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A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff
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 |
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.
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. |
[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. |
[3] | |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
E. Fehlberg,
Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme, Computing, 6 (1970), 61-71.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
B. Shizgal, Spectral Methods in Chemistry and Physics, Springer, 2015.
doi: 10.1007/978-94-017-9454-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[3] | |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
E. Fehlberg,
Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme, Computing, 6 (1970), 61-71.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
B. Shizgal, Spectral Methods in Chemistry and Physics, Springer, 2015.
doi: 10.1007/978-94-017-9454-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
























$K$ | $L$ | $|{I_{K, L}}|$ |
3 | 3 | 64 |
3 | 5 | 144 |
3 | 7 | 256 |
$K$ | $L$ | $|{I_{K, L}}|$ |
3 | 3 | 64 |
3 | 5 | 144 |
3 | 7 | 256 |
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 |
$K=1, L=2$ | $5.666437\cdot 10^{-2}$ |
$K=2, L=4$ | $6.201405\cdot 10^{-3}$ |
$K=3, L=6$ | $5.937974\cdot 10^{-4}$ |
$K=4, L=8$ | $7.978434\cdot 10^{-5}$ |
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 | |
$K=1, L=2$ | $9.886570 \cdot 10^{-2}$ | $9.875780 \cdot 10^{-2}$ | $9.874720 \cdot 10^{-2}$ | $9.874130 \cdot 10^{-2}$ |
$K=2, L=4$ | $4.842620 \cdot 10^{-2}$ | $4.839010 \cdot 10^{-2}$ | $4.835910 \cdot 10^{-2}$ | $4.834270 \cdot 10^{-2}$ |
$K=3, L=6$ | $2.402240 \cdot 10^{-2}$ | $2.379990 \cdot 10^{-2}$ | $2.367980 \cdot 10^{-2}$ | $2.361850 \cdot 10^{-2}$ |
$K=4, L=8$ | $1.729450 \cdot 10^{-2}$ | $1.706360 \cdot 10^{-2}$ | $1.694290 \cdot 10^{-2}$ | $1.688380 \cdot 10^{-2}$ |
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 | |
$K=1, L=2$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ | $3.226070 \cdot 10^{-2}$ |
$K=2, L=4$ | $1.175180 \cdot 10^{-2}$ | $1.176170 \cdot 10^{-2}$ | $1.174380 \cdot 10^{-2}$ | $1.172670 \cdot 10^{-2}$ |
$K=3, L=6$ | $2.435140 \cdot 10^{-3}$ | $2.343420 \cdot 10^{-3}$ | $2.277850 \cdot 10^{-3}$ | $2.233940 \cdot 10^{-3}$ |
$K=4, L=8$ | $2.415870 \cdot 10^{-3}$ | $2.309910 \cdot 10^{-3}$ | $2.242550 \cdot 10^{-3}$ | $2.197230 \cdot 10^{-3}$ |
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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