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Computational Methods for the Boltzmann Equation

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Applied and Industrial Mathematics

Part of the book series: Mathematics and Its Applications ((MAIA,volume 56))

Abstract

This paper contains the basic ideas and practical aspects for numerical methods for solving the Boltzmann Equation. The main field of application considered is the reentry of a Space Shuttle in the transition from free molecular flow to continuum flow. The method used will be called Finite Pointset Method (FPM) approximating the solution by finite sets of particles in a rigorously defined way. Convergence results are cited while practical aspects of the algorithm are emphasized. Ideas for the transition to the Navier Stokes domain are shortly discussed.

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References

  1. Aristov, V.V., Tscheremissine, F.G.: The conservative splitting method for solving the Boltzmann equation, USSR Comp, Math, and Math. Phys., Vol. 20, p. 208 (1980)

    MATH  Google Scholar 

  2. Babovsky, H.: A Convergence Proof for Nanbu’s Boltzmann Simulation Scheme, European Journal of Mechanics B/Fluids, 8, no. 1 (1989)

    Google Scholar 

  3. Babovsky, H., Gropengießer, F., Neunzert, H., Struckmeier, J., Wiesen, B.: Application of well-distributed sequences to the numerical simulation of the Boltzmann equation, to appear in Computational and Applied Mathematics

    Google Scholar 

  4. Babovsky, H., Illner, R.: A Convergence Proof for Nanbu’s Simulation Method for the full Boltzmann equation,SIAM Journal of Numerical Analysis, Vol. 26, no. 1, pp. 45–65 (1989)

    MathSciNet  MATH  Google Scholar 

  5. Bird, G., Moss, J.: Direct Simulation of Transitional Flow for Hypersonic Reentry Conditions, AIAA paper no. 84 - 0223, 1984

    Google Scholar 

  6. Caflish, R.: The fluid dynamic limit of the nonlinear Boltzmann equation, Comm. Pure & Appl. Math., Vol. 33, pp. 651–666 (1980)

    Article  MathSciNet  Google Scholar 

  7. Cercignani, C.: The Boltzmann Equation and its Applications, Springer 198

    Google Scholar 

  8. Cercignani, C.: Scattering Kernels for Gas-Surface Interaction, Workshop on Hypersonic Flows for Reentry Problems, Antibes Tome 1, 1990

    Google Scholar 

  9. Chapman, S., Cowlings, T.G.: The mathematical theory of non-uniform gases, Cambridge University Press, 1970

    Google Scholar 

  10. Coron, F.: Derivation of slip boundary conditions for the Navier Stokes System from the Boltzmann Equation, To appear in Journal of Statistical Physics

    Google Scholar 

  11. Coron, F.: Applications de la théorie cinétique à 1’aérodynamique hypersonique: line approche mathématique, Thèse de Doctorat de l’Université Paris Nord

    Google Scholar 

  12. Elizarova, T.G., Chetverushkin, B.N.: Kinetical consistent finite difference gasdynamic schemes, to appear in: Japanese Journal of Aerospace Science

    Google Scholar 

  13. Frezzotti, A., Parani, R.: Direct Numerical Solution of the Boltzmann Equation for a Binary Mixture of Hard Sphere Gases, to appear in Meccanica

    Google Scholar 

  14. Gropengießer, F., Neunzert, H., Struckmeier, J., Wiesen, B.: Rarefied gas flow around a disc with different angles of attack, in preparation

    Google Scholar 

  15. Gropengießer, F., Neunzert, H., Struckmeier, J., Wiesen, B.: Rarefied gas flow around a 3d-deltawing at low Knudsen number, in preparation

    Google Scholar 

  16. Hlavka, E.: Funktionen von beschränkter Variation in der Theorie der Gleichverteilung, Ann. Math, pura et appl. IV, 54 (1961)

    Google Scholar 

  17. Illner, R., Pulvirenti, M.: Global validity of the Boltzmann Equation for two- and three-dimensional rare gas in vacuum: Comm. Math. Phys. 121, 143–146 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaniel, S.: A kinetic model for the compressible flow equations, Indiana University Mathematics Journal, Vol. 37, No. 3, 1988

    Google Scholar 

  19. Körber, S., Wiesen, B.: A comparison of a Microscopic and a Phenomenological Model for a Polyatomic Gas, in preparation

    Google Scholar 

  20. Kuipers, L., Niederreiter, H.: Uniform distribution of sequences, John Wiley & Sons (1974)

    Google Scholar 

  21. Landford, O.E.: The evolution of large classical systems, Proceedings of the 1974 Battelle Rencontres on Dynamical Systems, J. Moser ed., Springer Lecture Notes in Physics no. 35, pp. 1–111 (1975)

    Google Scholar 

  22. Lécot, C.: Low Discrepancy sequences for solving the Boltzmann Equation, to appear

    Google Scholar 

  23. van der Mee, C.V.M.: Stationary Solutions of the Nonlinear Boltzmann Equation in a Bounded Spatial Domain, Math. Meth. in the Appl. Sciences, Vol. 11, pp. 471–481 (1989)

    Article  MATH  Google Scholar 

  24. Mißmahl, G.: Randwertprobleme bei der Boltzmann Simulation, Diplomarbeit, University of Kaiserslautern (1990)

    Google Scholar 

  25. Nanbu, K.: Direct Simulation Schemes derived from the Boltzmann Equation, J.Phys. Japan 49, p. 2042 (1980)

    Google Scholar 

  26. Neunzert, H., Wick, J.: Die Theorie der asymptotischen Verteilung und die numerische Losung von Integrodifferentialgleichungen, Num. Math. 21, pp. 234–243 (1973)

    MathSciNet  MATH  Google Scholar 

  27. Perthame, B.: Boltzmann type schemes and the entropy property, to appear in SIAM Journal on Numerical Analysis

    Google Scholar 

  28. Platkowski, T.: private communication

    Google Scholar 

  29. Weyl, H.: Über die Gleichverteilung von Zahlen mod 1, Math. Arm. 77 (1916)

    Google Scholar 

  30. Yen, S.M.: Numerical Solution of the Nonlinear Boltzmann Equation for Nonequilibrium Gas Flow Problems, Ann. Rev. Fluid Mech., Vol. 16, pp. 67–97 (1984)

    Article  Google Scholar 

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© 1991 Kluwer Academic Publishers

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Neunzert, H., Gropengießer, F., Struckmeier, J. (1991). Computational Methods for the Boltzmann Equation. In: Spigler, R. (eds) Applied and Industrial Mathematics. Mathematics and Its Applications, vol 56. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1908-2_10

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  • DOI: https://doi.org/10.1007/978-94-009-1908-2_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7351-6

  • Online ISBN: 978-94-009-1908-2

  • eBook Packages: Springer Book Archive

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