Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 5, pp. 1515-1531.

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

DOI: 10.1051/m2an/2013078
Classification: 65T50, 68Q25, 74S25, 76P05
Mots-clés : Boltzmann equation, discrete-velocity approximations, discrete-velocity methods, fast summation methods, farey series, convolutive decomposition
@article{M2AN_2013__47_5_1515_0,
     author = {Mouhot, Cl\'ement and Pareschi, Lorenzo and Rey, Thomas},
     title = {Convolutive decomposition and fast summation methods for discrete-velocity approximations of the {Boltzmann} equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1515--1531},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {5},
     year = {2013},
     doi = {10.1051/m2an/2013078},
     mrnumber = {3100773},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2013078/}
}
TY  - JOUR
AU  - Mouhot, Clément
AU  - Pareschi, Lorenzo
AU  - Rey, Thomas
TI  - Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 1515
EP  - 1531
VL  - 47
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2013078/
DO  - 10.1051/m2an/2013078
LA  - en
ID  - M2AN_2013__47_5_1515_0
ER  - 
%0 Journal Article
%A Mouhot, Clément
%A Pareschi, Lorenzo
%A Rey, Thomas
%T Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 1515-1531
%V 47
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2013078/
%R 10.1051/m2an/2013078
%G en
%F M2AN_2013__47_5_1515_0
Mouhot, Clément; Pareschi, Lorenzo; Rey, Thomas. Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 5, pp. 1515-1531. doi : 10.1051/m2an/2013078. http://www.numdam.org/articles/10.1051/m2an/2013078/

[1] A. Bobylev and S. Rjasanow, Difference scheme for the Boltzmann equation based on the fast Fourier transform. Eur. J. Mech. B Fluids 16 (1997) 293-306. | MR | Zbl

[2] A.V. Bobylev, Exact solutions of the Boltzmann equation. Dokl. Akad. Nauk SSSR 225 (1975) 1296-1299. | MR | Zbl

[3] A.V. Bobylev, A. Palczewski and J. Schneider, On approximation of the Boltzmann equation by discrete velocity models. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995) 639-644. | MR | Zbl

[4] A.V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres. Eur. J. Mech. B Fluids 18 (1999) 869-887. | MR | Zbl

[5] A.V. Bobylev and S. Rjasanow, 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. | MR | Zbl

[6] A.V. Bobylev and M.C. Vinerean, Construction of discrete kinetic models with given invariants. J. Statist. Phys. 132 (2008) 153-170. | MR | Zbl

[7] C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Trans. Theory Statist. Phys. 25 (1996) 33-60. | MR | Zbl

[8] H. Cabannes, R. Gatignol and L.-S. Luo, The Discrete Boltzmann Equation (Theory and Applications). University of California, College of engineering, Los-Angeles (2003). | MR

[9] 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). | MR | Zbl

[10] T. Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann. Acta Math. 60 (1933) 91-146. | JFM | MR

[11] C. Cercignani, Theory and application of the Boltzmann equation. Elsevier, New York (1975). | MR | Zbl

[12] 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). | MR | Zbl

[13] J.W. Cooley and J.W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19 (1965) 297-301. | MR | Zbl

[14] F. Coquel, F. Rogier and J. Schneider, A deterministic method for solving the homogeneous Boltzmann equation. Rech. Aérospat. 3 (1992) 1-10. | MR | Zbl

[15] F. Filbet, J.W. Hu and S. Jin, A numerical scheme for the quantum Boltzmann equation efficient in the fluid regime. ESAIM: M2AN 42 (2012) 443-463. | Numdam | Zbl

[16] F. Filbet and C. Mouhot, Analysis of spectral methods for the homogeneous Boltzmann equation. Trans. Amer. Math. Soc. 363 (2011) 1947-1980. | MR | Zbl

[17] F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in N log2N. SIAM J. Sci. Comput. 28 (2006) 1029. | MR | Zbl

[18] I. Gamba and S.H. Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states. J. Comput. Phys. 228 (2009) 2012-2036. | MR | Zbl

[19] 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. | MR | Zbl

[20] J. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator, to appear in CMS. Preprint (2011). | MR | Zbl

[21] I. Ibragimov and S. Rjasanow, Numerical solution of the Boltzmann equation on the uniform grid. Comput. 69 (2002) 163-186. | MR | Zbl

[22] P. Kowalczyk, A. Palczewski, G. Russo and Z. Walenta, Numerical solutions of the Boltzmann equation: comparison of different algorithms. Eur. J. Mech. B Fluids 27 (2008) 62-74. | MR | Zbl

[23] M. Krook and T.T. Wu, Exact solutions of the Boltzmann equation. Phys. Fluids 20 (1977) 1589. | Zbl

[24] P. Markowich and L. Pareschi, Fast, conservative and entropic numerical methods for the Boson Boltzmann equation. Numer. Math. 99 (2005) 509-532. | MR | Zbl

[25] Y.-L. Martin, F. Rogier and J. Schneider, Une méthode déterministe pour la résolution de l'équation de Boltzmann inhomogène. C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) 483-487. | MR | Zbl

[26] P. Michel and J. Schneider, 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. | MR | Zbl

[27] C. Mouhot and L. Pareschi, Fast methods for the Boltzmann collision integral. C. R. Acad. Sci. Paris Sér. I Math. 339 (2004) 71-76. | MR | Zbl

[28] C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75 (2006) 1833-1852. | MR | Zbl

[29] A. Nogueira and B. Sevennec, Multidimensional Farey partitions. Indag. Math. (N.S.) 17 (2006) 437-456. | MR | Zbl

[30] V.A. Panferov and A.G. Heintz, A new consistent discret-velocity model for the Boltzmann equation. Math. Models Methods Appl. Sci. 25 (2002) 571-593. | MR | Zbl

[31] 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. | MR | Zbl

[32] L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37 (2000) 1217-1245. | MR | Zbl

[33] 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. | MR | Zbl

[34] T. Płatkowski and R. Illner, Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory. SIAM Rev. 30 (1988) 213-255. | MR | Zbl

[35] T. Platkowski and W. Walús, An acceleration procedure for discrete velocity approximation of the Boltzmann collision operator. Comput. Math. Appl. 39 (2000) 151-163. | MR | Zbl

[36] F. Rogier and J. Schneider, A direct method for solving the Boltzmann equation. Trans. Theory Statist. Phys. 23 (1994) 313-338. | MR | Zbl

[37] D. Valougeorgis and S. Naris, Acceleration schemes of the discrete velocity method: Gaseous flows in rectangular microchannels. SIAM J. Sci. Comput. 25 (2003) 534-552. | MR | Zbl

[38] C. Villani, A review of mathematical topics in collisional kinetic theory. Elsevier Science (2002). | MR | Zbl

Cited by Sources: