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Convolutive decomposition and fast summation methods fordiscrete-velocity approximations of the Boltzmann equation

Published online by Cambridge University Press:  14 August 2013

Clément Mouhot ,
Lorenzo Pareschi  and
Thomas Rey
Clément Mouhot
Affiliation:
DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.. [email protected]
Lorenzo Pareschi
Affiliation:
DMI, Università di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy.; [email protected]
Thomas Rey
Affiliation:
CSCAMM, University of Maryland, CSIC Building, Paint Branch Drive, College Park, MD 20740, USA.; [email protected]
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Abstract

Discrete-velocity approximations represent a popular way for computing the Boltzmanncollision operator. The direct numerical evaluation of such methods involve a prohibitivecost, typically O(N2d + 1)where d is the dimension of the velocity space. In this paper, followingthe ideas introduced in [C. Mouhot and L. Pareschi, C. R. Acad. Sci. Paris Sér. IMath. 339 (2004) 71–76, C. Mouhot and L. Pareschi, Math.Comput. 75 (2006) 1833–1852], we derive fast summation techniquesfor the evaluation of discrete-velocity schemes which permits to reduce the computationalcost from O(N2d + 1) to O(N̅dNd log2N), N̅ ≪ N, with almost no loss of accuracy.

Keywords

Boltzmann equationdiscrete-velocity approximationsdiscrete-velocity methodsfast summation methodsfarey seriesconvolutive decomposition
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 5 , September 2013 , pp. 1515 - 1531
Copyright
© EDP Sciences, SMAI, 2013

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References

Bobylev, A. and Rjasanow, S., Difference scheme for the Boltzmann equation based on the fast Fourier transform. Eur. J. Mech. B Fluids 16 (1997) 293–306. Google Scholar
Bobylev, A.V., Exact solutions of the Boltzmann equation. Dokl. Akad. Nauk SSSR 225 (1975) 1296–1299. Google Scholar
Bobylev, A.V., Palczewski, A. and Schneider, J., On approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995) 639–644. Google Scholar
Bobylev, A.V. and Rjasanow, S., Fast deterministic method of solving the Boltzmann equation for hard spheres. Eur. J. Mech. B Fluids 18 (1999) 869–887. Google Scholar
Bobylev, A.V. and Rjasanow, S., Numerical solution of the Boltzmann equation using a fully conservative difference scheme based on the fast fourier transform. Trans. Theory Statist. Phys. 29 (2000) 289–310. Google Scholar
Bobylev, A.V. and Vinerean, M.C., Construction of discrete kinetic models with given invariants. J. Statist. Phys. 132 (2008) 153–170. Google Scholar
Buet, C., A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Trans. Theory Statist. Phys. 25 (1996) 33–60. Google Scholar
H. Cabannes, R. Gatignol and L.-S. Luo, The Discrete Boltzmann Equation (Theory and Applications). University of California, College of engineering, Los-Angeles (2003).
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods in fluid dynamics. Springer Series in Computational Physics. Springer-Verlag, New York (1988).
Carleman, T., Sur la théorie de l’équation intégrodifférentielle de Boltzmann. Acta Math. 60 (1933) 91–146. Google Scholar
C. Cercignani, Theory and application of the Boltzmann equation. Elsevier, New York (1975).
C. Cercignani, R. Illner and M. Pulvirenti, The mathematical theory of dilute gases, in vol. 106 of Appl. Math. Sci. Springer-Verlag, New York (1994).
Cooley, J.W. and Tukey, J.W., An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19 (1965) 297–301. Google Scholar
Coquel, F., Rogier, F. and Schneider, J., A deterministic method for solving the homogeneous Boltzmann equation. Rech. Aérospat. 3 (1992) 1–10. Google Scholar
Filbet, F., Hu, J.W. and Jin, S., A numerical scheme for the quantum Boltzmann equation efficient in the fluid regime. ESAIM: M2AN 42 (2012) 443–463. Google Scholar
Filbet, F. and Mouhot, C., Analysis of spectral methods for the homogeneous Boltzmann equation. Trans. Amer. Math. Soc. 363 (2011) 1947–1980. Google Scholar
Filbet, F., Mouhot, C. and Pareschi, L., Solving the Boltzmann equation in N log2N. SIAM J. Sci. Comput. 28 (2006) 1029. Google Scholar
Gamba, I. and Tharkabhushanam, S.H., Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states. J. Comput. Phys. 228 (2009) 2012–2036. Google Scholar
G.H. Hardy and E.M. Wright, An introduction to the theory of numbers, sixth ed. Oxford University Press, Oxford (2008). Revised by D.R. Heath-Brown and J.H. Silverman, with a foreword by Andrew Wiles.
J. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator, to appear in CMS. Preprint (2011).
Ibragimov, I. and Rjasanow, S., Numerical solution of the Boltzmann equation on the uniform grid. Comput. 69 (2002) 163–186. Google Scholar
Kowalczyk, P., Palczewski, A., Russo, G. and Walenta, Z., Numerical solutions of the Boltzmann equation: comparison of different algorithms. Eur. J. Mech. B Fluids 27 (2008) 62–74. Google Scholar
Krook, M. and Wu, T.T., Exact solutions of the Boltzmann equation. Phys. Fluids 20 (1977) 1589. Google Scholar
Markowich, P. and Pareschi, L., Fast, conservative and entropic numerical methods for the Boson Boltzmann equation. Numer. Math. 99 (2005) 509–532. Google Scholar
Martin, Y.-L., Rogier, F. and Schneider, J., Une méthode déterministe pour la résolution de l’équation de Boltzmann inhomogène. C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) 483–487. Google Scholar
Michel, P. and Schneider, J., Approximation simultanée de réels par des nombres rationnels et noyau de collision de l’équation de Boltzmann. C. R. Acad. Sci. Sér. I Math. 330 (2000) 857–862. Google Scholar
Mouhot, C. and Pareschi, L., Fast methods for the Boltzmann collision integral. C. R. Acad. Sci. Paris Sér. I Math. 339 (2004) 71–76. Google Scholar
Mouhot, C. and Pareschi, L., Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75 (2006) 1833–1852. Google Scholar
Nogueira, A. and Sevennec, B., Multidimensional Farey partitions. Indag. Math. (N.S.) 17 (2006) 437–456. Google Scholar
Panferov, V.A. and Heintz, A.G., A new consistent discret-velocity model for the Boltzmann equation. Math. Models Methods Appl. Sci. 25 (2002) 571–593. Google Scholar
L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations. In Proc. of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), Trans. Theory Statist. Phys. 25 (1996) 369–382.
Pareschi, L. and Russo, G., Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37 (2000) 1217–1245. Google Scholar
L. Pareschi and G. Russo, On the stability of spectral methods for the homogeneous Boltzmann equation. In Proc. of the Fifth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics (Maui, HI 1998). Trans. Theory Statist. Phys. 29 (2000) 431–447.
Płatkowski, T. and Illner, R., Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory. SIAM Rev. 30 (1988) 213–255. Google Scholar
Platkowski, T. and Walús, W., An acceleration procedure for discrete velocity approximation of the Boltzmann collision operator. Comput. Math. Appl. 39 (2000) 151–163. Google Scholar
Rogier, F. and Schneider, J., A direct method for solving the Boltzmann equation. Trans. Theory Statist. Phys. 23 (1994) 313–338. Google Scholar
Valougeorgis, D. and Naris, S., Acceleration schemes of the discrete velocity method: Gaseous flows in rectangular microchannels. SIAM J. Sci. Comput. 25 (2003) 534–552. Google Scholar
C. Villani, A review of mathematical topics in collisional kinetic theory. Elsevier Science (2002).