Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-25T08:50:29.290Z Has data issue: false hasContentIssue false
  • English
  • Français

Particle Methods for the Boltzmann Equation

Published online by Cambridge University Press:  07 November 2008

Helmut Neunzert  and
Jens Struckmeier
Helmut Neunzert
Affiliation:
Department of Mathematics, University of Kaiserslautern, Germany, E-mail: [email protected] and [email protected]
Jens Struckmeier
Affiliation:
Department of Mathematics, University of Kaiserslautern, Germany, E-mail: [email protected] and [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

In the following chapters we will discuss particle methods for the numerical simulation of rarefied gas flows.

We will mainly treat a billiard game, that is, our particles will be hard spheres. But we will also touch upon cases where particles have internal energies due to rotation or vibration, which they exchange in a collision, and we will talk about chemical reactions happening during a collision.

Due to the limited size of this paper, we are only able to mention the principles of these real-gas effects. On the other hand, the general concepts of particle methods to be presented may be used for other kinds of kinetic equations, such as the semiconductor device simulation. We leave this part of the research to subsequent papers.

Type
Research Article
Information
Acta Numerica , Volume 4 , January 1995 , pp. 417 - 457
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Babovsky, H. (1989), ‘Convergence proof for Nanbu's Boltzmann simulation scheme’, Europ. J. of Mech. B/Fluids 1, 41–55.Google Scholar
Bartel, T. J. and Plimpton, S. J. (1992), DSMC Simulation of Rarefied Gas Dynamics on a Large Hypercube Supercomputer, Paper-92–2860, Am. Inst. Aero. Astro., Washington.CrossRefGoogle Scholar
Bärwinkel, K. and Wolters, H. (1975), Reaktionskinetik und Transport in relaxierenden Gasen, Report BMF-FB W W 75–28, Dornier, Friedrichshafen.Google Scholar
Bird, G. A. (1976), Molecular Gas Dynamics, Clarendon Press, Oxford.Google Scholar
Bird, G. A. (1989), ‘Perception of Numerical Methods in Rarefied Gas Dynamics’, Progr. Astro, and Aero. 118, 211–226.Google Scholar
Bobylev, A. V. (1993), ‘The Boltzmann Equation and the Group Transformations’, Math. Models & Methods in Appl. Sciences 3, No. 4, 443–476.CrossRefGoogle Scholar
Borgnakke, C. and Larssen, P. S. (1975), ‘Statistical collision model for Monte Carlo simulation of polyatomic gas mixtures’, J. Comput. Phys. 18, 405–420.CrossRefGoogle Scholar
Bourgat, J. F., LeTallec, P., Perthame, B. and Qiu, Y. (1994), ‘Coupling Boltzmann and Euler Equations without Overlapping’, Contemporary Mathematics 157, 377–398.CrossRefGoogle Scholar
Cercignani, C. (1989), The Boltzmann Equation and Its Applications, Spinger, Berlin.Google Scholar
Dagum, L. (1991), Three Dimensional Direct Particle Simulation on the Connection Machine, Paper-91–1365, Am. Inst. Aero. Astro., Washington.CrossRefGoogle Scholar
Faure, H. (1982), ‘Discrepance des suites associées à une système de numération (en dimension s)’, Acta Arithmeticae 41, 337–351.CrossRefGoogle Scholar
Greengard, C. and Reyna, L. G. (1992), ‘Conservation of expected momentum and energy in Monte Carlo particle simulation’, Phys. Fluids A 4, 849–852.CrossRefGoogle Scholar
Hack, M. (1993), ‘Construction of Particlesets to Simulate Rarefied Gases’, Report No. 89, Lab. Technomathematics, University of Kaiserslautern.Google Scholar
Halton, J. H. (1960), ‘On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals’, Numer. Math. 2, 84–90.CrossRefGoogle Scholar
Hammersley, J. M. (1960), ‘Monte Carlo methods for solving multivariate problems’, Ann. New York Acad. Sci. 86, 844–874.CrossRefGoogle Scholar
Hlawka, E. and Mück, R. (1972), A transformation of equidistributed sequences, Appl. of Number Theory to Numerical Analysis, ed. Zaremba, S. K., Academic Press, New York, 371–388CrossRefGoogle Scholar
Illner, R. and Neunzert, H. (1993), ‘Domain Decomposition: Linking of Aerodynamic and Kinetic Descriptions’, Report No. 90, Lab. Technomathematics, University of Kaiserslautern.Google Scholar
Ivanov, M. S. and Rogasinsky, S. V. (1988), ‘Analysis of numerical techniques of the direct simulation Monte Carlo method in the rarefied gas dynamics’, Sov. J. Numer. Anal. Math. Modelling 3, No. 6, 453–465.CrossRefGoogle Scholar
Klar, A. (1994), ‘Domain Decomposition for Kinetic and Aerodynamic Equations’, PhD thesis, Dept. Math., University of Kaiserslautern.Google Scholar
Kuščer, I. (1991), ‘Dissociation and Recombination in an Inhomogeneous Gas’, Physica A 176, 542–556.CrossRefGoogle Scholar
Lord, G. (1991), ‘Some extensions to the Cercignani-Lampis gas-surface scattering kernel’, Phys. Fluids A 4, 706–710.CrossRefGoogle Scholar
Ludwig, G. and Heil, M. (1960), ‘Boundary-layer theory with dissociation and ionization’, in Advances of Applied Mechanics, vol. 6, Academic Press, New York.Google Scholar
Lukshin, A., Neunzert, H. and Struckmeier, J. (1992), ‘Interim Report for the Project DPH 6473/91: Coupling of Navier–Stokes and Boltzmann Regions’, Internal Report, Dept. Math., University of Kaiserslautern.Google Scholar
Missmahl, G. (1990), ‘Randwertprobleme bei der Boltzmann–Simulation’, Diploma thesis, Dept. Math., University of Kaiserslautern.Google Scholar
Nanbu, K. (1980), ‘Direct Simulation Scheme Derived from the Boltzmann Equation’, J. Phys. Japan 49, 2042–2049.CrossRefGoogle Scholar
Neunzert, H., Gropengiesser, F. and Struckmeier, J. (1991), ‘Computational methods for the Boltzmann equation’, in Venice 1989: The State of Art in Appl. and Ind. Math. (ed.), Spigler, R., Kluwer, , Dordrecht, 111–140.Google Scholar
Neunzert, H., Steiner, K. and Wick, J. (1993), ‘Entwicklung und Validierung eines Partikelverfahrens zur Berechnung von Strömungen um Raumfahrzeuge im Bereich verdünnter ionisierter Gase’, Report DFG-FB Ne 269/8–1, Lab. Technomathematics, University of Kaiserslautern.Google Scholar
Niclot, B. (1987), The Two Particle Boltzmann Collision Operator in Axisymmetric Geometry, Report No. 164, Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau.Google Scholar
Niederreiter, H. (1992), Random Number Generation and Quasi–Monte Carlo Methods, SIAM, Philadelphia.CrossRefGoogle Scholar
Nocilla, S. (1961), ‘On the Interactions between Stream and Body in Free-molecule Flow’, Proc. 2nd Int. Symp. on Rarefied Gas Mechanics, (Talbot, L., ed.), Academic Press, New York.Google Scholar
Pagés, G. (1992), ‘Van der Corput sequences, Kakutani transform and one-dimensional numerical integration’, J. Comp. & Appl. Math. 44, 21–39.CrossRefGoogle Scholar
Sobol, K. (1969), Multidimensional quadrature formulae and Haar functions, Nauka, Moscow.Google Scholar
Schreiner, M. (1991), ‘Weighted Particles in the Finite Pointset Method’, Report No. 62, Lab. Technomathematics, University of Kaiserslautern.Google Scholar
Schreiner, W. (1994), Partikelverfahren für kinetische Schemata zu den Eulergleichungen, PhD thesis, Dept. Math., University of Kaiserslautern.Google Scholar
Struckmeier, J. (1993), ‘Fast Generation of Low-Discrepancy Sequences’, Report No. 93, Lab. Technomathematics, University of Kaiserslautern, to appear in J. of Comp. & Appl. Math.Google Scholar
Struckmeier, J. (1994), ‘Die Methode der finiten Punktmengen: Neue Ideen und Anregungen’, PhD thesis, Dept. Math., University of Kaiserslautern.Google Scholar
Struckmeier, J. and Pfreundt, F. J. (1993), ‘On the efficiency of simulation methods for the Boltzmann equation on parallel computers’, Parallel Computing 19, 103–119.CrossRefGoogle Scholar
Struckmeier, J. and Steiner, K. (1993), ‘A Comparison of Simulation Methods for the Boltzmann Equation’, Report No. 91, Lab. Technomathematics, University of Kaiserslautern, submitted to J. Comp. Physics.Google Scholar
Van der Corput, J. G. (1935), ‘Verteilungsfunktionen I, II’, Nederl. Akad. Wetensch. Proc. Ser. B 38, 813–821; 1058–1066.Google Scholar
Wagner, W. (1992), ‘A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation’, J. Stat. Phys. 66, 689–722.CrossRefGoogle Scholar
Wong, B. C. and Long, L. N. (1992), Direct Simulation Monte Carlo (DSMC) on the Connection Machine, Paper-92–0564, Am. Inst. Aero. Astro., Washington.CrossRefGoogle Scholar