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Weighted Particle Methods Solving Kinetic Equations for Dilute Ionized Gases

  • K. Steiner
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NONUFM, volume 48)

Summary

The numerical treatment of kinetic equations for ionized gases has to handle two additional problems. The first one, the different time scales for the electrons and the heavy particle, is solved by asymptotic methods. The second one arises from small ionization rates and is treated with a weighted particle method. The derivation of a weighted particle method for general kinetic systems, the choice of the weights and the convergence properties are discussed in the first part of the paper. The use of variable weights in a particle scheme leads to nonconservative particle schemes which results in purely numerical instabilities. The second part of the paper presents possible modifications of the weighted particle method which conserve mass, charge, momentum and energy on the discrete level of description and lead to stable and convergent numerical methods. Numerical results for a reduced plasma system show the advantages in comparison to equiweighted particle method with respect to accuracy, computational time and computer memory.

Keywords

Boltzmann Equation Particle Method Collision Operator Vlasov Equation Simulation Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Arkeryd, L.; Cercignani, C.; Illner, R.: Measure Solutions of the Steady Boltzmann Equation in a Slab, Commun. Math. Phys. 142, 285–296 (1991).MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Babovsky, H.: On A Simulation Scheme for the Boltzmann Equation, M2AS 8, 223–233 (1986).MathSciNetzbMATHGoogle Scholar
  3. [3]
    Babovsky, H.: A Convergence Proof for Nanbu’s Boltzmann Simulation Scheme, European J. Mech., B/Fluids 8, No. 1, 41–55 (1989).MathSciNetGoogle Scholar
  4. [4]
    Birdsall, C.K.; Langdon, A.B.: Plasma Physics Via Computer Simulation, McGraw-Hill, New York (1981).Google Scholar
  5. [5]
    Bobylev, A.V.; Struckmeier, J.: Implicit and Iterative Methods for the Boltzmann Equation, AGTM-Report Nr. 123 (1994), printed in TTSP.Google Scholar
  6. [6]
    Cercignani, C.; Illner, R.; Pulvirenti, M.: The Mathematical Theory of Dilute Gases, Springer, New York (1994).zbMATHGoogle Scholar
  7. [7]
    Chapman, S.; Cowling, T.G.: The Mathematical Theory of Non-Uniform Gases, University Press, Cambridge (1970).Google Scholar
  8. [8]
    Cottet, G.-H.; Raviart, P.-A.: Particle Methods for the One-Dimensional Vlasov-Poisson Equations, SIAM J. Numer. Anal. Vol. 21, No.1, 52–76 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Degond, P.; Lucquin-Desreux, B.: The Asymptotics of Collision Operators for Two Species of Particles of Disparate Masses, submitted to TTSP.Google Scholar
  10. [10]
    Degond, P.; Lucquin-Desreux, B.: Transport Coefficients of Plasmas and Disparate Mass Binary Gases, submitted to M3AS.Google Scholar
  11. [11]
    Degond, P.; Raviart, P.A.: An Asymptotic Analysis of the One-Dimensional Vlasov-Poisson System: The Child-Langmuir Law, Asymp. Anal. 4, 187–214 (1991).MathSciNetzbMATHGoogle Scholar
  12. [12]
    Desvillettes, L.: On The Convergence of Splitting Algorithms for Some Kinetic Equations, Asymp. Anal. 6, 315–333 (1993).MathSciNetzbMATHGoogle Scholar
  13. [13]
    Desvillettes, L.; Mischler, S.: About the Splitting Algorithm for Boltzmann and B.G.K. Equations, submitted to Asymp. Anal.Google Scholar
  14. [14]
    Feng Kang; Qin Meng-shao: The Symplectic Methods for the Computation of Hamiltonian Equations, 1–37, in: Zhu you-lan; Gu Ben-yu (Eds.): Proc. of the 1st Chinese Conf. on Numerical Methods of PDE’s (1986), Springer, Berlin (1987).Google Scholar
  15. [15]
    Greengard, C.; Reyna, L.G.: Conservation of Expected Momentum and Energy in Monte Carlo Particle Simulation, Phys. Fluids A 4(4) 849–852 (1992).MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Großmann, C.; Roos, H.-G.: Numerik Partieller Differentialgleichungen, Teubner, Stuttgart (1992).zbMATHGoogle Scholar
  17. [17]
    Neunzert, H.: Mathematical Investigations on Particle-in-Cell Methods, Proc. of XIII Symp. advanc. Probl. Meth. Fluid Mech., Fluid Dyn. 9 (1978).Google Scholar
  18. [18]
    Neunzert, H.: An Introduction to the Nonlinear Boltzmann-Vlasov Equation, 60–110, in: Cercignani, C. (Ed.): Kinetic Theories and the Boltzmann Equation, Springer, Berlin (1984).CrossRefGoogle Scholar
  19. [19]
    Neunzert, H.; Steiner, K.; Wick, J.: Entwicklung und Validierung Eines Partikelverfahrens zur Berechnung von Strömungen um Raumfahrzeuge im Bereich Verdünnter Ionisierter Gase, DFG-Bericht Ne 269/8-1 (1993).Google Scholar
  20. [20]
    Neunzert, H.; Struckmeier, J.: Particle Methods for the Boltzmann Equation, Acta Numerica (1995).Google Scholar
  21. [21]
    Sack, W.: Modellierung und Numerische Berechnung von Reaktiven Strömungen in Verdünnten Gasen, PhD Thesis, University of Kaiserslautern (1995).Google Scholar
  22. [22]
    Schreiner, M.: Weighted Particles in the Finite Pointset Method, TTSP 22(6), 793–817 (1993).MathSciNetzbMATHGoogle Scholar
  23. [23]
    Spatschek, K.H.: Theoretische Plasmaphysik, B.G. Teubner, Stuttgart (1990).CrossRefGoogle Scholar
  24. [24]
    Steiner, K.: Kinetische Gleichungen zur Beschreibung Verdünnter Ionisierter Gase und Ihre Numerische Behandlung Mittels Gewichteter Partikelverfahren, PhD Thesis, University of Kaiserslautern (1995).Google Scholar
  25. [25]
    Struckmeier, J.: Die Methode der Finiten Punktmengen — Neue Ideen und Anregungen — PhD Thesis, University of Kaiserslautern (1994).Google Scholar
  26. [26]
    Struckmeier, J.; Steiner, K.: Boltzmann Simulations With Axisymmetric Geometry, AGTM-Report Nr. 83 (1992).Google Scholar
  27. [27]
    Struckmeier, J.; Steiner, K.: A Comparison of Simulation Methods for Rarefied Gas Flows, Physics of Fluids A (1995).Google Scholar
  28. [28]
    Struckmeier, J.; Steiner, K.: Second Order Scheme for the Spatially Homogeneous Boltzmann Equation With Maxwellian Molecules, AGTM-Report Nr. 127 (1995), printed in M3AS.Google Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden 1996

Authors and Affiliations

  • K. Steiner
    • 1
  1. 1.Arbeitsgruppe Technomathematik Fachbereich MathematikUniversität KaiserslauternGermany

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