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(N̅dNd log2N), N̅ ≪ N, with almost no loss of accuracy.
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 , Tome 47 (2013) no. 5, pp. 1515-1531. doi : 10.1051/m2an/2013078. http://www.numdam.org/articles/10.1051/m2an/2013078/
[1] and , Difference scheme for the Boltzmann equation based on the fast Fourier transform. Eur. J. Mech. B Fluids 16 (1997) 293-306. | MR | Zbl
[2] , Exact solutions of the Boltzmann equation. Dokl. Akad. Nauk SSSR 225 (1975) 1296-1299. | MR | Zbl
[3] , and , 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] and , Fast deterministic method of solving the Boltzmann equation for hard spheres. Eur. J. Mech. B Fluids 18 (1999) 869-887. | MR | Zbl
[5] and , 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] and , Construction of discrete kinetic models with given invariants. J. Statist. Phys. 132 (2008) 153-170. | MR | Zbl
[7] , A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Trans. Theory Statist. Phys. 25 (1996) 33-60. | MR | Zbl
[8] , and , The Discrete Boltzmann Equation (Theory and Applications). University of California, College of engineering, Los-Angeles (2003). | MR
[9] , , and , Spectral methods in fluid dynamics. Springer Series in Computational Physics. Springer-Verlag, New York (1988). | MR | Zbl
[10] , Sur la théorie de l'équation intégrodifférentielle de Boltzmann. Acta Math. 60 (1933) 91-146. | JFM | MR
[11] , Theory and application of the Boltzmann equation. Elsevier, New York (1975). | MR | Zbl
[12] , and , The mathematical theory of dilute gases, in vol. 106 of Appl. Math. Sci. Springer-Verlag, New York (1994). | MR | Zbl
[13] and , An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19 (1965) 297-301. | MR | Zbl
[14] , and , A deterministic method for solving the homogeneous Boltzmann equation. Rech. Aérospat. 3 (1992) 1-10. | MR | Zbl
[15] , and , A numerical scheme for the quantum Boltzmann equation efficient in the fluid regime. ESAIM: M2AN 42 (2012) 443-463. | Numdam | Zbl
[16] and , Analysis of spectral methods for the homogeneous Boltzmann equation. Trans. Amer. Math. Soc. 363 (2011) 1947-1980. | MR | Zbl
[17] , and , Solving the Boltzmann equation in N log2N. SIAM J. Sci. Comput. 28 (2006) 1029. | MR | Zbl
[18] and , Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states. J. Comput. Phys. 228 (2009) 2012-2036. | MR | Zbl
[19] and , 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] and , A fast spectral algorithm for the quantum Boltzmann collision operator, to appear in CMS. Preprint (2011). | MR | Zbl
[21] and , Numerical solution of the Boltzmann equation on the uniform grid. Comput. 69 (2002) 163-186. | MR | Zbl
[22] , , and , Numerical solutions of the Boltzmann equation: comparison of different algorithms. Eur. J. Mech. B Fluids 27 (2008) 62-74. | MR | Zbl
[23] and , Exact solutions of the Boltzmann equation. Phys. Fluids 20 (1977) 1589. | Zbl
[24] and , Fast, conservative and entropic numerical methods for the Boson Boltzmann equation. Numer. Math. 99 (2005) 509-532. | MR | Zbl
[25] , and , 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] and , 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] and , Fast methods for the Boltzmann collision integral. C. R. Acad. Sci. Paris Sér. I Math. 339 (2004) 71-76. | MR | Zbl
[28] and , Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75 (2006) 1833-1852. | MR | Zbl
[29] and , Multidimensional Farey partitions. Indag. Math. (N.S.) 17 (2006) 437-456. | MR | Zbl
[30] and , A new consistent discret-velocity model for the Boltzmann equation. Math. Models Methods Appl. Sci. 25 (2002) 571-593. | MR | Zbl
[31] and , 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] and , 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] and , 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] and , 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] and , An acceleration procedure for discrete velocity approximation of the Boltzmann collision operator. Comput. Math. Appl. 39 (2000) 151-163. | MR | Zbl
[36] and , A direct method for solving the Boltzmann equation. Trans. Theory Statist. Phys. 23 (1994) 313-338. | MR | Zbl
[37] and , Acceleration schemes of the discrete velocity method: Gaseous flows in rectangular microchannels. SIAM J. Sci. Comput. 25 (2003) 534-552. | MR | Zbl
[38] , A review of mathematical topics in collisional kinetic theory. Elsevier Science (2002). | MR | Zbl
Cité par Sources :
