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Bobylev, Alexander

Open this publication in new window or tab >>DSMC Modeling of a Single Hot Spot Evolution Using the Landau-Fokker-Planck Equation### Bobylev, Alexander

### Brantov, Andrei

### Bychenkov, Valerii

### Karpov, Stanislav

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_0_j_idt195_some",{id:"formSmash:j_idt191:0:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_0_j_idt195_some",multiple:true}); ### Potapenko, Irina

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_0_j_idt195_otherAuthors",{id:"formSmash:j_idt191:0:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_0_j_idt195_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt191_0_j_idt195_j_idt209",{id:"formSmash:j_idt191:0:j_idt195:j_idt209",widgetVar:"widget_formSmash_j_idt191_0_j_idt195_j_idt209",onLabel:"Hide others...",offLabel:"Show others..."}); 2014 (English)In: 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) Published
##### Abstract [en]

##### Keywords

Boltzmann kinetic equation, Coulomb collisions, Nonlinear Landau-Fokker-Planck equation, DSMC method, PiC method, Nonlocal heat transport
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-41509 (URN)10.1007/s10440-014-9940-x (DOI)000341706300010 ()
#####

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Available from: 2016-04-25 Created: 2016-04-11 Last updated: 2020-05-20Bibliographically approved

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).

RAS, Lebedev Phys Inst, Moscow 117901, Russia.;Dukhov All Russia Res Inst Automat, Moscow, Russia..

RAS, Lebedev Phys Inst, Moscow 117901, Russia.;Dukhov All Russia Res Inst Automat, Moscow, Russia..

Dukhov All Russia Res Inst Automat, Moscow, Russia..

Dukhov All Russia Res Inst Automat, Moscow, Russia.;RAS, Keldysh Inst Appl Math, Moscow 117901, Russia..

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.

Open this publication in new window or tab >>From Particle Systems to the Landau Equations: A Consistency Result### Bobylev, Alexander

### Pulvirenti, Mario

### Saffirio, Chiara

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_1_j_idt195_some",{id:"formSmash:j_idt191:1:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_1_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_1_j_idt195_otherAuthors",{id:"formSmash:j_idt191:1:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_1_j_idt195_otherAuthors",multiple:true}); 2013 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 319, no 3, p. 693-702Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2013
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-16083 (URN)10.1007/s00220-012-1633-6 (DOI)000318291500003 ()
#####

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##### Funder

Swedish Research Council, 621-2009-5751
Available from: 2012-12-04 Created: 2012-12-04 Last updated: 2017-12-07Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Dipartimento di Matematica Guido Castelnuovo, Università La Sapienza, Roma.

Dipartimento di Matematica Guido Castelnuovo, Università La Sapienza, Roma.

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.

Open this publication in new window or tab >>Monte Carlo methods and their analysis for Coulomb collisions in multicomponent plasmas### Bobylev, Alexander

### Potapenko, Irina

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_2_j_idt195_some",{id:"formSmash:j_idt191:2:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_2_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_2_j_idt195_otherAuthors",{id:"formSmash:j_idt191:2:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_2_j_idt195_otherAuthors",multiple:true}); 2013 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 246, p. 123-144Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2013
##### Keywords

Monte Carlo methods, Coulomb collisions, Landau-Fokker-Planck equations, Boltzmann equations, Error of approximation
##### National Category

Other Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-38602 (URN)10.1016/j.jcp.2013.03.024 (DOI)000320604000009 ()
#####

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Available from: 2015-11-30 Created: 2015-11-23 Last updated: 2017-12-01Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Russia.

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.

Open this publication in new window or tab >>Stochastic simulation of the nonlinear kinetic equation with high-frequency electromagnetic fields### Andriash, A. V.

### Bobylev, Alexander V.

### Brantov, A. V.

### Bychenkov, V. Yu.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_some",{id:"formSmash:j_idt191:3:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_some",multiple:true}); ### Karpov, S. A.

### Potapenko, I. F.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_otherAuthors",{id:"formSmash:j_idt191:3:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt191_3_j_idt195_j_idt209",{id:"formSmash:j_idt191:3:j_idt195:j_idt209",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_j_idt209",onLabel:"Hide others...",offLabel:"Show others..."}); 2013 (English)In: PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, ISSN 1562-6016, no 4, p. 233-237Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

KHARKOV INST PHYSICS & TECHNOLOGY, NATL SCIENCE CTR, 2013
##### Keywords

BASIC INTERACTIONS, CALCULATION METHODS, DIFFERENTIAL EQUATIONS, ELASTIC SCATTERING, ELECTROMAGNETIC INTERACTIONS, EQUATIONS, FUNCTIONS, INTEGRO-DIFFERENTIAL EQUATIONS, INTERACTIONS, KINETIC EQUATIONS
##### National Category

Mathematics Mathematical Analysis
##### Research subject

Materials Science
##### Identifiers

urn:nbn:se:kau:diva-38666 (URN)000324081600054 ()
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_j_idt378",{id:"formSmash:j_idt191:3:j_idt195:j_idt378",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_j_idt378",multiple:true});
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Available from: 2015-11-23 Created: 2015-11-23 Last updated: 2016-08-12Bibliographically approved

All Russia Res Inst Automat VNIIA, Moscow, Russia.

Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science.

All Russia Res Inst Automat VNIIA, Moscow, Russia.;RAS, PN Lebedev Phys Inst, Moscow 117901, Russia.

All Russia Res Inst Automat VNIIA, Moscow, Russia.;RAS, PN Lebedev Phys Inst, Moscow 117901, Russia.

All Russia Res Inst Automat VNIIA, Moscow, Russia.

RAS, MV Keldysh Appl Math Inst, Moscow 117901, Russia.

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.

Open this publication in new window or tab >>Transport Coefficients in the 2-dimensional Boltzmann Equation### Bobylev, Alexander

### Esposito, Raffaele

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_4_j_idt195_some",{id:"formSmash:j_idt191:4:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_4_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_4_j_idt195_otherAuthors",{id:"formSmash:j_idt191:4:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_4_j_idt195_otherAuthors",multiple:true}); 2013 (English)In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 6, no 4, p. 789-800Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Boltzmann equation, transport coefficients
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-38569 (URN)10.3934/krm.2013.6.789 (DOI)000327733900008 ()
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Available from: 2016-01-22 Created: 2015-11-23 Last updated: 2017-11-30Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

University Aquila, Italy .

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.

Open this publication in new window or tab >>Boltzmann equation and hydrodynamics at the Burnett level### Bobylev, Alexander

### Windfäll, Åsa

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_5_j_idt195_some",{id:"formSmash:j_idt191:5:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_5_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_5_j_idt195_otherAuthors",{id:"formSmash:j_idt191:5:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_5_j_idt195_otherAuthors",multiple:true}); 2012 (English)In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 5, no 2, p. 237-260Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Institute of Mathematical Sciences, 2012
##### Keywords

Hydrodynamics, regularized Burnett equations, Stability, sound propagation.
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-8710 (URN)10.3934/krm.2012.5.237 (DOI)000302962700002 ()
#####

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Available from: 2011-11-03 Created: 2011-11-03 Last updated: 2020-01-03Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

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.

Open this publication in new window or tab >>DSMC Methods for Multicomponent Plasmas### Bobylev, Alexander

### Potapenko,, Irina F.

### Karpov, Stanislav A.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_6_j_idt195_some",{id:"formSmash:j_idt191:6:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_6_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_6_j_idt195_otherAuthors",{id:"formSmash:j_idt191:6:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_6_j_idt195_otherAuthors",multiple:true}); 2012 (English)In: 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, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

New York: American Institute of Physics (AIP), 2012 Edition: 1
##### Series

AIP Conference Proceedings, ISSN 0094-243X ; 1501
##### Keywords

Boltzmann equations, Coulomb collisions, Landau-Fokker-Planck equations, Monte Carlo methods
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-16082 (URN)10.1063/1.4769589 (DOI)000312411200070 ()978-0-7354-1115-9 (ISBN)
##### Conference

28th International Symposium on Rarefied Gas Dynamics, Zaragoza, July 9-13th, 2012
#####

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##### Funder

Swedish Research Council, 621-2009-5751
##### Note

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Keldysh Institute for Applied Mathematics.

Keldysh Institute for Applied Mathematics.

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

28th International Symposium on Rarefied Gas Dynamics, Zaragoza, July 9-13th, 2012

Available from: 2012-12-04 Created: 2012-12-04 Last updated: 2016-04-25Bibliographically approvedOpen this publication in new window or tab >>Monte-Carlo method for two component plasmas### Bobylev, Alexander

### Karpov, S.A.

### Potapenko, I.F.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_7_j_idt195_some",{id:"formSmash:j_idt191:7:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_7_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_7_j_idt195_otherAuthors",{id:"formSmash:j_idt191:7:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_7_j_idt195_otherAuthors",multiple:true}); 2012 (Russian)In: Matematicheskoe Modelirovanie, ISSN 0234-0879, Vol. 24, no 9, p. 35-49Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Moskva: Steklov Mathematical Institute, Russian Academy of Sciences, 2012
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-16081 (URN)
#####

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Available from: 2012-12-03 Created: 2012-12-03 Last updated: 2017-06-30Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Keldysh Institute for Applied Mathematics.

Keldysh Institute for Applied Mathematics.

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.

Open this publication in new window or tab >>Solutions of the linear Boltzmann equation and some Dirichlet series### Bobylev, Alexander

### Gamba, Irene

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_some",{id:"formSmash:j_idt191:8:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_otherAuthors",{id:"formSmash:j_idt191:8:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_otherAuthors",multiple:true}); 2012 (English)In: Forum Mathematicum, ISSN 1435-5337, Vol. 24, no 2, p. 239-251Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Walter de Gruyter, 2012
##### Keywords

Boltzmann equation, Dirichlet series and functional equations, Riemann Zeta and L$L$-functions
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-10556 (URN)10.1515/form.2011.058 (DOI)000303419000002 ()
#####

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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_j_idt378",{id:"formSmash:j_idt191:8:j_idt195:j_idt378",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_j_idt378",multiple:true});
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##### Funder

Swedish Research Council, 2006-3404
Available from: 2012-02-08 Created: 2012-02-08 Last updated: 2012-12-04Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Department of Mathematics, The University of Texas at Austin.

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.

Open this publication in new window or tab >>Symmetric extensions of normal discrete velocity models### Bobylev, Alexander

### Vinerean (Bernhoff), Mirela

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_9_j_idt195_some",{id:"formSmash:j_idt191:9:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_9_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_9_j_idt195_otherAuthors",{id:"formSmash:j_idt191:9:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_9_j_idt195_otherAuthors",multiple:true}); 2012 (English)In: 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, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Institute of Physics (AIP), 2012 Edition: 1
##### Series

AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616 ; 1501
##### Keywords

Kinetic theory, discrete kinetic (velocity) models, conservation laws
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:kau:diva-16072 (URN)10.1063/1.4769516 (DOI)000312411200032 ()978-0-7354-1115-9 (ISBN)
##### Conference

28th International Symposium on Rarefied Gas Dynamics 2012, July 9 - 13, Zaragoza
#####

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Available from: 2012-12-03 Created: 2012-12-03 Last updated: 2017-10-30Bibliographically approved

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

Karlstad University, Faculty of Technology and Science, Department of Mathematics.

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.