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On some properties of linear and linearized Boltzmann collision operators for hard spheres
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
2008 (English)In: Kinetic and related models, ISSN 1937-5093, Vol. 1, no 4, 521-555 p.Article in journal (Refereed) Published
Abstract [en]

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

Place, publisher, year, edition, pages
Springfield, MO: American Institute of Mathematical Sciences, 2008. Vol. 1, no 4, 521-555 p.
National Category
Natural Sciences
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-1847DOI: 10.3934/krm.2008.1.521OAI: oai:DiVA.org:kau-1847DiVA: diva2:351
Available from: 2008-09-10 Created: 2008-09-10 Last updated: 2013-10-30Bibliographically approved
In thesis
1. Some numerical and analytical methods for equations of wave propagation and kinetic theory
Open this publication in new window or tab >>Some numerical and analytical methods for equations of wave propagation and kinetic theory
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.

 

The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.

 

The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.

 

Place, publisher, year, edition, pages
Karlstad: Karlstads universitet, 2008. 21 p.
Series
Karlstad University Studies, ISSN 1403-8099 ; 2008:33
Keyword
wave propagation, finite difference metods high order methods, Landau-Fokker-Planck equation, Monte-Carlo simulations, Boltzmann equation, hard sphere model, eigenvalue problem
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-1848 (URN)978-91-7063-192-4 (ISBN)
Public defence
2008-10-04, 21A 342, Karlstads universitet, Karlstad, 13:15 (English)
Opponent
Supervisors
Available from: 2008-09-10 Created: 2008-09-10 Last updated: 2011-12-19Bibliographically approved

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Bobylev, Alexander V.Mossberg, Eva

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