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A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff
June  2019, 12(3): 507-549. doi: 10.3934/krm.2019021

## Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation

 Saarland University, Department of Mathematics, P.O. Box 15 11 50, 66041 Saarbrücken, Germany

* Corresponding author: Torsten Keßler

Received  December 2017 Revised  July 2018 Published  February 2019

In this paper, we present an application of a Galerkin-Petrov method to the spatially one-dimensional Boltzmann equation. The three-dimensional velocity space is discretised by a spectral method. The space of the test functions is spanned by polynomials, which includes the collision invariants. This automatically insures the exact conservation of mass, momentum and energy. The resulting system of hyperbolic PDEs is solved with a finite volume method. We illustrate our method with two standard tests, namely the Fourier and the Sod shock tube problems. Our results are validated with the help of a stochastic particle method.

Citation: Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic & Related Models, 2019, 12 (3) : 507-549. doi: 10.3934/krm.2019021
##### References:
 [1] 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.  doi: 10.1016/j.jcp.2014.03.031.  Google Scholar [2] A. Alekseenko and J. Limbacher, Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $\mathcal{O}(N^2)$ operations using the discrete Fourier transform, 2018, arXiv: 1801.05892v1. Google Scholar [3] G. A. Bird, Molecular Gas Dynamics, Clarendon Press, 1976. Google Scholar [4] A. V. Bobylev and S. Rjasanow, Difference scheme for the Boltzmann equation based on fast Fourier transform, European Journal of Mechanics - B/Fluids, 16 (1997), 293-306.   Google Scholar [5] A. V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann for hard spheres, European Journal of Mechanics - B/Fluids, 18 (1999), 869-887.  doi: 10.1016/S0997-7546(99)00121-1.  Google Scholar [6] A. V. Bobylev and S. Rjasanow, Numerical solution of the Boltzmann equation using fully conservative difference scheme based on the Fast Fourier Transform, Transport Theory and Statistical Physics, 29 (2000), 289-310.  doi: 10.1080/00411450008205876.  Google Scholar [7] D. Burnett, The distribution of velocities in a slightly non-uniform gas, Proceedings of the London Mathematical Society, 39 (1935), 385-430.  doi: 10.1112/plms/s2-39.1.385.  Google Scholar [8] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied mathematical sciences, Springer-Verlag New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar [9] F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls, no. ⅩⅧ in Springer Monographs in Mathematics, Springer-Verlag New York, 2013. doi: 10.1007/978-1-4614-6660-4.  Google Scholar [10] G. Dimarco, R. Loubère, J. Narski and T. Rey, An efficient numerical method for solving the Boltzmann equation in multidimensions, Journal of Computational Physics, 353 (2018), 46-81.  doi: 10.1016/j.jcp.2017.10.010.  Google Scholar [11] G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, in Acta Numerica, Cambridge University Press (CUP), 23 (2014), 369–520. doi: 10.1017/S0962492914000063.  Google Scholar [12] E. Fehlberg, Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme, Computing, 6 (1970), 61-71.   Google Scholar [13] F. Filbet and C. Mouhot, Analysis of spectral methods for the homogeneous Boltzmann equation, Transactions of the American Mathematical Society, 363 (2011), 1947-1980.  doi: 10.1090/S0002-9947-2010-05303-6.  Google Scholar [14] F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in $N \log_2 N$, SIAM Journal on Scientific Computing, 28 (2006), 1029-1053.  doi: 10.1137/050625175.  Google Scholar [15] F. Filbet and G. Russo, High order numerical methods for the space non-homogeneous Boltzmann equation, Journal of Computational Physics, 186 (2003), 457-480.  doi: 10.1016/S0021-9991(03)00065-2.  Google Scholar [16] E. Fonn, P. Grohs and R. Hiptmair, Polar Spectral Scheme for the Spatially Homogeneous Boltzmann Equation, Technical Report 2014-13, Seminar for Applied Mathematics, ETH Zürich, Switzerland, 2014. Google Scholar [17] I. M. Gamba and J. R. Haack, A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit, Journal of Computational Physics, 270 (2014), 40-57.  doi: 10.1016/j.jcp.2014.03.035.  Google Scholar [18] I. M. Gamba and S. Rjasanow, Galerkin Petrov approach for the Boltzmann equation, Journal of Computational Physics, 366 (2018), 341-365.  doi: 10.1016/j.jcp.2018.04.017.  Google Scholar [19] 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.  doi: 10.1016/j.jcp.2008.09.033.  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.  doi: 10.4208/jcm.1003-m0011.  Google Scholar [21] G. P. Ghiroldi and L. Gibelli, A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization, Journal of Computational Physics, 258 (2014), 568-584.  doi: 10.1016/j.jcp.2013.10.055.  Google Scholar [22] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar [23] P. Grohs, R. Hiptmair and S. Pintarelli, Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in two dimensions, SMAI-Journal of Computational Mathematics, 3 (2017), 219-248.  doi: 10.5802/smai-jcm.26.  Google Scholar [24] I. Ibragimov and S. Rjasanow, Numerical solution of the Boltzmann equation on the uniform grid, Computing, 69 (2002), 163-186.  doi: 10.1007/s00607-002-1458-9.  Google Scholar [25] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126 (1996), 202-228.  doi: 10.1006/jcph.1996.0130.  Google Scholar [26] G. Kitzler and J. Schöberl, A high order space omentum discontinuous Galerkin method for the Boltzmann equation, Mathematics with Applications, 70 (2015), 1539-1554.  doi: 10.1016/j.camwa.2015.06.011.  Google Scholar [27] V. I. Lebedev, Values of the nodes and weights of ninth to seventeenth order Gauss-Markov quadrature formulae invariant under the octahedron group with inversion, USSR Computational Mathematics and Mathematical Physics, 15 (1975), 48-54.   Google Scholar [28] X. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115 (1994), 200-212.  doi: 10.1006/jcph.1994.1187.  Google Scholar [29] C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator, Mathematics of Computation, 75 (2006), 1833-1852.  doi: 10.1090/S0025-5718-06-01874-6.  Google Scholar [30] 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 [31] 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 [32] L. Pareschi and G. Russo, On the stability of spectral methods for the homogeneous Boltzmann equation, Transport Theory and Statistical Physics, 29 (2000), 431-447.  doi: 10.1080/00411450008205883.  Google Scholar [33] B. Shizgal, Spectral Methods in Chemistry and Physics, Springer, 2015. doi: 10.1007/978-94-017-9454-1.  Google Scholar [34] G. Sod, Survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, Journal of Computational Physics, 27 (1978), 1-31.  doi: 10.1016/0021-9991(78)90023-2.  Google Scholar [35] L. Wu, H. Liu, Y. Zhang and J. M. Reese, Influence of intermolecular potentials on rarefied gas flows: Fast spectral solutions of the Boltzmann equation, Physics of Fluids, 27 (2015), 082002.  doi: 10.1063/1.4929485.  Google Scholar [36] L. Wu, J. M. Reese and Y. Zhang, Oscillatory rarefied gas flow inside rectangular cavities, Journal of Fluid Mechanics, 748 (2014), 350-367.  doi: 10.1017/jfm.2014.183.  Google Scholar [37] L. Wu, J. M. Reese and Y. Zhang, Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows, Journal of Fluid Mechanics, 746 (2014), 53-84.  doi: 10.1017/jfm.2014.79.  Google Scholar [38] 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.  doi: 10.1016/j.jcp.2013.05.003.  Google Scholar [39] L. Wu, C. White, T. J. Scanlon, J. M. Reese and Y. Zhang, A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases, Journal of Fluid Mechanics, 763 (2015), 24-50.  doi: 10.1017/jfm.2014.632.  Google Scholar

show all references

##### References:
 [1] 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.  doi: 10.1016/j.jcp.2014.03.031.  Google Scholar [2] A. Alekseenko and J. Limbacher, Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $\mathcal{O}(N^2)$ operations using the discrete Fourier transform, 2018, arXiv: 1801.05892v1. Google Scholar [3] G. A. Bird, Molecular Gas Dynamics, Clarendon Press, 1976. Google Scholar [4] A. V. Bobylev and S. Rjasanow, Difference scheme for the Boltzmann equation based on fast Fourier transform, European Journal of Mechanics - B/Fluids, 16 (1997), 293-306.   Google Scholar [5] A. V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann for hard spheres, European Journal of Mechanics - B/Fluids, 18 (1999), 869-887.  doi: 10.1016/S0997-7546(99)00121-1.  Google Scholar [6] A. V. Bobylev and S. Rjasanow, Numerical solution of the Boltzmann equation using fully conservative difference scheme based on the Fast Fourier Transform, Transport Theory and Statistical Physics, 29 (2000), 289-310.  doi: 10.1080/00411450008205876.  Google Scholar [7] D. Burnett, The distribution of velocities in a slightly non-uniform gas, Proceedings of the London Mathematical Society, 39 (1935), 385-430.  doi: 10.1112/plms/s2-39.1.385.  Google Scholar [8] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, vol. 106 of Applied mathematical sciences, Springer-Verlag New York, 1994. doi: 10.1007/978-1-4419-8524-8.  Google Scholar [9] F. Dai and Y. Xu, Approximation Theory and Harmonic Analysis on Spheres and Balls, no. ⅩⅧ in Springer Monographs in Mathematics, Springer-Verlag New York, 2013. doi: 10.1007/978-1-4614-6660-4.  Google Scholar [10] G. Dimarco, R. Loubère, J. Narski and T. Rey, An efficient numerical method for solving the Boltzmann equation in multidimensions, Journal of Computational Physics, 353 (2018), 46-81.  doi: 10.1016/j.jcp.2017.10.010.  Google Scholar [11] G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, in Acta Numerica, Cambridge University Press (CUP), 23 (2014), 369–520. doi: 10.1017/S0962492914000063.  Google Scholar [12] E. Fehlberg, Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme, Computing, 6 (1970), 61-71.   Google Scholar [13] F. Filbet and C. Mouhot, Analysis of spectral methods for the homogeneous Boltzmann equation, Transactions of the American Mathematical Society, 363 (2011), 1947-1980.  doi: 10.1090/S0002-9947-2010-05303-6.  Google Scholar [14] F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in $N \log_2 N$, SIAM Journal on Scientific Computing, 28 (2006), 1029-1053.  doi: 10.1137/050625175.  Google Scholar [15] F. Filbet and G. Russo, High order numerical methods for the space non-homogeneous Boltzmann equation, Journal of Computational Physics, 186 (2003), 457-480.  doi: 10.1016/S0021-9991(03)00065-2.  Google Scholar [16] E. Fonn, P. Grohs and R. Hiptmair, Polar Spectral Scheme for the Spatially Homogeneous Boltzmann Equation, Technical Report 2014-13, Seminar for Applied Mathematics, ETH Zürich, Switzerland, 2014. Google Scholar [17] I. M. Gamba and J. R. Haack, A conservative spectral method for the Boltzmann equation with anisotropic scattering and the grazing collisions limit, Journal of Computational Physics, 270 (2014), 40-57.  doi: 10.1016/j.jcp.2014.03.035.  Google Scholar [18] I. M. Gamba and S. Rjasanow, Galerkin Petrov approach for the Boltzmann equation, Journal of Computational Physics, 366 (2018), 341-365.  doi: 10.1016/j.jcp.2018.04.017.  Google Scholar [19] 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.  doi: 10.1016/j.jcp.2008.09.033.  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.  doi: 10.4208/jcm.1003-m0011.  Google Scholar [21] G. P. Ghiroldi and L. Gibelli, A direct method for the Boltzmann equation based on a pseudo-spectral velocity space discretization, Journal of Computational Physics, 258 (2014), 568-584.  doi: 10.1016/j.jcp.2013.10.055.  Google Scholar [22] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar [23] P. Grohs, R. Hiptmair and S. Pintarelli, Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in two dimensions, SMAI-Journal of Computational Mathematics, 3 (2017), 219-248.  doi: 10.5802/smai-jcm.26.  Google Scholar [24] I. Ibragimov and S. Rjasanow, Numerical solution of the Boltzmann equation on the uniform grid, Computing, 69 (2002), 163-186.  doi: 10.1007/s00607-002-1458-9.  Google Scholar [25] G.-S. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126 (1996), 202-228.  doi: 10.1006/jcph.1996.0130.  Google Scholar [26] G. Kitzler and J. Schöberl, A high order space omentum discontinuous Galerkin method for the Boltzmann equation, Mathematics with Applications, 70 (2015), 1539-1554.  doi: 10.1016/j.camwa.2015.06.011.  Google Scholar [27] V. I. Lebedev, Values of the nodes and weights of ninth to seventeenth order Gauss-Markov quadrature formulae invariant under the octahedron group with inversion, USSR Computational Mathematics and Mathematical Physics, 15 (1975), 48-54.   Google Scholar [28] X. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115 (1994), 200-212.  doi: 10.1006/jcph.1994.1187.  Google Scholar [29] C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator, Mathematics of Computation, 75 (2006), 1833-1852.  doi: 10.1090/S0025-5718-06-01874-6.  Google Scholar [30] 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 [31] 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 [32] L. Pareschi and G. Russo, On the stability of spectral methods for the homogeneous Boltzmann equation, Transport Theory and Statistical Physics, 29 (2000), 431-447.  doi: 10.1080/00411450008205883.  Google Scholar [33] B. Shizgal, Spectral Methods in Chemistry and Physics, Springer, 2015. doi: 10.1007/978-94-017-9454-1.  Google Scholar [34] G. Sod, Survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, Journal of Computational Physics, 27 (1978), 1-31.  doi: 10.1016/0021-9991(78)90023-2.  Google Scholar [35] L. Wu, H. Liu, Y. Zhang and J. M. Reese, Influence of intermolecular potentials on rarefied gas flows: Fast spectral solutions of the Boltzmann equation, Physics of Fluids, 27 (2015), 082002.  doi: 10.1063/1.4929485.  Google Scholar [36] L. Wu, J. M. Reese and Y. Zhang, Oscillatory rarefied gas flow inside rectangular cavities, Journal of Fluid Mechanics, 748 (2014), 350-367.  doi: 10.1017/jfm.2014.183.  Google Scholar [37] L. Wu, J. M. Reese and Y. Zhang, Solving the Boltzmann equation deterministically by the fast spectral method: application to gas microflows, Journal of Fluid Mechanics, 746 (2014), 53-84.  doi: 10.1017/jfm.2014.79.  Google Scholar [38] 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.  doi: 10.1016/j.jcp.2013.05.003.  Google Scholar [39] L. Wu, C. White, T. J. Scanlon, J. M. Reese and Y. Zhang, A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases, Journal of Fluid Mechanics, 763 (2015), 24-50.  doi: 10.1017/jfm.2014.632.  Google Scholar
Generalised spectrum of $D$ with respect to $M$ for $K = 9$, $L = 9$, i.e. $n = 1000$
Approximation of the solution by a piecewise constant function. To fulfil boundary conditions, the dashed cells, called ghost cells, are added to the discretisation
Sketch of the one-dimensional Fourier problem. We seek the particle density function along the axis labeled by $x$. $T_l$, $T_r$ are the temperatures of the walls, $T_0$ is the initial temperature of the gas
Course of the total mass for $K = 3, L = 6$, 256 spatial cells and ${\rm{Kn}} = 0.1$
Sketch of the initial situation of the shock tube problem. Two areas of same bulk velocities and temperatures but different densities are separated by diaphragm (dashed line), which is removed at $t = 0$
Contour plot of the final particle density function for ${\rm{Kn}} = 0.25$ and $K = 3$, $L = 3$ at $x = 0.25$ in the $(v_1,v_2)$-plane
shows the density at the time $t_f$, whereas in Figure 7b, the temperature at the time $t_f$ is shown">Figure 7.  Comparison of different sets of basis functions for ${\rm{Kn}} = 1.0$. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figure 7a shows the density at the time $t_f$, whereas in Figure 7b, the temperature at the time $t_f$ is shown
and 8b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 8c and 8d show the temperature at the time $t_f$ near the left and the right wall, respectively">Figure 8.  Comparison of different sets of basis functions for ${\rm{Kn}} = 1.0$ near the walls. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figures 8a and 8b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 8c and 8d show the temperature at the time $t_f$ near the left and the right wall, respectively
shows the density at the time $t_f$, whereas in Figure 9b, the temperature at the time $t_f$ is shown">Figure 9.  Comparison of different sets of basis functions for ${\rm{Kn}} = 0.25$. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figure 9a shows the density at the time $t_f$, whereas in Figure 9b, the temperature at the time $t_f$ is shown
and 10b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 10c and 10d show the temperature at the time $t_f$ near the left and the right wall, respectively">Figure 10.  Comparison of different sets of basis functions for ${\rm{Kn}} = 0.25$ near the walls. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figures 10a and 10b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 10c and 10d show the temperature at the time $t_f$ near the left and the right wall, respectively
shows the density at the time $t_f$, whereas in Figure 11b, the temperature at the time $t_f$ is shown">Figure 11.  Comparison of different sets of basis functions for ${\rm{Kn}} = 0.025$. $K$ is set to 3; $L$ the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figure 11a shows the density at the time $t_f$, whereas in Figure 11b, the temperature at the time $t_f$ is shown
and 12b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 12c and 12d show the temperature at the time $t_f$ near the left and the right wall, respectively">Figure 12.  Comparison of different sets of basis functions for ${\rm{Kn}} = 0.025$ near the walls. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figures 12a and 12b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 12c and 12d show the temperature at the time $t_f$ near the left and the right wall, respectively
Comparison of the final particle density function for $K = 3$, $L = 3$ with the stochastic particle density function at $x = 0.25$ for the Knudsen numbers $1.0$, $0.25$ and $0.025$
Numerical solution of the shock tube problem at $t_f$ obtained with DSMC and the Galerkin–Petrov method for ${\rm{Kn}} = 0.1$. The exact solution of the Euler equations is shown by dashed lines
Numerical solution of the shock tube problem at $t_f$ obtained with DSMC and the Galerkin–Petrov method for ${\rm{Kn}} = 0.01$. The exact solution of the Euler equations is shown by dashed lines
Numerical solution of the shock tube problem at $t_f$ obtained with DSMC and the Galerkin--Petrov method for ${\rm{Kn}} = 0.01$. The exact solution of the Euler equations is shown by dashed lines
Particle density functions for ${\rm{Kn}} = 0.1$ shortly after the diaphragm is removed
Particle density functions for ${\rm{Kn}} = 0.1$ when the shock discontinuity reaches the third evaluation point
Particle density functions for ${\rm{Kn}} = 0.1$ when the contact discontinuity reaches the third evaluation point at time $t_f$
Particle density functions for ${\rm{Kn}} = 0.01$ shortly after the diaphragm is removed
Particle density functions for ${\rm{Kn}} = 0.01$ when the shock discontinuity reaches the third evaluation point
Particle density functions for ${\rm{Kn}} = 0.01$ when the contact discontinuity reaches the third evaluation point at time $t_f$
Particle density functions for ${\rm{Kn}} = 0.001$ shortly after the diaphragm is removed
Particle density functions for ${\rm{Kn}} = 0.001$ when the shock discontinuity reaches the third evaluation point
Particle density functions for ${\rm{Kn}} = 0.001$ when the contact discontinuity reaches the third evaluation point at time $t_f$
Number of basis and test functions for different choices of parameters. The set $I_{K,L}$ is defined in equation (8)
 $K$ $L$ $|{I_{K, L}}|$ 3 3 64 3 5 144 3 7 256
 $K$ $L$ $|{I_{K, L}}|$ 3 3 64 3 5 144 3 7 256
Relative $L^2$-error for the mixture from equation (18) with parameters given in equation (19)
 basis functions $L^2$-error $K=1, L=2$ $5.666437\cdot 10^{-2}$ $K=2, L=4$ $6.201405\cdot 10^{-3}$ $K=3, L=6$ $5.937974\cdot 10^{-4}$ $K=4, L=8$ $7.978434\cdot 10^{-5}$
 basis functions $L^2$-error $K=1, L=2$ $5.666437\cdot 10^{-2}$ $K=2, L=4$ $6.201405\cdot 10^{-3}$ $K=3, L=6$ $5.937974\cdot 10^{-4}$ $K=4, L=8$ $7.978434\cdot 10^{-5}$
Relative error of mass conservation for ${\rm{Kn}} = 1$
 Spatial cells 64 128 256 512 $K=1, L=2$ $9.886570 \cdot 10^{-2}$ $9.875780 \cdot 10^{-2}$ $9.874720 \cdot 10^{-2}$ $9.874130 \cdot 10^{-2}$ $K=2, L=4$ $4.842620 \cdot 10^{-2}$ $4.839010 \cdot 10^{-2}$ $4.835910 \cdot 10^{-2}$ $4.834270 \cdot 10^{-2}$ $K=3, L=6$ $2.402240 \cdot 10^{-2}$ $2.379990 \cdot 10^{-2}$ $2.367980 \cdot 10^{-2}$ $2.361850 \cdot 10^{-2}$ $K=4, L=8$ $1.729450 \cdot 10^{-2}$ $1.706360 \cdot 10^{-2}$ $1.694290 \cdot 10^{-2}$ $1.688380 \cdot 10^{-2}$
 Spatial cells 64 128 256 512 $K=1, L=2$ $9.886570 \cdot 10^{-2}$ $9.875780 \cdot 10^{-2}$ $9.874720 \cdot 10^{-2}$ $9.874130 \cdot 10^{-2}$ $K=2, L=4$ $4.842620 \cdot 10^{-2}$ $4.839010 \cdot 10^{-2}$ $4.835910 \cdot 10^{-2}$ $4.834270 \cdot 10^{-2}$ $K=3, L=6$ $2.402240 \cdot 10^{-2}$ $2.379990 \cdot 10^{-2}$ $2.367980 \cdot 10^{-2}$ $2.361850 \cdot 10^{-2}$ $K=4, L=8$ $1.729450 \cdot 10^{-2}$ $1.706360 \cdot 10^{-2}$ $1.694290 \cdot 10^{-2}$ $1.688380 \cdot 10^{-2}$
Relative error of mass conservation for ${\rm{Kn}} = 0.1$
 Spatial cells 64 128 256 512 $K=1, L=2$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $K=2, L=4$ $1.175180 \cdot 10^{-2}$ $1.176170 \cdot 10^{-2}$ $1.174380 \cdot 10^{-2}$ $1.172670 \cdot 10^{-2}$ $K=3, L=6$ $2.435140 \cdot 10^{-3}$ $2.343420 \cdot 10^{-3}$ $2.277850 \cdot 10^{-3}$ $2.233940 \cdot 10^{-3}$ $K=4, L=8$ $2.415870 \cdot 10^{-3}$ $2.309910 \cdot 10^{-3}$ $2.242550 \cdot 10^{-3}$ $2.197230 \cdot 10^{-3}$
 Spatial cells 64 128 256 512 $K=1, L=2$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $K=2, L=4$ $1.175180 \cdot 10^{-2}$ $1.176170 \cdot 10^{-2}$ $1.174380 \cdot 10^{-2}$ $1.172670 \cdot 10^{-2}$ $K=3, L=6$ $2.435140 \cdot 10^{-3}$ $2.343420 \cdot 10^{-3}$ $2.277850 \cdot 10^{-3}$ $2.233940 \cdot 10^{-3}$ $K=4, L=8$ $2.415870 \cdot 10^{-3}$ $2.309910 \cdot 10^{-3}$ $2.242550 \cdot 10^{-3}$ $2.197230 \cdot 10^{-3}$
Relative error of mass conservation for ${\rm{Kn}} = 0.01$
 Spatial cells 64 128 256 512 $K=1, L=2$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $K=2, L=4$ $7.244610 \cdot 10^{-3}$ $7.324720 \cdot 10^{-3}$ $7.358720 \cdot 10^{-3}$ $7.393080 \cdot 10^{-3}$ $K=3, L=6$ $8.215370 \cdot 10^{-4}$ $8.215370 \cdot 10^{-4}$ $8.215370 \cdot 10^{-4}$ $8.215370 \cdot 10^{-4}$ $K=4, L=8$ $2.432010 \cdot 10^{-4}$ $2.791020 \cdot 10^{-4}$ $2.729200 \cdot 10^{-4}$ $2.519770 \cdot 10^{-4}$
 Spatial cells 64 128 256 512 $K=1, L=2$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $3.226070 \cdot 10^{-2}$ $K=2, L=4$ $7.244610 \cdot 10^{-3}$ $7.324720 \cdot 10^{-3}$ $7.358720 \cdot 10^{-3}$ $7.393080 \cdot 10^{-3}$ $K=3, L=6$ $8.215370 \cdot 10^{-4}$ $8.215370 \cdot 10^{-4}$ $8.215370 \cdot 10^{-4}$ $8.215370 \cdot 10^{-4}$ $K=4, L=8$ $2.432010 \cdot 10^{-4}$ $2.791020 \cdot 10^{-4}$ $2.729200 \cdot 10^{-4}$ $2.519770 \cdot 10^{-4}$
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