Figure 1.
Functions to be approximated
-
Previous Article
Strong solutions for the Alber equation and stability of unidirectional wave spectra
- KRM Home
- This Issue
-
Next Article
On the Cucker-Smale ensemble with $ q $-closest neighbors under time-delayed communications
August
2020, 13(4): 677-702.
doi: 10.3934/krm.2020023
A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions
| 1. | Department of Mathematics, Purdue University, West Lafayette, IN 47907 USA |
| 2. | SAS Institute, Inc., Cary, NC 27513 USA |
We develop in this paper a Petrov-Galerkin spectral method for the inelastic Boltzmann equation in one dimension. Solutions to such equations typically exhibit heavy tails in the velocity space so that domain truncation or Fourier approximation would suffer from large truncation errors. Our method is based on the mapped Chebyshev functions on unbounded domains, hence requires no domain truncation. Furthermore, the test and trial function spaces are carefully chosen to obtain desired convergence and conservation properties. Through a series of examples, we demonstrate that the proposed method performs better than the Fourier spectral method and yields highly accurate results.
Keywords:
Inelastic Boltzmann equation,
heavy tail,
granular material,
Petrov-Galerkin spectral method,
mapped Chebyshev function,
unbounded domain.
Mathematics Subject Classification:
Primary: 65M70, 35Q20, 41A50; Secondary: 65R20.
Citation:
Jingwei Hu, Jie Shen, Yingwei Wang. A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions. Kinetic and Related Models,
2020, 13
(4)
: 677-702.
doi: 10.3934/krm.2020023
![]()
References:
| [1] |
A. Baldassarri, A. Puglisi and U. Marconi,
Kinetics models of inelastic gases, Math. Models Methods Appl. Sci., 12 (2002), 965-983.
doi: 10.1142/S0218202502001982. |
| [2] |
A. Barrat, E. Trizac, and M. H. Ernst, Granular gases: Dynamics and collective effects, J. Phys.: Condens. Matter, 17 (2005), S2429–S2437.
doi: 10.1088/0953-8984/17/24/004. |
| [3] |
D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti,
A non-Maxwellian steady distribution for one-dimensional granular media, J. Stat. Phys., 91 (1998), 979-990.
doi: 10.1023/A:1023032000560. |
| [4] |
D. Benedetto and M. Pulvirenti,
On the one-dimensional Boltzmann equation for granular flows, M2AN: Math. Model. Numer. Anal., 35 (2001), 899-905.
doi: 10.1051/m2an:2001141. |
| [5] |
A. V. Bobylev, J. A. Carrillo and I. M. Gamba,
On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys., 98 (2000), 743-773.
doi: 10.1023/A:1018627625800. |
| [6] |
A. V. Bobylev and C. Cercignani,
Moment equations for a granular material in a thermal bath, J. Statist. Phys., 106 (2002), 547-567.
doi: 10.1023/A:1013754205008. |
| [7] |
A. V. Bobylev and C. Cercignani,
Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys., 110 (2003), 333-375.
doi: 10.1023/A:1021031031038. |
| [8] |
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.
doi: 10.1016/S0997-7546(99)00121-1. |
| [9] |
J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2$^nd$ edition, Dover Publications, Inc., Mineola, NY, 2001. |
| [10] |
J. Bremer and H. Yang, Fast algorithms for Jacobi expansions via nonoscillatory phase functions, preprint, 2018, arXiv: 1803.03889. |
| [11] |
N. V. Brilliantov and T. Pöschel, Kinetic Theory of Granular Gases, Oxford University Press, Oxford, UK, 2004.
doi: 10.1093/acprof:oso/9780198530381.001.0001.
|
| [12] |
J. A. Carrillo, C. Cercignani and I. M. Gamba,
Steady states of a Boltzmann equation for driven granular media, Phys. Rev. E, 62 (2000), 7700-7707.
doi: 10.1103/PhysRevE.62.7700. |
| [13] |
J. A. Carrillo and G. Toscani,
Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198.
|
| [14] |
C. Cercignani, Recent developments in the mechanics of granular materials, in Fisica Matematica e Ingeneria Delle Strutture, Pitagora Editrice, Bologna, 1995,119–132. |
| [15] |
G. Chai and T.-J. Wang,
Generalized hermite spectral method for nonlinear Fokker-Planck equations on the whole line, J. Math. Study, 51 (2018), 177-195.
doi: 10.4208/jms.v51n2.18.04. |
| [16] |
G. Dimarco and L. Pareschi,
Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.
doi: 10.1017/S0962492914000063. |
| [17] |
F. Filbet,
On deterministic approximation of the Boltzmann equation in a bounded domain, Multiscale Model. Simul., 10 (2012), 792-817.
doi: 10.1137/11082419X. |
| [18] |
F. Filbet, C. Mouhot and L. Pareschi,
Solving the Boltzmann equation in NlogN, SIAM J. Sci. Comput., 28 (2006), 1029-1053.
doi: 10.1137/050625175. |
| [19] |
F. Filbet, L. Pareschi and G. Toscani,
Accurate numerical methods for the collisional motion of (heated) granular flows, J. Comput. Phys., 202 (2005), 216-235.
doi: 10.1016/j.jcp.2004.06.023. |
| [20] |
F. Filbet and T. Rey,
A rescaling velocity method for dissipative kinetic equations. Applications to granular media, J. Comput. Phys., 248 (2013), 177-199.
doi: 10.1016/j.jcp.2013.04.023. |
| [21] |
F. Filbet and G. Russo,
High order numerical methods for the space non-homogeneous Boltzmann equation, J. Comput. Phys., 186 (2003), 457-480.
doi: 10.1016/S0021-9991(03)00065-2. |
| [22] |
I. M. Gamba, J. R. Haack, C. D. Hauck and J. Hu, A fast spectral method for the Boltzmann collision operator with general collision kernels, SIAM J. Sci. Comput., 39 (2017), B658–B674.
doi: 10.1137/16M1096001. |
| [23] |
I. M. Gamba, V. Panferov and C. Villani,
On the Boltzmann equation for diffusively excited granular media, Commun. Math. Phys., 246 (2004), 503-541.
doi: 10.1007/s00220-004-1051-5. |
| [24] |
I. M. Gamba and S. Rjasanow,
Galerkin-Petrov approach for the Boltzmann equation, J. Comput. Phys., 366 (2018), 341-365.
doi: 10.1016/j.jcp.2018.04.017. |
| [25] |
I. M. Gamba, S. Rjasanow and W. Wagner,
Direct simulation of the uniformly heated granular Boltzmann equation, Math. Comput. Modelling, 42 (2005), 683-700.
doi: 10.1016/j.mcm.2004.02.047. |
| [26] |
I. M. Gamba and S. H. Tharkabhushanam,
Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., 228 (2009), 2012-2036.
doi: 10.1016/j.jcp.2008.09.033. |
| [27] |
J. Hu and Z. Ma,
A fast spectral method for the inelastic Boltzmann collision operator and application to heated granular gases, J. Comput. Phys., 385 (2019), 119-134.
doi: 10.1016/j.jcp.2019.01.049. |
| [28] |
J. Hu and L. Ying,
A fast spectral algorithm for the quantum Boltzmann collision operator, Commun. Math. Sci., 10 (2012), 989-999.
doi: 10.4310/CMS.2012.v10.n3.a13. |
| [29] |
G. Kizler and J. Schröberl, A polynomial spectral method for the spatially homogenenous Boltzmann equation, SIAM J. Sci. Comput., 41 (2019), B27–B49.
doi: 10.1137/17M1160240. |
| [30] |
J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, FL, 2003. |
| [31] |
C. Mouhot and L. Pareschi,
Fast algorithms for computing the Boltzmann collision operator, Math. Comp., 75 (2006), 1833-1852.
doi: 10.1090/S0025-5718-06-01874-6. |
| [32] |
G. Naldi, L. Pareschi and G. Toscani,
Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit, M2AN Math. Model. Numer. Anal., 37 (2003), 73-90.
doi: 10.1051/m2an:2003019. |
| [33] |
L. Pareschi and B. Perthame,
A Fourier spectral method for homogeneous Boltzmann equations, Transport Theory Statist. Phys., 25 (1996), 369-382.
doi: 10.1080/00411459608220707. |
| [34] |
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.
doi: 10.1137/S0036142998343300. |
| [35] |
L. Pareschi and G. Toscani, Interacting Multiagent Systems, Oxford University Press, UK, 2014.
|
| [36] |
S. Rjasanow and W. Wagner,
Time splitting error in DSMC schemes for the spatially homogeneous inelastic Boltzmann equation, SIAM J. Numer. Anal., 45 (2007), 54-67.
doi: 10.1137/050643842. |
| [37] |
J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin-Heidelberg, 2011. |
| [38] |
J. Shen, L.-L. Wang and H. Yu,
Approximations by orthonormal mapped Chebyshev functions for higher-dimensional problems in unbounded domains, J. Comput. Appl. Math., 265 (2014), 264-275.
doi: 10.1016/j.cam.2013.09.024. |
| [39] |
G. Toscani,
One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.
doi: 10.1051/m2an:2000127. |
| [40] |
T. P. C. van Noije and M. H. Ernst,
Velocity distributions in homogeneous granular fluids: The free and the heated case, Granular Matter, 1 (1998), 57-64.
doi: 10.1007/s100350050009. |
| [41] |
C. Villani,
Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.
doi: 10.1007/s10955-006-9038-6. |
| [42] |
Y. Wang and Z. Cai, Approximation of the Boltzmann collision operator based on Hermite spectral method, J.Comput. Phys, 397 (2019), 108815, 23 pp.
doi: 10.1016/j.jcp.2019.07.014. |
| [43] |
D. Williams and F. Mackintosh, Driven granular media in one dimension: Correlations and equation of state, Phys. Rev. E, 54 (1996), R9–R12.
doi: 10.1103/PhysRevE.54.R9. |
| [44] |
L. Wu, C. White, T. J. Scanlon, J. M. Reese and Y. Zhang,
Deterministic numerical solutions of the Boltzmann equation using the fast spectral method, J. Comput. Phys., 250 (2013), 27-52.
doi: 10.1016/j.jcp.2013.05.003. |
| [45] |
L. Wu, Y. Zhang and J. M. Reese,
Fast spectral solution of the generalized Enskog equation for dense gases, J. Comput. Phys., 303 (2015), 66-79.
doi: 10.1016/j.jcp.2015.09.034. |
| [46] |
H. Yang,
A unified framework for oscillatory integral transform: When to use NUFFT or butterfly factorization, J. Comput. Phys., 388 (2019), 103-122.
doi: 10.1016/j.jcp.2019.02.044. |
| [47] |
X. Yu, Y. Zhao and Z. Wang,
A diagonalized Legendre rational spectral method for problems on the whole line, J. Math. Study, 51 (2018), 196-213.
doi: 10.4208/jms.v51n2.18.05. |
show all references
References:
| [1] |
A. Baldassarri, A. Puglisi and U. Marconi,
Kinetics models of inelastic gases, Math. Models Methods Appl. Sci., 12 (2002), 965-983.
doi: 10.1142/S0218202502001982. |
| [2] |
A. Barrat, E. Trizac, and M. H. Ernst, Granular gases: Dynamics and collective effects, J. Phys.: Condens. Matter, 17 (2005), S2429–S2437.
doi: 10.1088/0953-8984/17/24/004. |
| [3] |
D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti,
A non-Maxwellian steady distribution for one-dimensional granular media, J. Stat. Phys., 91 (1998), 979-990.
doi: 10.1023/A:1023032000560. |
| [4] |
D. Benedetto and M. Pulvirenti,
On the one-dimensional Boltzmann equation for granular flows, M2AN: Math. Model. Numer. Anal., 35 (2001), 899-905.
doi: 10.1051/m2an:2001141. |
| [5] |
A. V. Bobylev, J. A. Carrillo and I. M. Gamba,
On some properties of kinetic and hydrodynamic equations for inelastic interactions, J. Statist. Phys., 98 (2000), 743-773.
doi: 10.1023/A:1018627625800. |
| [6] |
A. V. Bobylev and C. Cercignani,
Moment equations for a granular material in a thermal bath, J. Statist. Phys., 106 (2002), 547-567.
doi: 10.1023/A:1013754205008. |
| [7] |
A. V. Bobylev and C. Cercignani,
Self-similar asymptotics for the Boltzmann equation with inelastic and elastic interactions, J. Statist. Phys., 110 (2003), 333-375.
doi: 10.1023/A:1021031031038. |
| [8] |
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.
doi: 10.1016/S0997-7546(99)00121-1. |
| [9] |
J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2$^nd$ edition, Dover Publications, Inc., Mineola, NY, 2001. |
| [10] |
J. Bremer and H. Yang, Fast algorithms for Jacobi expansions via nonoscillatory phase functions, preprint, 2018, arXiv: 1803.03889. |
| [11] |
N. V. Brilliantov and T. Pöschel, Kinetic Theory of Granular Gases, Oxford University Press, Oxford, UK, 2004.
doi: 10.1093/acprof:oso/9780198530381.001.0001.
|
| [12] |
J. A. Carrillo, C. Cercignani and I. M. Gamba,
Steady states of a Boltzmann equation for driven granular media, Phys. Rev. E, 62 (2000), 7700-7707.
doi: 10.1103/PhysRevE.62.7700. |
| [13] |
J. A. Carrillo and G. Toscani,
Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma, 6 (2007), 75-198.
|
| [14] |
C. Cercignani, Recent developments in the mechanics of granular materials, in Fisica Matematica e Ingeneria Delle Strutture, Pitagora Editrice, Bologna, 1995,119–132. |
| [15] |
G. Chai and T.-J. Wang,
Generalized hermite spectral method for nonlinear Fokker-Planck equations on the whole line, J. Math. Study, 51 (2018), 177-195.
doi: 10.4208/jms.v51n2.18.04. |
| [16] |
G. Dimarco and L. Pareschi,
Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.
doi: 10.1017/S0962492914000063. |
| [17] |
F. Filbet,
On deterministic approximation of the Boltzmann equation in a bounded domain, Multiscale Model. Simul., 10 (2012), 792-817.
doi: 10.1137/11082419X. |
| [18] |
F. Filbet, C. Mouhot and L. Pareschi,
Solving the Boltzmann equation in NlogN, SIAM J. Sci. Comput., 28 (2006), 1029-1053.
doi: 10.1137/050625175. |
| [19] |
F. Filbet, L. Pareschi and G. Toscani,
Accurate numerical methods for the collisional motion of (heated) granular flows, J. Comput. Phys., 202 (2005), 216-235.
doi: 10.1016/j.jcp.2004.06.023. |
| [20] |
F. Filbet and T. Rey,
A rescaling velocity method for dissipative kinetic equations. Applications to granular media, J. Comput. Phys., 248 (2013), 177-199.
doi: 10.1016/j.jcp.2013.04.023. |
| [21] |
F. Filbet and G. Russo,
High order numerical methods for the space non-homogeneous Boltzmann equation, J. Comput. Phys., 186 (2003), 457-480.
doi: 10.1016/S0021-9991(03)00065-2. |
| [22] |
I. M. Gamba, J. R. Haack, C. D. Hauck and J. Hu, A fast spectral method for the Boltzmann collision operator with general collision kernels, SIAM J. Sci. Comput., 39 (2017), B658–B674.
doi: 10.1137/16M1096001. |
| [23] |
I. M. Gamba, V. Panferov and C. Villani,
On the Boltzmann equation for diffusively excited granular media, Commun. Math. Phys., 246 (2004), 503-541.
doi: 10.1007/s00220-004-1051-5. |
| [24] |
I. M. Gamba and S. Rjasanow,
Galerkin-Petrov approach for the Boltzmann equation, J. Comput. Phys., 366 (2018), 341-365.
doi: 10.1016/j.jcp.2018.04.017. |
| [25] |
I. M. Gamba, S. Rjasanow and W. Wagner,
Direct simulation of the uniformly heated granular Boltzmann equation, Math. Comput. Modelling, 42 (2005), 683-700.
doi: 10.1016/j.mcm.2004.02.047. |
| [26] |
I. M. Gamba and S. H. Tharkabhushanam,
Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., 228 (2009), 2012-2036.
doi: 10.1016/j.jcp.2008.09.033. |
| [27] |
J. Hu and Z. Ma,
A fast spectral method for the inelastic Boltzmann collision operator and application to heated granular gases, J. Comput. Phys., 385 (2019), 119-134.
doi: 10.1016/j.jcp.2019.01.049. |
| [28] |
J. Hu and L. Ying,
A fast spectral algorithm for the quantum Boltzmann collision operator, Commun. Math. Sci., 10 (2012), 989-999.
doi: 10.4310/CMS.2012.v10.n3.a13. |
| [29] |
G. Kizler and J. Schröberl, A polynomial spectral method for the spatially homogenenous Boltzmann equation, SIAM J. Sci. Comput., 41 (2019), B27–B49.
doi: 10.1137/17M1160240. |
| [30] |
J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, FL, 2003. |
| [31] |
C. Mouhot and L. Pareschi,
Fast algorithms for computing the Boltzmann collision operator, Math. Comp., 75 (2006), 1833-1852.
doi: 10.1090/S0025-5718-06-01874-6. |
| [32] |
G. Naldi, L. Pareschi and G. Toscani,
Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit, M2AN Math. Model. Numer. Anal., 37 (2003), 73-90.
doi: 10.1051/m2an:2003019. |
| [33] |
L. Pareschi and B. Perthame,
A Fourier spectral method for homogeneous Boltzmann equations, Transport Theory Statist. Phys., 25 (1996), 369-382.
doi: 10.1080/00411459608220707. |
| [34] |
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.
doi: 10.1137/S0036142998343300. |
| [35] |
L. Pareschi and G. Toscani, Interacting Multiagent Systems, Oxford University Press, UK, 2014.
|
| [36] |
S. Rjasanow and W. Wagner,
Time splitting error in DSMC schemes for the spatially homogeneous inelastic Boltzmann equation, SIAM J. Numer. Anal., 45 (2007), 54-67.
doi: 10.1137/050643842. |
| [37] |
J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin-Heidelberg, 2011. |
| [38] |
J. Shen, L.-L. Wang and H. Yu,
Approximations by orthonormal mapped Chebyshev functions for higher-dimensional problems in unbounded domains, J. Comput. Appl. Math., 265 (2014), 264-275.
doi: 10.1016/j.cam.2013.09.024. |
| [39] |
G. Toscani,
One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.
doi: 10.1051/m2an:2000127. |
| [40] |
T. P. C. van Noije and M. H. Ernst,
Velocity distributions in homogeneous granular fluids: The free and the heated case, Granular Matter, 1 (1998), 57-64.
doi: 10.1007/s100350050009. |
| [41] |
C. Villani,
Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.
doi: 10.1007/s10955-006-9038-6. |
| [42] |
Y. Wang and Z. Cai, Approximation of the Boltzmann collision operator based on Hermite spectral method, J.Comput. Phys, 397 (2019), 108815, 23 pp.
doi: 10.1016/j.jcp.2019.07.014. |
| [43] |
D. Williams and F. Mackintosh, Driven granular media in one dimension: Correlations and equation of state, Phys. Rev. E, 54 (1996), R9–R12.
doi: 10.1103/PhysRevE.54.R9. |
| [44] |
L. Wu, C. White, T. J. Scanlon, J. M. Reese and Y. Zhang,
Deterministic numerical solutions of the Boltzmann equation using the fast spectral method, J. Comput. Phys., 250 (2013), 27-52.
doi: 10.1016/j.jcp.2013.05.003. |
| [45] |
L. Wu, Y. Zhang and J. M. Reese,
Fast spectral solution of the generalized Enskog equation for dense gases, J. Comput. Phys., 303 (2015), 66-79.
doi: 10.1016/j.jcp.2015.09.034. |
| [46] |
H. Yang,
A unified framework for oscillatory integral transform: When to use NUFFT or butterfly factorization, J. Comput. Phys., 388 (2019), 103-122.
doi: 10.1016/j.jcp.2019.02.044. |
| [47] |
X. Yu, Y. Zhao and Z. Wang,
A diagonalized Legendre rational spectral method for problems on the whole line, J. Math. Study, 51 (2018), 196-213.
doi: 10.4208/jms.v51n2.18.05. |
Figure 2.
$ L_2 $ errors in log-log scale
Figure 3.
$ L_2 $ errors in linear-log scale
Figure 9.
Numerical solutions of heated case at time $ t = 20 $
Figure 10.
Numerical moments of $ f $ from $ t = 0 $ to $ t = 20 $ with $ e = 0.25, \varepsilon = 0.01 $
Figure 11.
Errors of moments of at $ t = 20 $ with $ e = 0.25, \varepsilon = 0.01 $ with respect to different $ N $
Figure 12.
Numerical moments of $ f $ from $ t = 0 $ to $ t = 20 $ with $ e = 0.75, \varepsilon = 0.01 $
Figure 13.
Errors of moments of at $ t = 20 $ with $ e = 0.75, \varepsilon = 0.01 $ with respect to different $ N $
| [1] |
Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic and Related Models, 2019, 12 (3) : 507-549. doi: 10.3934/krm.2019021 |
| [2] |
Qiyi Gao, Hua Wu. Legendre-Petrov-Galerkin Chebyshev spectral collocation method for second-order nonlinear differential equations. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022166 |
| [3] |
Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327 |
| [4] |
Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic and Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044 |
| [5] |
Zhonghui Li, Xiangyong Chen, Jianlong Qiu, Tongshui Xia. A novel Chebyshev-collocation spectral method for solving the transport equation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2519-2526. doi: 10.3934/jimo.2020080 |
| [6] |
Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic and Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139 |
| [7] |
Fei Meng, Fang Liu. On the inelastic Boltzmann equation for soft potentials with diffusion. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5197-5217. doi: 10.3934/cpaa.2020233 |
| [8] |
Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic and Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445 |
| [9] |
Rafael Sanabria. Inelastic Boltzmann equation driven by a particle thermal bath. Kinetic and Related Models, 2021, 14 (4) : 639-679. doi: 10.3934/krm.2021018 |
| [10] |
Guo Ben-Yu, Wang Zhong-Qing. Modified Chebyshev rational spectral method for the whole line. Conference Publications, 2003, 2003 (Special) : 365-374. doi: 10.3934/proc.2003.2003.365 |
| [11] |
Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic and Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034 |
| [12] |
Shi Jin, Yingda Li. Local sensitivity analysis and spectral convergence of the stochastic Galerkin method for discrete-velocity Boltzmann equations with multi-scales and random inputs. Kinetic and Related Models, 2019, 12 (5) : 969-993. doi: 10.3934/krm.2019037 |
| [13] |
Netra Khanal, Ramjee Sharma, Jiahong Wu, Juan-Ming Yuan. A dual-Petrov-Galerkin method for extended fifth-order Korteweg-de Vries type equations. Conference Publications, 2009, 2009 (Special) : 442-450. doi: 10.3934/proc.2009.2009.442 |
| [14] |
Juan-Ming Yuan, Jiahong Wu. A dual-Petrov-Galerkin method for two integrable fifth-order KdV type equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1525-1536. doi: 10.3934/dcds.2010.26.1525 |
| [15] |
S.V. Zelik. The attractor for a nonlinear hyperbolic equation in the unbounded domain. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 593-641. doi: 10.3934/dcds.2001.7.593 |
| [16] |
Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159 |
| [17] |
Shaofei Wu, Mingqing Wang, Maozhu Jin, Yuntao Zou, Lijun Song. Uniform $L^1$ stability of the inelastic Boltzmann equation with large external force for hard potentials. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1005-1013. doi: 10.3934/dcdss.2019068 |
| [18] |
Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216 |
| [19] |
Yuling Guo, Zhongqing Wang. A multi-domain Chebyshev collocation method for nonlinear fractional delay differential equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (12) : 7521-7545. doi: 10.3934/dcdsb.2022052 |
| [20] |
Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651 |
2021 Impact Factor: 1.398
Tools
Metrics
Other articles
by authors
[Back to Top]