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August  2019, 12(4): 703-726. doi: 10.3934/krm.2019027

Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform

Department of Mathematics, California State University Northridge, Northridge, CA 91330, USA

* Corresponding author: alexander.alekseenko@csun.edu

Received  January 2018 Revised  September 2018 Published  May 2019

Fund Project: Authors acknowledge support of NSF grant DMS-1620497. The first author was supported by the AFRL/AFIT MOA Small Grant Program. Computer resources were provided by the Extreme Science and Engineering Discovery Environment, supported by National Science Foundation Grant No. OCI-1053575

We present a numerical algorithm for evaluating the Boltzmann collision operator with $O(N^2)$ operations based on high order discontinuous Galerkin discretizations in the velocity variable. To formulate the approach, Galerkin projection of the collision operator is written in the form of a bilinear circular convolution. An application of the discrete Fourier transform allows to rewrite the six fold convolution sum as a three fold weighted convolution sum in the frequency space. The new algorithm is implemented and tested in the spatially homogeneous case, and results in a considerable improvement in speed as compared to the direct evaluation. Split and non-split forms of the collision operator are considered, which are forms of the collision operator that have separate and simultaneous evaluations of the gain and loss terms, respectively. Smaller numerical errors are observed in the conserved quantities in simulations using the non-split form.

Citation: Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027
References:
[1]

A. Alekseenko and E. Josyula, Deterministic solution of the Boltzmann equation using a discontinuous Galerkin velocity discretization, in 28th International Symposium on Rarefied Gas Dynamics, 9-13 July 2012, Zaragoza, Spain, AIP Conference Proceedings, American Institute of Physics, 2012, 8.

[2]

A. Alekseenko and E. Josyula, Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space, Journal of Computational Physics, 272 (2014), 170-188, URL http://www.sciencedirect.com/science/article/pii/S0021999114002186. doi: 10.1016/j.jcp.2014.03.031.

[3]

A. Alekseenko, T. Nguyen and A. Wood, A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(mn)$ operations, Kinetic & Related Models, 11 (2018), 1211-1234, URL http://aimsciences.org//article/id/140be380-22db-45cb-930b-c66b94ae3ca3. doi: 10.3934/krm.2018047.

[4]

R. Alexandre, A review of Boltzmann equation with singular kernels, Kinetic & Related Models, 2 (2009), 551-646, URL http://aimsciences.org//article/id/aae0536d-b7d8-422f-91d4-abe0c0f89f9d. doi: 10.3934/krm.2009.2.551.

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I. global existence for soft potential, Journal of Functional Analysis, 262 (2012), 915-1010, URL http://www.sciencedirect.com/science/article/pii/S0022123611003752. doi: 10.1016/j.jfa.2011.10.007.

[6]

V. V. Aristov and S. A. Zabelok, A deterministic method for the solution of the Boltzmann equation with parallel computations, Zhurnal Vychislitel'noi Tekhniki i Matematicheskoi Physiki, 42 (2002), 425-437.

[7]

V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Fluid Mechanics and Its Applications, Kluwer Academic Publishers, 2001. doi: 10.1007/978-94-010-0866-2.

[8]

H. Babovsky, Kinetic models on orthogonal groups and the simulation of the Boltzmann equation, in 26th International Symposium on Rarefied Gas Dynamics, Kyoto, Japan, 20-25 July 2008 (ed. T. Abe), vol. 1084 of AIP Conference Series, American Institute of Physics, 2008,415-420.

[9]

A. V. Bobylev and S. Rjasanow, Difference scheme for the Boltzmann equation based on the fast Fourier transform., European Journal of Mechanics - B/Fluids, 16 (1997), 293-306.

[10]

A. V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres, European Journal of Mechanics - B/Fluids, 18 (1999), 869-887, URL http://www.sciencedirect.com/science/article/pii/S0997754699001211. doi: 10.1016/S0997-7546(99)00121-1.

[11]

I. D. Boyd, Vectorization of a Monte Carlo simulation scheme for nonequilibrium gas dynamics, Journal of Computational Physics, 96 (1991), 411-427, URL http://www.sciencedirect.com/science/article/pii/002199919190243E.

[12]

C. Cercignani, On Boltzmann equation with cutoff potentials, The Physics of Fluids, 10 (1967), 2097-2104, URL https://aip.scitation.org/doi/abs/10.1063/1.1762004.

[13] C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Caclulations, Cambridge University Press, Cambridge, UK, 2000.
[14]

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Communications in Partial Differential Equations, 29 (2004), 133-155, URL https://doi.org/10.1081/PDE-120028847. doi: 10.1081/PDE-120028847.

[15]

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

[16]

F. Filbet and C. Mouhot, Analysis of spectral methods for the homogeneous Boltzmann equation, Transactions of the American Mathematical Society, 363 (2011), 1947-1980, URL http://www.jstor.org/stable/41104652. doi: 10.1090/S0002-9947-2010-05303-6.

[17]

F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in N log2 N, SIAM Journal on Scientific Computing, 28 (2006), 1029-1053, URL http://epubs.siam.org/doi/abs/10.1137/050625175. doi: 10.1137/050625175.

[18]

F. Filbet, L. Pareschi and T. Rey, On steady-state preserving spectral methods for homogeneous Boltzmann equations, Comptes Rendus Mathematique, 353 (2015), 309-314, URL http://www.sciencedirect.com/science/article/pii/S1631073X15000412. doi: 10.1016/j.crma.2015.01.015.

[19]

E. Fonn, P. Grohs and R. Hiptmair, Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation, IMA Journal of Numerical Analysis, 35 (2015), 1533-1567, URL http://dx.doi.org/10.1093/imanum/dru042. doi: 10.1093/imanum/dru042.

[20]

I. M. Gamba and S. H. Tharkabhushanam, Shock and boundary structure formation by spectral-Lagrangian methods for the inhomogeneous Boltzmann transport equation, Journal of Computational Mathematics, 28 (2010), 430-460, URL http://dx.doi.org/10.4208/jcm.1003-m0011. doi: 10.4208/jcm.1003-m0011.

[21]

I. Gamba and C. Zhang, A conservative discontinuous Galerkin scheme with O(N2) operations in computing Boltzmann collision weight matrix, in 29th International Symposium on Rarefied Gas Dynamics, July 2014, China, AIP Conference Proceedings, American Institute of Physics, 2014, 8.

[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 Journal on Scientific Computing, 39 (2017), B658-B674, URL https://doi.org/10.1137/16M1096001. doi: 10.1137/16M1096001.

[23]

I. M. Gamba and S. H. Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, Journal of Computational Physics, 228 (2009), 2012-2036, URL http://dx.doi.org/10.1016/j.jcp.2008.09.033. doi: 10.1016/j.jcp.2008.09.033.

[24]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, Journal of the American Mathematical Society, 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8.

[25]

P. GrohsR. Hiptmair and S. Pintarelli, Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in 2D, SMAI Journal of Computational Mathematics, 3 (2017), 219-248. doi: 10.5802/smai-jcm.26.

[26]

J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Texts in Applied Mathematics, 54. Springer, New York, 2008. doi: 10.1007/978-0-387-72067-8.

[27]

J. Hu, Q. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy, Journal of Scientific Computing, 62 (2015), 555-574, URL https://doi.org/10.1007/s10915-014-9869-2. doi: 10.1007/s10915-014-9869-2.

[28]

J. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator, Communications in Mathematical Sciences, 10 (2012), 989-999. doi: 10.4310/CMS.2012.v10.n3.a13.

[29]

S. Jaiswal, A. A. Alexeenko and J. Hu, A discontinuous galerkin fast spectral method for the full boltzmann equation with general collision kernels, Journal of Computational Physics, 378 (2019), 178-208, URL http://www.sciencedirect.com/science/article/pii/S0021999118307198. doi: 10.1016/j.jcp.2018.11.001.

[30]

R. Kirsch and S. Rjasanow, A weak formulation of the Boltzmann equation based on the Fourier transform, Journal of Statistical Physics, 129 (2007), 483-492, URL http://dx.doi.org/10.1007/s10955-007-9374-1. doi: 10.1007/s10955-007-9374-1.

[31] M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, USA, 1969. doi: 10.1007/978-1-4899-6381-9.
[32]

P. L. Lions, On Boltzmann and Landau equations, Philosophical Transactions: Physical Sciences and Engineering, 346 (1994), 191-204, URL http://www.jstor.org/stable/54323. doi: 10.1098/rsta.1994.0018.

[33]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Physica D: Nonlinear Phenomena, 188 (2004), 178-192, URL http://www.sciencedirect.com/science/article/pii/S0167278903003142. doi: 10.1016/j.physd.2003.07.011.

[34]

A. Majorana, A numerical model of the Boltzmann equation related to the discontinuous Galerkin method, Kinetic & Related Models, 4 (2011), 139-151, URL http://dx.doi.org/10.3934/krm.2011.4.139. doi: 10.3934/krm.2011.4.139.

[35]

C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator, Mathematics of Computation, 75 (2006), 1833-1852, URL http://www.jstor.org/stable/4100126. doi: 10.1090/S0025-5718-06-01874-6.

[36]

A. Munafò, J. R. Haack, I. M. Gamba and T. E. Magin, A spectral-Lagrangian Boltzmann solver for a multi-energy level gas, Journal of Computational Physics, 264 (2014), 152-176, URL http://www.sciencedirect.com/science/article/pii/S0021999114000631. doi: 10.1016/j.jcp.2014.01.036.

[37]

A. Narayan and A. Klöckner, Deterministic numerical schemes for the Boltzmann equation, preprint, arXiv: 0911.3589.

[38]

H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer Series in Information Sciences, 2. Springer-Verlag, Berlin-New York, 1981.

[39]

V. A. Panferov and A. G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Mathematical Methods in the Applied Sciences, 25 (2002), 571-593. doi: 10.1002/mma.303.

[40]

L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Transport Theory and Statistical Physics, 25 (1996), 369-382. doi: 10.1080/00411459608220707.

[41]

L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator, SIAM Journal on Numerical Analysis, 37 (2000), 1217-1245. doi: 10.1137/S0036142998343300.

[42]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics Series, Springer, Heidelberg, 2005.

[43]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Archive for Rational Mechanics and Analysis, 143 (1998), 273-307. doi: 10.1007/s002050050106.

[44]

C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, North-Holland, 1 (2002), 71-305, URL http://www.sciencedirect.com/science/article/pii/S1874579202800040. doi: 10.1016/S1874-5792(02)80004-0.

[45]

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, Journal of Computational Physics, 250 (2013), 27-52, URL http://www.sciencedirect.com/science/article/pii/S0021999113003276. doi: 10.1016/j.jcp.2013.05.003.

[46]

L. Wu, J. Zhang, J. M. Reese and Y. Zhang, A fast spectral method for the Boltzmann equation for monatomic gas mixtures, Journal of Computational Physics, 298 (2015), 602-621, URL http://www.sciencedirect.com/science/article/pii/S0021999115004167. doi: 10.1016/j.jcp.2015.06.019.

show all references

References:
[1]

A. Alekseenko and E. Josyula, Deterministic solution of the Boltzmann equation using a discontinuous Galerkin velocity discretization, in 28th International Symposium on Rarefied Gas Dynamics, 9-13 July 2012, Zaragoza, Spain, AIP Conference Proceedings, American Institute of Physics, 2012, 8.

[2]

A. Alekseenko and E. Josyula, Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space, Journal of Computational Physics, 272 (2014), 170-188, URL http://www.sciencedirect.com/science/article/pii/S0021999114002186. doi: 10.1016/j.jcp.2014.03.031.

[3]

A. Alekseenko, T. Nguyen and A. Wood, A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(mn)$ operations, Kinetic & Related Models, 11 (2018), 1211-1234, URL http://aimsciences.org//article/id/140be380-22db-45cb-930b-c66b94ae3ca3. doi: 10.3934/krm.2018047.

[4]

R. Alexandre, A review of Boltzmann equation with singular kernels, Kinetic & Related Models, 2 (2009), 551-646, URL http://aimsciences.org//article/id/aae0536d-b7d8-422f-91d4-abe0c0f89f9d. doi: 10.3934/krm.2009.2.551.

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I. global existence for soft potential, Journal of Functional Analysis, 262 (2012), 915-1010, URL http://www.sciencedirect.com/science/article/pii/S0022123611003752. doi: 10.1016/j.jfa.2011.10.007.

[6]

V. V. Aristov and S. A. Zabelok, A deterministic method for the solution of the Boltzmann equation with parallel computations, Zhurnal Vychislitel'noi Tekhniki i Matematicheskoi Physiki, 42 (2002), 425-437.

[7]

V. Aristov, Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Fluid Mechanics and Its Applications, Kluwer Academic Publishers, 2001. doi: 10.1007/978-94-010-0866-2.

[8]

H. Babovsky, Kinetic models on orthogonal groups and the simulation of the Boltzmann equation, in 26th International Symposium on Rarefied Gas Dynamics, Kyoto, Japan, 20-25 July 2008 (ed. T. Abe), vol. 1084 of AIP Conference Series, American Institute of Physics, 2008,415-420.

[9]

A. V. Bobylev and S. Rjasanow, Difference scheme for the Boltzmann equation based on the fast Fourier transform., European Journal of Mechanics - B/Fluids, 16 (1997), 293-306.

[10]

A. V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres, European Journal of Mechanics - B/Fluids, 18 (1999), 869-887, URL http://www.sciencedirect.com/science/article/pii/S0997754699001211. doi: 10.1016/S0997-7546(99)00121-1.

[11]

I. D. Boyd, Vectorization of a Monte Carlo simulation scheme for nonequilibrium gas dynamics, Journal of Computational Physics, 96 (1991), 411-427, URL http://www.sciencedirect.com/science/article/pii/002199919190243E.

[12]

C. Cercignani, On Boltzmann equation with cutoff potentials, The Physics of Fluids, 10 (1967), 2097-2104, URL https://aip.scitation.org/doi/abs/10.1063/1.1762004.

[13] C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Caclulations, Cambridge University Press, Cambridge, UK, 2000.
[14]

L. Desvillettes and B. Wennberg, Smoothness of the solution of the spatially homogeneous Boltzmann equation without cutoff, Communications in Partial Differential Equations, 29 (2004), 133-155, URL https://doi.org/10.1081/PDE-120028847. doi: 10.1081/PDE-120028847.

[15]

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

[16]

F. Filbet and C. Mouhot, Analysis of spectral methods for the homogeneous Boltzmann equation, Transactions of the American Mathematical Society, 363 (2011), 1947-1980, URL http://www.jstor.org/stable/41104652. doi: 10.1090/S0002-9947-2010-05303-6.

[17]

F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in N log2 N, SIAM Journal on Scientific Computing, 28 (2006), 1029-1053, URL http://epubs.siam.org/doi/abs/10.1137/050625175. doi: 10.1137/050625175.

[18]

F. Filbet, L. Pareschi and T. Rey, On steady-state preserving spectral methods for homogeneous Boltzmann equations, Comptes Rendus Mathematique, 353 (2015), 309-314, URL http://www.sciencedirect.com/science/article/pii/S1631073X15000412. doi: 10.1016/j.crma.2015.01.015.

[19]

E. Fonn, P. Grohs and R. Hiptmair, Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation, IMA Journal of Numerical Analysis, 35 (2015), 1533-1567, URL http://dx.doi.org/10.1093/imanum/dru042. doi: 10.1093/imanum/dru042.

[20]

I. M. Gamba and S. H. Tharkabhushanam, Shock and boundary structure formation by spectral-Lagrangian methods for the inhomogeneous Boltzmann transport equation, Journal of Computational Mathematics, 28 (2010), 430-460, URL http://dx.doi.org/10.4208/jcm.1003-m0011. doi: 10.4208/jcm.1003-m0011.

[21]

I. Gamba and C. Zhang, A conservative discontinuous Galerkin scheme with O(N2) operations in computing Boltzmann collision weight matrix, in 29th International Symposium on Rarefied Gas Dynamics, July 2014, China, AIP Conference Proceedings, American Institute of Physics, 2014, 8.

[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 Journal on Scientific Computing, 39 (2017), B658-B674, URL https://doi.org/10.1137/16M1096001. doi: 10.1137/16M1096001.

[23]

I. M. Gamba and S. H. Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, Journal of Computational Physics, 228 (2009), 2012-2036, URL http://dx.doi.org/10.1016/j.jcp.2008.09.033. doi: 10.1016/j.jcp.2008.09.033.

[24]

P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, Journal of the American Mathematical Society, 24 (2011), 771-847. doi: 10.1090/S0894-0347-2011-00697-8.

[25]

P. GrohsR. Hiptmair and S. Pintarelli, Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in 2D, SMAI Journal of Computational Mathematics, 3 (2017), 219-248. doi: 10.5802/smai-jcm.26.

[26]

J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Texts in Applied Mathematics, 54. Springer, New York, 2008. doi: 10.1007/978-0-387-72067-8.

[27]

J. Hu, Q. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy, Journal of Scientific Computing, 62 (2015), 555-574, URL https://doi.org/10.1007/s10915-014-9869-2. doi: 10.1007/s10915-014-9869-2.

[28]

J. Hu and L. Ying, A fast spectral algorithm for the quantum Boltzmann collision operator, Communications in Mathematical Sciences, 10 (2012), 989-999. doi: 10.4310/CMS.2012.v10.n3.a13.

[29]

S. Jaiswal, A. A. Alexeenko and J. Hu, A discontinuous galerkin fast spectral method for the full boltzmann equation with general collision kernels, Journal of Computational Physics, 378 (2019), 178-208, URL http://www.sciencedirect.com/science/article/pii/S0021999118307198. doi: 10.1016/j.jcp.2018.11.001.

[30]

R. Kirsch and S. Rjasanow, A weak formulation of the Boltzmann equation based on the Fourier transform, Journal of Statistical Physics, 129 (2007), 483-492, URL http://dx.doi.org/10.1007/s10955-007-9374-1. doi: 10.1007/s10955-007-9374-1.

[31] M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, USA, 1969. doi: 10.1007/978-1-4899-6381-9.
[32]

P. L. Lions, On Boltzmann and Landau equations, Philosophical Transactions: Physical Sciences and Engineering, 346 (1994), 191-204, URL http://www.jstor.org/stable/54323. doi: 10.1098/rsta.1994.0018.

[33]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Physica D: Nonlinear Phenomena, 188 (2004), 178-192, URL http://www.sciencedirect.com/science/article/pii/S0167278903003142. doi: 10.1016/j.physd.2003.07.011.

[34]

A. Majorana, A numerical model of the Boltzmann equation related to the discontinuous Galerkin method, Kinetic & Related Models, 4 (2011), 139-151, URL http://dx.doi.org/10.3934/krm.2011.4.139. doi: 10.3934/krm.2011.4.139.

[35]

C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator, Mathematics of Computation, 75 (2006), 1833-1852, URL http://www.jstor.org/stable/4100126. doi: 10.1090/S0025-5718-06-01874-6.

[36]

A. Munafò, J. R. Haack, I. M. Gamba and T. E. Magin, A spectral-Lagrangian Boltzmann solver for a multi-energy level gas, Journal of Computational Physics, 264 (2014), 152-176, URL http://www.sciencedirect.com/science/article/pii/S0021999114000631. doi: 10.1016/j.jcp.2014.01.036.

[37]

A. Narayan and A. Klöckner, Deterministic numerical schemes for the Boltzmann equation, preprint, arXiv: 0911.3589.

[38]

H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer Series in Information Sciences, 2. Springer-Verlag, Berlin-New York, 1981.

[39]

V. A. Panferov and A. G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Mathematical Methods in the Applied Sciences, 25 (2002), 571-593. doi: 10.1002/mma.303.

[40]

L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Transport Theory and Statistical Physics, 25 (1996), 369-382. doi: 10.1080/00411459608220707.

[41]

L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator, SIAM Journal on Numerical Analysis, 37 (2000), 1217-1245. doi: 10.1137/S0036142998343300.

[42]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics Series, Springer, Heidelberg, 2005.

[43]

C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations, Archive for Rational Mechanics and Analysis, 143 (1998), 273-307. doi: 10.1007/s002050050106.

[44]

C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics, North-Holland, 1 (2002), 71-305, URL http://www.sciencedirect.com/science/article/pii/S1874579202800040. doi: 10.1016/S1874-5792(02)80004-0.

[45]

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, Journal of Computational Physics, 250 (2013), 27-52, URL http://www.sciencedirect.com/science/article/pii/S0021999113003276. doi: 10.1016/j.jcp.2013.05.003.

[46]

L. Wu, J. Zhang, J. M. Reese and Y. Zhang, A fast spectral method for the Boltzmann equation for monatomic gas mixtures, Journal of Computational Physics, 298 (2015), 602-621, URL http://www.sciencedirect.com/science/article/pii/S0021999115004167. doi: 10.1016/j.jcp.2015.06.019.

Figure 1.  Evaluation of the collision operator using split and non-split forms: (a) and (b) the split form evaluated using the Fourier transform; (c) and (d) the split form evaluated directly; (e) and (f) the non-split form evaluated using the Fourier transform
Figure 2.  Relaxation of moments $f_{\varphi_{i, p}} = \int_{R^3} (u_{i}-\bar{u}_{i})^p f(t, \vec{u})\, du$, $i = 1, 2$, $p = 2, 3, 4, 6$ in a mix of Maxwellian streams corresponding to a shock wave with Mach number 3.0 obtained by solving the Boltzmann equation using Fourier and direct evaluations of the collision integral. In the case of $p = 2$, the relaxation of moments is also compared to moments of a DSMC solution [11]
Figure 3.  Relaxation of moments $f_{\varphi_{i, p}}$, $i = 1, 2$, $p = 2, 3, 4, 6$ in a mix of Maxwellian streams corresponding to a shock wave with Mach number 1.55 obtained by solving the Boltzmann equation using Fourier and direct evaluations of the collision integral
Table 1.  CPU times for evaluating the collision operator directly and using the Fourier transform
DFTDirectSpeedup
$M$time, s $\alpha$time, s $\alpha$
91.47E-021.25E-018.5
153.94E-016.434.91E+007.1812.5
213.09E+006.147.80E+018.2125.2
271.64E+016.656.05E+028.1536.7
DFTDirectSpeedup
$M$time, s $\alpha$time, s $\alpha$
91.47E-021.25E-018.5
153.94E-016.434.91E+007.1812.5
213.09E+006.147.80E+018.2125.2
271.64E+016.656.05E+028.1536.7
Table 2.  Absolute errors in conservation of mass and temperature in the discrete collision integral computed using split and non-split formulations
Error in Conservation of MassError in Conservation of Temperature
SplitNon-splitSplitNon-split
$n$FourierDirectFourierDirectFourierDirectFourierDirect
90.371.261.71E-51.92E-53.511.691.71E-21.84E-2
150.101.201.45E-51.71E-50.291.251.64E-33.15E-3
210.181.180.67E-50.93E-51.381.245.61E-51.75E-3
270.181.180.61E-50.86E-51.371.245.40E-41.05E-3
Error in Conservation of MassError in Conservation of Temperature
SplitNon-splitSplitNon-split
$n$FourierDirectFourierDirectFourierDirectFourierDirect
90.371.261.71E-51.92E-53.511.691.71E-21.84E-2
150.101.201.45E-51.71E-50.291.251.64E-33.15E-3
210.181.180.67E-50.93E-51.381.245.61E-51.75E-3
270.181.180.61E-50.86E-51.371.245.40E-41.05E-3
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