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  • 1.
    Bernhoff, Niclas
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Boundary Layers and Shock Profiles for the Broadwell Model2016In: International Journal of Differential Equations, ISSN 1687-9643, E-ISSN 1687-9651, Vol. 2016, p. 1-8, article id 5801728Article in journal (Refereed)
    Abstract [en]

    We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.

  • 2.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Boundary Layers and Shock Profiles for the Discrete Boltzmann Equation for Mixtures2012In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 5, no 1, p. 1-19Article in journal (Refereed)
    Abstract [en]

    We consider the discrete Boltzmann equation for binary gas mixtures. Some known results for half-space problems and shock profile solutions of the discrete Boltzmann for single-component gases are extended to the case of two-component gases. These results include well-posedness results for half-space problems for the linearized discrete Boltzmann equation, existence results for half-space problems for the weakly non-linear discrete Boltzmann equation, and existence results for shock profile solutions of the discrete Boltzmann equation. A characteristic number, corresponding to the speed of sound in the continuous case, is calculated for a class of symmetric models. Some explicit calculations are also made for a simplified 6+4 -velocity model.

  • 3.
    Bernhoff, Niclas
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Boundary layers for discrete kinetic models: Multicomponent mixtures, polyatomic molecules, bimolecular reactions, and quantum kinetic equations2017In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 10, no 4, p. 925-955Article in journal (Refereed)
    Abstract [en]

    We consider some extensions of the classical discrete Boltzmann equation to the cases of multicomponent mixtures, polyatomic molecules (with a finite number of different internal energies), and chemical reactions, but also general discrete quantum kinetic Boltzmann-like equations; discrete versions of the Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation for bosons and fermions and a kinetic equation for excitations in a Bose gas interacting witha Bose-Einstein condensate. In each case we have an H-theorem and so for the planar stationary half-space problem, we have convergence to an equilibrium distribution at infinity (or at least a manifold of equilibrium distributions). In particular, we consider the nonlinear half-space problem of condensation and evaporation for these discrete Boltzmann-like equations. We assume that the flow tends to a stationary point at infinity and that the outgoing flow is known at the wall, maybe also partly linearly depending on the incoming flow. We find that the systems we obtain are of similar structures as for the classical discrete Boltzmann equation (for single species), and that previously obtained results for the discrete Boltzmann equation can be applied after being generalized. Then the number of conditions on the assigned data at the wall needed for existence of a unique solution is found. The number of parameters to be specified in the boundary conditions depends on if we have subsonic or supersonic condensation or evaporation. All our results are valid for any finite number of velocities.

  • 4.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Boundary layers for the nonlinear discrete Boltzmann equation: Condensing vapor flow in the presence of a non-condensable gas2012In: Proceedings of 28th International Symposium on Rarefied Gas Dynamics 2012 / [ed] Michel Mareschal, Andrés Santos, Melville, New York: American Institute of Physics (AIP), 2012, 1, p. 223-230Conference paper (Refereed)
    Abstract [en]

    Half-space problems for the Boltzmann equation are of great importance in the study of the asymptotic behaviorof the solutions of boundary value problems of the Boltzmann equation for small Knudsen numbers. Half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. Here we consider a half-space problem of condensation for apure vapor in the presence of a non-condensable gas by using discrete velocity models (DVMs) of the Boltzmann equation. The Boltzmann equation can be approximated by DVMs up to any order, and these DVMs can be applied for numerical methods,but also for mathematical studies to bring deeper understanding and new ideas. For one-dimensional half-space problems,the discrete Boltzmann equation (the general DVM) reduces to a system of ODEs. We obtain that the number of parametersto be specified in the boundary conditions depends on whether the condensing vapor flow is subsonic or supersonic. Thisbehavior has earlier been found numerically. We want to stress that our results are valid for any finite number of velocities.This is an extension of known results for single-component gases (and for binary mixtures of two vapors) to the case when a non-condensable gas is present. The vapor is assumed to tend to an assigned Maxwellian, with a flow velocity towards thecondensed phase, at infinity, while the non-condensable gas tends to zero at infinity. Steady condensation of the vapor takes place at the condensed phase, which is held at a constant temperature. We assume that the vapor is completely absorbed, that the non-condensable gas is diffusively reflected at the condensed phase, and that vapor molecules leaving the condensed phase are distributed according to a given distribution. The conditions, on the given distribution at the condensed phase, needed for the existence of a unique solution of the problem are investigated, assuming that the given distribution at the condensed phase is sufficiently close to the Maxwellian at infinity and that the total mass of the non-condensable gas is sufficiently small. Exact solutions and solvability conditions are found for a specific simplified discrete velocity model (with few velocities).

  • 5.
    Bernhoff, Niclas
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Discrete quantum Boltzmann equation2019In: AIP Conference Proceedings, American Institute of Physics (AIP), 2019, Vol. 2132, p. 1-9, article id 130011Conference paper (Refereed)
    Abstract [en]

    In this work, we consider a Boltzmann equation for anyons. In particular, we study a general discrete velocity model of the equation, where the velocity variable is assumed to only take values from a given finite-such that the (finite) number of velocities is arbitrary-set of velocities. Included, as two limiting cases, is the discrete quantum Boltzmann equation (Nordheim-Boltzmann/Uehling-Uhlenbeck equation) for bosons and fermions. Mass, momentum, and energy are assumed to be conserved during collisions, and considering suitable discrete velocity models, they will also be the only collision invariants. The equilibrium distributions will be given by a transcendental equation, and only in some few cases-including the two limiting cases where they are Planckians-they will be explicitly expressed. However, there is an H-theorem, and therefore one can prove that for the spatially homogeneous equation, as time tends to infinity, as well as, for the steady equation in a half-space with slab-symmetry, as the space variable tends to infinity, the distribution function converges to an equilibrium distribution. Linearizing around an equilibrium distribution in a suitable way, we find that the obtained linearized operator has similar properties as the corresponding linearized operator for the discrete Boltzmann equation: E.g. it is symmetric and positive semi-definite. Hence, previously obtained results for the spatially homogeneous Cauchy problem and the steady half-space problem in a slab symmetry for the discrete Boltzmann equation, can be applied also in the considered quantum case.

  • 6.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Discrete Velocity Models and Half-Space Problems2003Licentiate thesis, monograph (Other academic)
    Abstract [en]

    We study some questions related to discrete velocity 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 this paper 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. 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 limiting case of corresponding discrete models. The main results of the paper can be also used for moment approximations and other versions of discretizised kinetic equations

  • 7.
    Bernhoff, Niclas
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Discrete velocity models for multicomponent mixtures and polyatomic molecules without nonphysical collision invariants and shock profiles2016In: 30th International Symposium on Rarefied Gas Dynamics: RGD 30 / [ed] Andrew Ketsdever, Henning Struchtrup, American Institute of Physics (AIP), 2016, p. 040005-1-040005-8, article id 040005Conference paper (Refereed)
    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.

  • 8.
    Bernhoff, Niclas
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Discrete Velocity Models for Polyatomic Molecules Without Nonphysical Collision Invariants2018In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 172, no 3, p. 742-761Article in journal (Refereed)
    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.

  • 9.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    Half-Space Problem for the Discrete Boltzmann Equation: Condensing Vapor Flow in the Presence of a Non-condensable Gas2012In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 147, no 6, p. 1156-1181Article in journal (Refereed)
    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.

  • 10.
    Bernhoff, Niclas
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation2015In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 159, no 2, p. 358-379Article in journal (Refereed)
    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.

  • 11.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On Half-Space and Shock-Wave Problems for Discrete Velocity Models of the Boltzmann Equation2005Doctoral thesis, comprehensive summary (Other academic)
    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.

  • 12.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On half-space problems for the discrete Boltzmann equation2010In: Il Nuovo Cimento C, ISSN 2037-4909, 1826-9885, Vol. 33, no 1, p. 47-54Article in journal (Refereed)
    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

  • 13.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On half-space problems for the linearized discrete Boltzmann equation2008In: Rivista di Matematica della Universita' di Parma, Vol. (7)9, p. 73-124Article in journal (Refereed)
    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.

  • 14.
    Bernhoff, Niclas
    Karlstad University, Faculty of Technology and Science, Department of Mathematics.
    On half-space problems for the weakly non-linear discrete Boltzmann equation2010In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, no 2, p. 195-222Article in journal (Refereed)
    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

  • 15.
    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

  • 16.
    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

  • 17.
    Bernhoff, Niclas
    et al.
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Vinerean, Mirela
    Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).
    Discrete Velocity Models for Mixtures Without Nonphysical Collision Invariants2016In: Journal of statistical physics, ISSN 0022-4715, E-ISSN 1572-9613, Vol. 165, no 2, p. 434-453Article in journal (Refereed)
    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.

  • 18.
    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)
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