American Institute of Mathematical Sciences

Advanced
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

Jingwei Hu 1,, , Jie Shen 1, and  Yingwei Wang 2,

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.

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2$^nd$ edition, Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[10]

J. Bremer and H. Yang, Fast algorithms for Jacobi expansions via nonoscillatory phase functions, preprint, 2018, arXiv: 1803.03889. Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[14]

C. Cercignani, Recent developments in the mechanics of granular materials, in Fisica Matematica e Ingeneria Delle Strutture, Pitagora Editrice, Bologna, 1995,119–132. Google Scholar

[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.  Google Scholar

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G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

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F. Filbet, On deterministic approximation of the Boltzmann equation in a bounded domain, Multiscale Model. Simul., 10 (2012), 792-817.  doi: 10.1137/11082419X.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35] L. Pareschi and G. Toscani, Interacting Multiagent Systems, Oxford University Press, UK, 2014.   Google Scholar
[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.  Google Scholar

[37]

J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin-Heidelberg, 2011. Google Scholar

[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.  Google Scholar

[39]

G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.  doi: 10.1051/m2an:2000127.  Google Scholar

[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.  Google Scholar

[41]

C. Villani, Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.  doi: 10.1007/s10955-006-9038-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2$^nd$ edition, Dover Publications, Inc., Mineola, NY, 2001.  Google Scholar

[10]

J. Bremer and H. Yang, Fast algorithms for Jacobi expansions via nonoscillatory phase functions, preprint, 2018, arXiv: 1803.03889. Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[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.   Google Scholar

[14]

C. Cercignani, Recent developments in the mechanics of granular materials, in Fisica Matematica e Ingeneria Delle Strutture, Pitagora Editrice, Bologna, 1995,119–132. Google Scholar

[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.  Google Scholar

[16]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman and Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35] L. Pareschi and G. Toscani, Interacting Multiagent Systems, Oxford University Press, UK, 2014.   Google Scholar
[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.  Google Scholar

[37]

J. Shen, T. Tang, and L.-L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin-Heidelberg, 2011. Google Scholar

[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.  Google Scholar

[39]

G. Toscani, One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal., 34 (2000), 1277-1291.  doi: 10.1051/m2an:2000127.  Google Scholar

[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.  Google Scholar

[41]

C. Villani, Mathematics of granular materials, J. Stat. Phys., 124 (2006), 781-822.  doi: 10.1007/s10955-006-9038-6.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Functions to be approximated
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Figure 2.  $ L_2 $ errors in log-log scale
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Figure 3.  $ L_2 $ errors in linear-log scale
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Figure 4.  Numerical solutions of $ f $ at time $ t = 0, 5, 10, 20 $ with $ e = 0.25 $
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Figure 5.  Numerical moments of $ f $ from $ t = 0 $ to $ t = 20 $ with $ e = 0.25 $
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Figure 6.  Numerical solutions in the rescaled case at time $ t = 0, 5, 10, 20 $ with $ e = 0.25 $
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Figure 7.  Numerical solutions in the rescaled case at time $ t = 0, 5, 10, 20 $ with $ e = 0.75 $
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Figure 8.  Errors of numerical moments of $ g $ from $ t = 0 $ to $ t = 20 $
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Figure 9.  Numerical solutions of heated case at time $ t = 20 $
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Figure 10.  Numerical moments of $ f $ from $ t = 0 $ to $ t = 20 $ with $ e = 0.25, \varepsilon = 0.01 $
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Figure 11.  Errors of moments of at $ t = 20 $ with $ e = 0.25, \varepsilon = 0.01 $ with respect to different $ N $
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Figure 12.  Numerical moments of $ f $ from $ t = 0 $ to $ t = 20 $ with $ e = 0.75, \varepsilon = 0.01 $
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Figure 13.  Errors of moments of at $ t = 20 $ with $ e = 0.75, \varepsilon = 0.01 $ with respect to different $ N $
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