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Weak Shock Wave Solutions for the Discrete Boltzmann Equation
Karlstad University, Faculty of Technology and Science, Department of Mathematics. (Kinetisk teori)ORCID iD: 0000-0003-1232-3272
Karlstad University, Faculty of Technology and Science, Department of Mathematics. (Kinetisk teori)
2007 (English)In: 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, Published 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

Place, publisher, year, edition, pages
Novosibirsk: Publishing House of the Siberian Branch of the Russian Academy of Sciences , 2007. p. 173-178
Keywords [en]
Boltzmann equation, discrete velocity models, shock profiles
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-25700ISBN: 9785769209246 (print)OAI: oai:DiVA.org:kau-25700DiVA, id: diva2:599479
Conference
25th International Symposium on Rarefied Gas Dynamics, Saint-Petersburg, Russia, July 21-28, 2006
Funder
Swedish Research Council, 20035357Available from: 2013-01-22 Created: 2013-01-22 Last updated: 2017-12-06Bibliographically approved

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Bernhoff, NiclasBobylev, Alexander

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