocitymodel.

  • 104.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation2015Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 159, nr 2, s. 358-379Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for a general discrete model of a quantum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation. Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calculated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.

  • 105.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    On Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation2005Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    We study some questions related to general discrete velocity (with arbitrarily number of velocities) models (DVMs) of the Boltzmann equation. In the case of plane stationary problems the typical DVM reduces to a dynamical system (system of ODEs). Properties of such systems are studied in the most general case. In particular, a topological classification of their singular points is made and dimensions of the corresponding stable, unstable and center manifolds are computed.

    These results are applied to typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer. A classification of well-posed half-space problems for linearized DVMs is made. Exact solutions of a (simplified) linearized kinetic model of BGK type are found as a limiting case of the corresponding discrete models.

    Existence of solutions of weakly non-linear half-space problems for general DVMs are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. Both implicit, in the non-degenerate cases, and sometimes, in both degenerate and non-degenerate cases, explicit conditions are found.

    Shock-waves can be seen as heteroclinic orbits connecting two singular points (Maxwellians) for DVMs. We give a constructive proof for the existence of solutions of the shock-wave problem for the general DVM. This is worked out for shock speeds close to a typical speed, corresponding to the sound speed in the continuous case. We clarify how close the shock speed must be for our theorem to hold, and present an iteration scheme for obtaining the solution.

    The main results of the paper can be used for DVMs for mixtures as well as for DVMs for one species.

  • 106.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    On half-space problems for the discrete Boltzmann equation2010Ingår i: Il Nuovo Cimento C, ISSN 2037-4909, 1826-9885, Vol. 33, nr 1, s. 47-54Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for the discrete Boltzmann equation (a general discrete velocity model, DVM, with an arbitrary finite number of velocities). Then the discrete Boltzmann equation reduces to a system of ODEs. The data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. A classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete Boltzmann equation is made. In the non-linear case the solutions are assumed to tend to an assigned Maxwellian at infinity. The conditions on the data at the boundary needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at the Maxwellian at infinity is different of -1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. An application to axially symmetric models is also studied

  • 107.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    On half-space problems for the linearized discrete Boltzmann equation2008Ingår i: Rivista di Matematica della Universita' di Parma, Vol. (7)9, s. 73-124Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for the discrete Boltzmann equation. The discrete Boltzmann equation reduces to a system of ODEs for plane stationary problems. These systems are studied, and for general boundary conditions at the "wall" a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete Boltzmann equation is made. Applications for axially symmetric models are studied in more detail. Exact solutions of a (simplified) linearized kinetic model of BGK type are also found as a limiting case of the corresponding discrete models.

  • 108.
    Bernhoff, Niclas
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    On half-space problems for the weakly non-linear discrete Boltzmann equation2010Ingår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, nr 2, s. 195-222Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Existence of solutions of weakly non-linear half-space problems for the general discrete velocity (with arbitrarily finite number of velocities) model of the Boltzmann equation are studied. The solutions are assumed to tend to an assigned Maxwellian at infinity, and the data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. The conditions, on the data at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at infinity is different of -1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. Applications to axially symmetric models are studied in more detail

  • 109.
    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 Equation2007Ingår 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. 173-178Konferensbidrag (Refereegranskat)
    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

  • 110.
    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 equation2007Ingår i: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 5, nr 4, s. 815-832Artikel i tidskrift (Refereegranskat)
    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

  • 111.
    Bernhoff, Niclas
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Vinerean, Mirela
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap (from 2013).
    Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants2016Ingår i: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 165, nr 2, s. 434-453Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    An important aspect of constructing discrete velocity models (DVMs) for the Boltzmann equation is to obtain the right number of collision invariants. It is a well-known fact that DVMs can also have extra collision invariants, so called spurious collision invariants, in plus to the physical ones. A DVM with only physical collision invariants, and so without spurious ones, is called normal. For binary mixtures also the concept of supernormal DVMs was introduced, meaning that in addition to the DVM being normal, the restriction of the DVM to any single species also is normal. Here we introduce generalizations of this concept to DVMs for multicomponent mixtures. We also present some general algorithms for constructing such models and give some concrete examples of such constructions. One of our main results is that for any given number of species, and any given rational mass ratios we can construct a supernormal DVM. The DVMs are constructed in such a way that for half-space problems, as the Milne and Kramers problems, but also nonlinear ones, we obtain similar structures as for the classical discrete Boltzmann equation for one species, and therefore we can apply obtained results for the classical Boltzmann equation.

  • 112.
    Berthold, Stefan
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Inter-temporal Privacy Metrics2014Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    Informational privacy of individuals has significantly gained importance after information technology has become widely deployed. Data, once digitalised, can be copied, distributed, and long-term stored at negligible costs. This has dramatic consequences for individuals that leave traces in the form of personal data whenever they interact with information technology, for instance, computers and phones; or even when information technology is recording the personal data of aware or unaware individuals. The right of individuals for informational privacy, in particular to control the flow and use of their personal data, is easily undermined by those controlling the information technology.

    The objective of this thesis is to study the measurement of informational privacy with a particular focus on scenarios where an individual discloses personal data to a second party which uses this data for re-identifying the individual within a set of other individuals. We contribute with privacy metrics for several instances of this scenario in the publications included in this thesis, most notably one which adds a time dimension to the scenario for modelling the effects of the time passed between data disclosure and usage. The result is a new framework for inter-temporal privacy metrics.

  • 113.
    Berthold, Stefan
    et al.
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Lundin, Reine
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    Re-identification revisitedManuskript (preprint) (Övrigt vetenskapligt)
  • 114.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Boltzmann Equation and Hydrodynamics Beyond the Navier-Stokes Level (Harold Grad Lecture)2007Konferensbidrag (Refereegranskat)
  • 115.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Generalized Burnett Hydrodynamics2008Ingår i: J. Statist. Phys, Vol. 132:3Artikel i tidskrift (Refereegranskat)
    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.

  • 116.
    Bobylev, Alexander
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Instabilities in the Chapman-Enskog expansion and hyperbolic Burnett equations2006Ingår i: J. Statist. PhysArtikel i tidskrift (Refereegranskat)
  • 117.
    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 Systems2003Ingår i: Lecture Notes on the Discretization of the Boltzmann Equation / [ed] N. Bellomo, R. Gatignol, Singapore: World Scientific, 2003, s. 203-222Kapitel i bok, del av antologi (Övrigt vetenskapligt)
  • 118.
    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 equations2011Ingår i: Physics of fluids, ISSN 1070-6631, E-ISSN 1089-7666, Vol. 23, nr 3, s. 030607-030607-10Artikel i tidskrift (Refereegranskat)
    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

  • 119.
    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 Equation2014Ingår 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. 107-116Artikel i tidskrift (Refereegranskat)
    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.

  • 120.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Exact eternal solutions of the Boltzmann equation2002Ingår i: J. Statist. Phys, Vol. 106:5-6Artikel i tidskrift (Refereegranskat)
  • 121.
    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 bath2002Ingår i: J. Statist. Phys, Vol. 106:3-4Artikel i tidskrift (Refereegranskat)
  • 122.
    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 interactions2003Ingår i: J. Statist. Phys, Vol. 110:1-2Artikel i tidskrift (Refereegranskat)
  • 123.
    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 Dynamics2004Ingår i: Lecture Notes in Physics, 2004Kapitel i bok, del av antologi (Övrigt vetenskapligt)
  • 124.
    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 applications2002Ingår i: J. Statist. Phys, Vol. 106:5-6Artikel i tidskrift (Refereegranskat)
  • 125.
    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 molecules2002Ingår i: J. Statist. Phys, Vol. 108:3-4Artikel i tidskrift (Refereegranskat)
  • 126.
    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 equation2002Ingår i: Appl. Math. Lett, Vol. 15:7Artikel i tidskrift (Refereegranskat)
  • 127.
    Bobylev, Alexander
    et al.
    Karlstads universitet, Fakulteten för teknik- och naturvetenskap, Avdelningen för matematik.
    Cercignani, C.
    Weak eternal solutions of the Boltzmann equation2003Konferensbidrag (Refereegranskat)
  • 128.
    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 gases2008Ingår i: Mathematical Models of Granular Matter, Berlin: Springer , 2008Kapitel i bok, del av antologi (Övrigt vetenskapligt)
  • 129.
    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 Models2009Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 291, nr 3, s. 599-644Artikel i tidskrift (Refereegranskat)
    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

  • 130.
    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 materials2003Ingår i: J. Statist. Phys, Vol. 111:1-2Artikel i tidskrift (Refereegranskat)
  • 131.
    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 equation2009Ingår i: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 24, nr 1, s. 35-57Artikel i tidskrift (Refereegranskat)
    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.               

  • 132.
    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 Equation2013Ingår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 6, nr 4, s. 789-800Artikel i tidskrift (Refereegranskat)
    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.

  • 133.
    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 tails2006Ingår i: J. Statist. PhysArtikel i tidskrift (Refereegranskat)
  • 134.
    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 interactions2004Ingår i: J. Statist. Phys, Vol. 116:5-6Artikel i tidskrift (Refereegranskat)
  • 135.
    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 series2012Ingår i: Forum Mathematicum, ISSN 1435-5337, Vol. 24, nr 2, s. 239-251Artikel i tidskrift (Refereegranskat)
    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.

  • 136.
    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 materials2002Ingår i: Eur. J. Mech. B Fluids, Vol. 21:1Artikel i tidskrift (Refereegranskat)
  • 137.
    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 dynamics2001Ingår i: J. Statist. Phys, Vol. 102:5-6Artikel i tidskrift (Refereegranskat)
  • 138.
    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 model2001Ingår i: J. Statist. Phys, Vol. 102:5-6Artikel i tidskrift (Refereegranskat)
  • 139.
    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 plasmas2012Ingår i: Matematicheskoe Modelirovanie, ISSN 0234-0879, Vol. 24, nr 9, s. 35-49Artikel i tidskrift (Refereegranskat)
    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.

  • 140.
    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 forces2007Konferensbidrag (Refereegranskat)
  • 141.
    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 equations2001Ingår i: Appl. Math. Lett, Vol. 14:1Artikel i tidskrift (Refereegranskat)
  • 142.
    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 plasmas2013Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 246, s. 123-144Artikel i tidskrift (Refereegranskat)
    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.

  • 143.
    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 Plasmas2012Ingår 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. 541-548Konferensbidrag (Refereegranskat)
    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

  • 144.
    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 Result2013Ingår i: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 319, nr 3, s. 693-702Artikel i tidskrift (Refereegranskat)
    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.

  • 145.
    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 models2012Ingår 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. 254-261Konferensbidrag (Refereegranskat)
    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.

  • 146.
    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 laws2010Ingår i: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, nr 1, s. 35-58Artikel i tidskrift (Refereegranskat)
  • 147.
    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 Invariants2007Ingår i: Riv. Mat. Univ. Parma (7) 7 (2007), pp.1-80, Vol. 7Artikel i tidskrift (Refereegranskat)
  • 148.
    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 Invariants2008Ingår i: Journal of Statistical PhysicsArtikel i tidskrift (Refereegranskat)
  • 149.
    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 Laws2006Ingår 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 , 2006Kapitel i bok, del av antologi (Övrigt vetenskapligt)
  • 150.
    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 Laws2007Konferensbidrag (Refereegranskat)
    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)

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