On the stability of spectral methods for the homogeneous Boltzmann equation

References

• Bird , G. A. 1994 . Molecular gas dynamics , Oxford : Clarendon Press .
   Google Scholar
• Bobylev , A. V. 1975 . The method of the Fourier transform in the theory of the Boltzmann equation for Maxwell molecules . Dokl. Akad. Nauk SSSR , 225 ( 6 ) : 1041 – 1044 .
   Google Scholar
• Bobylev , A. V. and Rjasanov . 1997 . Difference scheme for the Boltzmann equation based on the Fast Fourier Transform . Eur. J. Mech., B/Fluids , 16 ( 2 ) : 293 – 306 .
   Web of Science ®Google Scholar
• Buet , C. 1996 . A discrete velocity scheme for the Boltzmann operator of rarefied gas dynamics . Trans. Theo. Stat. Phys. , 25 ( 1 ) : 33 – 60 .
   Google Scholar
• Canuto , C. , Hussaini , M. Y. , Quarteroni , A. and Zang , T. A. 1988 . Spectral methods in fluid dynamics , New York : Springer Verlag .
   Google Scholar
• Cercignani , C. 1988 . The Boltzmann equation and its applications , Berlin : Springer-Verlag .
   Google Scholar
• Cercignani , C. , Illenr , R. and Pulvirenti , M. 1995 . The mathematical theory of dilute gases , New York : Springer-Verlag .
   Google Scholar
• Gabetta , E. and Pareschi , L. 1994 . The Maxwell gas and its Fourier transform towards a numerical approximation . Waves and stability in continuous media, Bologna 1993, Ser. Adv. Math. Appl. Sci. , 23 : 197 – 201 .
   Google Scholar
• Gabetta , E. , Pareschi , L. and Toscani , G. 1997 . Relaxation schemes for nonlinear kinetic equations . SIAM J. Numer. Anal. , 34 ( 6 ) : 1967 – 2011 .
   Web of Science ®Google Scholar
• Goldstein , D. , Sturtevant , B. and Broadwell , J. E. 1989 . “ Investigation of the motion of discrete velocity gases ” . In Rarefied Gas Dynamics: Theoretical and Computational Techniques , Edited by: Munz , E. P. , Weaver , D. P. and Campbell , H. D. Vol. 118 , 100 Washington , DC : AIAA .
   Google Scholar
• Gottlieb , D. and Orszag , S. A. 1977 . Numerical Analysis of Spectral Methods: Theory and Applications , SIAM CBMS-NSF . Series
   Google Scholar
• Grigoriev , Y. N. and Mikhalitsyn , A. N. 1983 . A spectral method of solving Boltzmann's kinetic equation numerically . U.S.S.R. Comput. Maths. Math. Phys. , 23 ( 6 ) : 105 – 111 .
   Web of Science ®Google Scholar
• Ernst , M. H. 1983 . “ Exact solutions of the nonlinear Boltzmann equation and related kinetic models ” . In Nonequilibrium Phenomena I. The Boltzmann equation , 52 – 119 . North-Holland .
   Google Scholar
• Pareschi , L. and Perthame , B. 1996 . A Fourier spectral method for homogeneous Boltzmann equations . Trans. Theo. and Stat. Phys. , 25 ( 5 ) : 369 – 383 .
   Google Scholar
• Pareschi , L. and Russo , G. Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator . SIAM J. Num. Anal , to appear
   Web of Science ®Google Scholar
• Pareschi , L. and Russo , G. Numerical solution of the Boltzmann equation It: Uniformly accurate second order schemes , work in preparation
   Google Scholar
• Rogier , F. and Schneider , J. 1994 . A direct method for solving the Boltzmann equation . Trans. Theo. Stat. Phys. , 23 ( 1–3 ) : 313 – 338 .
   Google Scholar
• Takata , S. , Sone , Y. and Aoki , K. 1993 . Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules . Phys. Fluids A , 5 ( 3 ) : 716 – 737 .
   Web of Science ®Google Scholar