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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:
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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.  Google Scholar [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. Google Scholar [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. Google Scholar [13] C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Caclulations, Cambridge University Press, Cambridge, UK, 2000.   Google Scholar [14] L. Desvillettes and B. 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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. 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 Journal on Scientific Computing, 39 (2017), B658-B674, URL https://doi.org/10.1137/16M1096001. doi: 10.1137/16M1096001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] P. Grohs, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, USA, 1969.  doi: 10.1007/978-1-4899-6381-9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [37] A. Narayan and A. Klöckner, Deterministic numerical schemes for the Boltzmann equation, preprint, arXiv: 0911.3589. Google Scholar [38] H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer Series in Information Sciences, 2. Springer-Verlag, Berlin-New York, 1981.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [42] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics Series, Springer, Heidelberg, 2005.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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. Google Scholar [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.   Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [13] C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Actual Caclulations, Cambridge University Press, Cambridge, UK, 2000.   Google Scholar [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.  Google Scholar [15] G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numerica, 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. 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 Journal on Scientific Computing, 39 (2017), B658-B674, URL https://doi.org/10.1137/16M1096001. doi: 10.1137/16M1096001.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] P. Grohs, R. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] M. N. Kogan, Rarefied Gas Dynamics, Plenum Press, New York, USA, 1969.  doi: 10.1007/978-1-4899-6381-9.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [37] A. Narayan and A. Klöckner, Deterministic numerical schemes for the Boltzmann equation, preprint, arXiv: 0911.3589. Google Scholar [38] H. J. Nussbaumer, Fast Fourier Transform and Convolution Algorithms, Springer Series in Information Sciences, 2. Springer-Verlag, Berlin-New York, 1981.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [42] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows Approximation Methods in Kinetic Theory, Interaction of Mechanics and Mathematics Series, Springer, Heidelberg, 2005.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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
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]
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
CPU times for evaluating the collision operator directly and using the Fourier transform
 DFT Direct Speedup $M$ time, s $\alpha$ time, s $\alpha$ 9 1.47E-02 1.25E-01 8.5 15 3.94E-01 6.43 4.91E+00 7.18 12.5 21 3.09E+00 6.14 7.80E+01 8.21 25.2 27 1.64E+01 6.65 6.05E+02 8.15 36.7
 DFT Direct Speedup $M$ time, s $\alpha$ time, s $\alpha$ 9 1.47E-02 1.25E-01 8.5 15 3.94E-01 6.43 4.91E+00 7.18 12.5 21 3.09E+00 6.14 7.80E+01 8.21 25.2 27 1.64E+01 6.65 6.05E+02 8.15 36.7
Absolute errors in conservation of mass and temperature in the discrete collision integral computed using split and non-split formulations
 Error in Conservation of Mass Error in Conservation of Temperature Split Non-split Split Non-split $n$ Fourier Direct Fourier Direct Fourier Direct Fourier Direct 9 0.37 1.26 1.71E-5 1.92E-5 3.51 1.69 1.71E-2 1.84E-2 15 0.10 1.20 1.45E-5 1.71E-5 0.29 1.25 1.64E-3 3.15E-3 21 0.18 1.18 0.67E-5 0.93E-5 1.38 1.24 5.61E-5 1.75E-3 27 0.18 1.18 0.61E-5 0.86E-5 1.37 1.24 5.40E-4 1.05E-3
 Error in Conservation of Mass Error in Conservation of Temperature Split Non-split Split Non-split $n$ Fourier Direct Fourier Direct Fourier Direct Fourier Direct 9 0.37 1.26 1.71E-5 1.92E-5 3.51 1.69 1.71E-2 1.84E-2 15 0.10 1.20 1.45E-5 1.71E-5 0.29 1.25 1.64E-3 3.15E-3 21 0.18 1.18 0.67E-5 0.93E-5 1.38 1.24 5.61E-5 1.75E-3 27 0.18 1.18 0.61E-5 0.86E-5 1.37 1.24 5.40E-4 1.05E-3
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