Fast algorithms for computing the Boltzmann collision operator
HTML articles powered by AMS MathViewer
- by Clément Mouhot and Lorenzo Pareschi PDF
- Math. Comp. 75 (2006), 1833-1852
Abstract:
The development of accurate and fast numerical schemes for the five-fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed.References
- Leif Arkeryd, On the Boltzmann equation. I. Existence, Arch. Rational Mech. Anal. 45 (1972), 1–16. MR 339665, DOI 10.1007/BF00253392
- D. Benedetto, E. Caglioti, and M. Pulvirenti, A kinetic equation for granular media, RAIRO Modél. Math. Anal. Numér. 31 (1997), no. 5, 615–641 (English, with English and French summaries). MR 1471181, DOI 10.1051/m2an/1997310506151
- Dario Benedetto, Emanuele Caglioti, and Mario Pulvirenti, Erratum: “A kinetic equation for granular media” [RAIRO Modél. Math. Anal. Numér. 31 (1997), no. 5, 615–641; MR1471181 (98k:82145)], M2AN Math. Model. Numer. Anal. 33 (1999), no. 2, 439–441. MR 1700043, DOI 10.1051/m2an:1999118
- G. A. Bird, Molecular gas dynamics and the direct simulation of gas flows, Oxford Engineering Science Series, vol. 42, The Clarendon Press, Oxford University Press, New York, 1995. Corrected reprint of the 1994 original; With 1 IBM-PC floppy disk (3.5 inch; DD); Oxford Science Publications. MR 1352466
- A. Bobylev and S. Rjasanow, Difference scheme for the Boltzmann equation based on the fast Fourier transform, European J. Mech. B Fluids 16 (1997), no. 2, 293–306. MR 1439069
- A. V. Bobylëv, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Mathematical physics reviews, Vol. 7, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., vol. 7, Harwood Academic Publ., Chur, 1988, pp. 111–233. MR 1128328
- A. V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres, Eur. J. Mech. B Fluids 18 (1999), no. 5, 869–887. MR 1728639, DOI 10.1016/S0997-7546(99)00121-1
- A. V. Bobylev and S. Rjasanow, Numerical solution of the Boltzmann equation using a fully conservative difference scheme based on the fast Fourier transform, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998), 2000, pp. 289–310. MR 1770434, DOI 10.1080/00411450008205876
- Alexandre Vasiljévitch Bobylev, Andrzej Palczewski, and Jacques Schneider, On approximation of the Boltzmann equation by discrete velocity models, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 5, 639–644 (English, with English and French summaries). MR 1322351
- C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics, Transport Theory Statist. Phys. 25 (1996), no. 1, 33–60. MR 1380030, DOI 10.1080/00411459608204829
- Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 917480, DOI 10.1007/978-3-642-84108-8
- Torsten Carleman, Sur la théorie de l’équation intégrodifférentielle de Boltzmann, Acta Math. 60 (1932).
- Carlo Cercignani, Theory and application of the Boltzmann equation, Elsevier, New York, 1975. MR 0406273
- Carlo Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. MR 1313028, DOI 10.1007/978-1-4612-1039-9
- Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620, DOI 10.1007/978-1-4419-8524-8
- James W. Cooley and John W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp. 19 (1965), 297–301. MR 178586, DOI 10.1090/S0025-5718-1965-0178586-1
- F. Coquel, F. Rogier, and J. Schneider, A deterministic method for solving the homogeneous Boltzmann equation, Rech. Aérospat. 3 (1992), 1–10 (French, with English and French summaries); English transl., Rech. Aérospat. (English Edition) 3 (1992), 1–10. MR 1192070
- P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, Math. Models Methods Appl. Sci. 2 (1992), no. 2, 167–182. MR 1167768, DOI 10.1142/S0218202592000119
- Pierre Degond, Lorenzo Pareschi, and Giovanni Russo (eds.), Modeling and computational methods for kinetic equations, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2069221, DOI 10.1007/978-0-8176-8200-2
- Miguel Escobedo and Stephane Mischler, On a quantum Boltzmann equation for a gas of photons, J. Math. Pures Appl. (9) 80 (2001), no. 5, 471–515 (English, with English and French summaries). MR 1831432, DOI 10.1016/S0021-7824(00)01201-0
- Francis Filbet, Clément Mouhot, and Lorenzo Pareschi, Solving the Boltzmann equation in ${N}\log _2 {N}$, SIAM J. Sci. Comp. (submitted).
- Francis Filbet and Lorenzo Pareschi, A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the nonhomogeneous case, J. Comput. Phys. 179 (2002), no. 1, 1–26. MR 1906573, DOI 10.1006/jcph.2002.7010
- Francis Filbet, Lorenzo Pareschi, and Giuseppe Toscani, Accurate numerical methods for the collisional motion of (heated) granular flows, J. Comput. Phys. 202 (2005), no. 1, 216–235. MR 2102883, DOI 10.1016/j.jcp.2004.06.023
- Francis Filbet and Giovanni Russo, High order numerical methods for the space non-homogeneous Boltzmann equation, J. Comput. Phys. 186 (2003), no. 2, 457–480. MR 1973198, DOI 10.1016/S0021-9991(03)00065-2
- E. Gabetta, L. Pareschi, and G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal. 34 (1997), no. 6, 2168–2194. MR 1480374, DOI 10.1137/S0036142995287768
- Tommy Gustafsson, $L^p$-estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 92 (1986), no. 1, 23–57. MR 816620, DOI 10.1007/BF00250731
- Tommy Gustafsson, Global $L^p$-properties for the spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 103 (1988), no. 1, 1–38. MR 946968, DOI 10.1007/BF00292919
- I. Ibragimov and S. Rjasanow, Numerical solution of the Boltzmann equation on the uniform grid, Computing 69 (2002), no. 2, 163–186. MR 1954793, DOI 10.1007/s00607-002-1458-9
- P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. MR 1063852, DOI 10.1007/978-3-7091-6961-2
- Yves-Loïc Martin, François Rogier, and Jacques Schneider, Une méthode déterministe pour la résolution de l’équation de Boltzmann inhomogène, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 6, 483–487 (French, with English summary). MR 1154392
- Clément Mouhot, Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions, Comm. Partial Differential Equations 30 (2005), no. 4-6, 881–917. MR 2153518, DOI 10.1081/PDE-200059299
- Clément Mouhot and Lorenzo Pareschi, An ${O}({N}\log _2 {N})$ algorithm for computing discrete velocity models, (Work in progress).
- Clément Mouhot and Cédric Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Ration. Mech. Anal. 173 (2004), no. 2, 169–212. MR 2081030, DOI 10.1007/s00205-004-0316-7
- Giovanni Naldi, Lorenzo Pareschi, and Giuseppe Toscani, Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit, M2AN Math. Model. Numer. Anal. 37 (2003), no. 1, 73–90. MR 1972651, DOI 10.1051/m2an:2003019
- Kenichi Nanbu, Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent gases, J. Phys. Soc. Japan 52 (1983), 2042–2049.
- Andrzej Palczewski and Jacques Schneider, Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation, J. Statist. Phys. 91 (1998), no. 1-2, 307–326. MR 1632506, DOI 10.1023/A:1023000406921
- Andrzej Palczewski, Jacques Schneider, and Alexandre V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal. 34 (1997), no. 5, 1865–1883. MR 1472201, DOI 10.1137/S0036142995289007
- Vladislav A. Panferov and Alexei G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Math. Methods Appl. Sci. 25 (2002), no. 7, 571–593. MR 1895119, DOI 10.1002/mma.303
- L. Pareschi, Computational methods and fast algorithms for Boltzmann equations, Chapter 7 Lecture Notes on the discretization of the Boltzmann equation, 2003, pp. 527–548.
- L. Pareschi, G. Russo, and G. Toscani, Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comput. Phys. 165 (2000), no. 1, 216–236. MR 1795398, DOI 10.1006/jcph.2000.6612
- L. Pareschi, G. Toscani, and C. Villani, Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit, Numer. Math. 93 (2003), no. 3, 527–548. MR 1953752, DOI 10.1007/s002110100384
- Lorenzo Pareschi and Benoit Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), 1996, pp. 369–382. MR 1407541, DOI 10.1080/00411459608220707
- Lorenzo Pareschi and Giovanni Russo, Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal. 37 (2000), no. 4, 1217–1245. MR 1756425, DOI 10.1137/S0036142998343300
- Lorenzo Pareschi and Giovanni Russo, On the stability of spectral methods for the homogeneous Boltzmann equation, Proceedings of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI, 1998), 2000, pp. 431–447. MR 1770438, DOI 10.1080/00411450008205883
- Lorenzo Pareschi, Giovanni Russo, and Giuseppe Toscani, Méthode spectrale rapide pour l’équation de Fokker-Planck-Landau, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 517–522 (French, with English and French summaries). MR 1756970, DOI 10.1016/S0764-4442(00)00212-3
- Ada Pulvirenti and Bernt Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys. 183 (1997), no. 1, 145–160. MR 1461954, DOI 10.1007/BF02509799
- François Rogier and Jacques Schneider, A direct method for solving the Boltzmann equation, Transport Theory Statist. Phys. 23 (1994), no. 1-3, 313–338. MR 1257657, DOI 10.1080/00411459408203868
- Michelle Schatzman, Analyse numérique, InterEditions, Paris, 1991 (French, with French summary). Cours et exercices pour la licence. [Course and exercises for the bachelor’s degree]. MR 1200897
- Cédric Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I, North-Holland, Amsterdam, 2002, pp. 71–305. MR 1942465, DOI 10.1016/S1874-5792(02)80004-0
Additional Information
- Clément Mouhot
- Affiliation: UMPA, ENS Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France
- Email: cmouhot@umpa.ens-lyon.fr
- Lorenzo Pareschi
- Affiliation: Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy
- Email: lorenzo.pareschi@unife.it
- Received by editor(s): February 7, 2004
- Received by editor(s) in revised form: March 13, 2005
- Published electronically: July 12, 2006
- Additional Notes: The first author was supported by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282
- © Copyright 2006 by the authors
- Journal: Math. Comp. 75 (2006), 1833-1852
- MSC (2000): Primary 65T50, 68Q25, 74S25, 76P05
- DOI: https://doi.org/10.1090/S0025-5718-06-01874-6
- MathSciNet review: 2240637