# American Institute of Mathematical Sciences

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

* Corresponding author: Jingwei Hu

Received  June 2019 Revised  January 2020 Published  May 2020

Fund Project: J. Hu's research was supported by NSF grant DMS-1620250 and NSF CAREER grant DMS-1654152. J. Shen's research was supported in part by NSF DMS-1720442 and AFOSR FA9550-16-1-0102

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.

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.

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##### 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.
Functions to be approximated
$L_2$ errors in log-log scale
$L_2$ errors in linear-log scale
Numerical solutions of $f$ at time $t = 0, 5, 10, 20$ with $e = 0.25$
Numerical moments of $f$ from $t = 0$ to $t = 20$ with $e = 0.25$
Numerical solutions in the rescaled case at time $t = 0, 5, 10, 20$ with $e = 0.25$
Numerical solutions in the rescaled case at time $t = 0, 5, 10, 20$ with $e = 0.75$
Errors of numerical moments of $g$ from $t = 0$ to $t = 20$
Numerical solutions of heated case at time $t = 20$
Numerical moments of $f$ from $t = 0$ to $t = 20$ with $e = 0.25, \varepsilon = 0.01$
Errors of moments of at $t = 20$ with $e = 0.25, \varepsilon = 0.01$ with respect to different $N$
Numerical moments of $f$ from $t = 0$ to $t = 20$ with $e = 0.75, \varepsilon = 0.01$
Errors of moments of at $t = 20$ with $e = 0.75, \varepsilon = 0.01$ with respect to different $N$
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