October  2018, 11(5): 1211-1234. doi: 10.3934/krm.2018047

A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations

1. 

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

2. 

Department of Mathematics & Statistics, Wright State University, Dayton, OH 45435, USA

3. 

Department of Mathematics & Statistics, Air Force Institute of Technology, WPAFB, OH 45433, USA

* Corresponding author: alexander.alekseenko@csun.edu

Received  March 2016 Revised  September 2017 Published  May 2018

Fund Project: The first author was supported by NSF grant DMS-1620497 and HPTi PETTT grant PP-SASKY06-001. The third author was supported by AFOSR Grant No. F4FGA04296J00

We developed and implemented a numerical algorithm for evaluating the Boltzmann collision integral with $O(MN)$ operations, where $N$ is the number of the discrete velocity points and $M <N$. At the base of the algorithm are nodal-discontinuous Galerkin discretizations of the collision operator on uniform grids and a bilinear convolution form of the Galerkin projection of the collision operator. Efficiency of the algorithm is achieved by applying singular value decomposition compression of the discrete collision kernel and by approximating the kinetic solution by a sum of Maxwellian streams using a stochastic likelihood maximization algorithm. Accuracy of the method is established on solutions to the problem of spatially homogeneous relaxation.

Citation: Alexander Alekseenko, Truong Nguyen, Aihua Wood. A deterministic-stochastic method for computing the Boltzmann collision integral in $\mathcal{O}(MN)$ operations. Kinetic & Related Models, 2018, 11 (5) : 1211-1234. doi: 10.3934/krm.2018047
References:
[1]

A. Alekseenko and C. Euler, A Bhatnagar-Gross-Krook kinetic model with velocity-dependent collision frequency and corrected relaxation of moments, Continuum Mechanics and Thermodynamics, 28 (2016), 751-763. doi: 10.1007/s00161-014-0407-0. Google Scholar

[2]

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, 8pp.Google Scholar

[3]

A. Alekseenko and E. Josyula, Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space, emphJournal 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

[4]

L. Andallah and H. Babovsky, A discrete Boltzmann equation based on a cub-octahedron in $\mathbb{R}^3$, SIAM Journal on Scientific Computing, 31 (2009), 799-825. doi: 10.1137/060673850. Google Scholar

[5]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide, 3rd edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999.Google Scholar

[6]

V. 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

[7]

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

[8]

H. Babovsky, Kinetic models on orthogonal groups and the simulation of the Boltzmann equation, AIP Conference Proceedings, 1084 (2008), 415–420, URL http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.3076513.Google Scholar

[9]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525. Google Scholar

[10]

G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Engineering Science Series, Oxford University Press, New York, USA, 1995. Google Scholar

[11]

C. M. Bishop, Neural Networks for Pattern Recognition, Advanced Texts in Econometrics, Clarendon Press, 1995. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. M. BurtE. Josyula and I. D. Boyd, Novel Cartesian implementation of the direct simulation Monte Carlo method, Journal of Thermophysics and Heat Transfer, 26 (2012), 258-270. Google Scholar

[16]

L. Devroye, General principles in random variate generation, in Non-Uniform Random Variate Generation, Springer New York, 1986, 27–82. doi: 10.1007/978-1-4613-8643-8_2. Google Scholar

[17]

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

[18]

I. D. Dinov, Expectation maximization and mixture modeling tutorial, in Statistics Online Computational Resource, UCLA: Statistics Online Computational Resource, 2008, URL http://escholarship.org/uc/item/1rb70972.Google Scholar

[19]

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

[20]

F. FilbetC. 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

[21]

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

[22]

E. FonnP. Grohs and R. Hiptmair, Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation, IMA Journal of Numerical Analysis, 35 (2015), 1533-1567. doi: 10.1093/imanum/dru042. Google Scholar

[23]

R. O. Fox and P. Vedula, Quadrature-based moment model for moderately dense polydisperse gas-particle flows, Industrial and Engineering Chemistry Research, 49 (2010), 5174-5187. doi: 10.1021/ie9013138. Google Scholar

[24]

I. M. Gamba and S. H. Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., 228 (2009), 2012-2036. doi: 10.1016/j.jcp.2008.09.033. Google Scholar

[25]

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

[26]

I. M. Gamba and C. Zhang, A conservative discontinuous Galerkin scheme with $O(n^2)$ 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, 8pp.Google Scholar

[27]

B. I. Green and P. Vedula, Validation of a collisional lattice Boltzmann method, in 20th AIAA Computational Fluid Dynamics Conference, 27-30 June 2011, Honolulu Hawaii, AIP Conference Proceedings, American Institute of Physics, 2011, 14pp.Google Scholar

[28]

L. H. Holway, New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9 (1966), 1658-1673. doi: 10.1063/1.1761920. Google Scholar

[29]

M. Ivanov, A. Kashkovsky, S. Gimelshein, G. Markelov, A. Alexeenko, Y. Bondar, G. Zhukova and S. Nikiforov, SMILE system for 2D/3D DSMC computations, in 25th International Symposium on Rarefied Gas Dynamics, 21–28 July 2006, St. Petersburg, Russia, AIP Conference Proceedings, Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia, 2007, 8pp.Google Scholar

[30]

D. Kalman, A singularly valuable decomposition: The SVD of a matrix, College Math Journal, 27 (1996), 2-23. Google Scholar

[31]

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. doi: 10.1007/s10955-007-9374-1. Google Scholar

[32]

Y. Y. KlossF. G. Tcheremissine and P. V. Shuvalov, Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels, Computational Mathematics and Mathematical Physics, 50 (2010), 1093-1103. doi: 10.1134/S096554251006014X. Google Scholar

[33]

R. Larsen, PROPACK: Computing the singular value decomposition of large and sparse or structured matrices. Computer software, 2005.Google Scholar

[34]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065. doi: 10.1007/BF02179552. Google Scholar

[35]

A. Majorana, A numerical model of the Boltzmann equation related to the discontinuous Galerkin method, Kinetic and Related Models, 4 (2011), 139-151. doi: 10.3934/krm.2011.4.139. Google Scholar

[36]

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

[37]

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

[38]

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

C. R. Schrock and A. W. Wood, Convergence of a distributional Monte Carlo method for the Boltzmann equation, Advances in Applied Mathematics and Mechanics, 4 (2012), 102-121. doi: 10.4208/aamm.10-m11113. Google Scholar

[42]

C. R. Schrock and A. W. Wood, Distributional Monte Carlo solution technique for rarefied gasdynamics, Journal of Thermophysics and Heat Transfer, 26 (2012), 185-189. Google Scholar

[43]

N. Selden, C. Ngalande, N. Gimelshein, S. Gimelshein and A. Ketsdever, Origins of radiometric forces on a circular vane with a temperature gradient, Journal of Fluid Mechanics, 634 (2009), 419–431, URL http://journals.cambridge.org/article_S0022112009007976. doi: 10.1017/S0022112009007976. Google Scholar

[44]

E. M. Shakhov, Approximate kinetic equations in rarefied gas theory, Fluid Dynamics, 3 (1968), 112-115. doi: 10.1007/BF01016254. Google Scholar

[45]

E. M. Shakhov, Generalization of the Krook kinetic relaxation equation, Fluid Dynamics, 3 (1968), 95-96. doi: 10.1007/BF01029546. Google Scholar

[46]

K. Stephani, D. Goldstein and P. Varghese, Generation of a hybrid DSMC/CFD solution for gas mixtures with internal degrees of freedom, in 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Aerospace Sciences Meetings, American Institute of Aeronautics and Astronautics, 2012, p648. doi: 10.2514/6.2012-648. Google Scholar

[47]

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

[48]

S. Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford, 2013, URL https://books.google.com/books?id=OC0Sj_xgnhAC. Google Scholar

[49]

F. G. Tcheremissine, Solution to the Boltzmann kinetic equation for high-speed flows, Computational Mathematics and Mathematical Physics, 46 (2006), 315-329. Google Scholar

[50]

F. G. Tcheremissine, Method for solving the Boltzmann kinetic equation for polyatomic gases, Computational Mathematics and Mathematical Physics, 52 (2012), 252-268. doi: 10.1134/S0965542512020054. Google Scholar

[51]

V. A. Titarev, Efficient deterministic modelling of three-dimensional rarefied gas flows, Communications in Computational Physics, 12 (2012), 162-192. doi: 10.4208/cicp.220111.140711a. Google Scholar

[52]

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. Google Scholar

[53]

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 C. Euler, A Bhatnagar-Gross-Krook kinetic model with velocity-dependent collision frequency and corrected relaxation of moments, Continuum Mechanics and Thermodynamics, 28 (2016), 751-763. doi: 10.1007/s00161-014-0407-0. Google Scholar

[2]

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, 8pp.Google Scholar

[3]

A. Alekseenko and E. Josyula, Deterministic solution of the spatially homogeneous Boltzmann equation using discontinuous Galerkin discretizations in the velocity space, emphJournal 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

[4]

L. Andallah and H. Babovsky, A discrete Boltzmann equation based on a cub-octahedron in $\mathbb{R}^3$, SIAM Journal on Scientific Computing, 31 (2009), 799-825. doi: 10.1137/060673850. Google Scholar

[5]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users' Guide, 3rd edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1999.Google Scholar

[6]

V. 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

[7]

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

[8]

H. Babovsky, Kinetic models on orthogonal groups and the simulation of the Boltzmann equation, AIP Conference Proceedings, 1084 (2008), 415–420, URL http://scitation.aip.org/content/aip/proceeding/aipcp/10.1063/1.3076513.Google Scholar

[9]

P. L. BhatnagarE. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525. Google Scholar

[10]

G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford Engineering Science Series, Oxford University Press, New York, USA, 1995. Google Scholar

[11]

C. M. Bishop, Neural Networks for Pattern Recognition, Advanced Texts in Econometrics, Clarendon Press, 1995. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

J. M. BurtE. Josyula and I. D. Boyd, Novel Cartesian implementation of the direct simulation Monte Carlo method, Journal of Thermophysics and Heat Transfer, 26 (2012), 258-270. Google Scholar

[16]

L. Devroye, General principles in random variate generation, in Non-Uniform Random Variate Generation, Springer New York, 1986, 27–82. doi: 10.1007/978-1-4613-8643-8_2. Google Scholar

[17]

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

[18]

I. D. Dinov, Expectation maximization and mixture modeling tutorial, in Statistics Online Computational Resource, UCLA: Statistics Online Computational Resource, 2008, URL http://escholarship.org/uc/item/1rb70972.Google Scholar

[19]

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

[20]

F. FilbetC. 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

[21]

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

[22]

E. FonnP. Grohs and R. Hiptmair, Hyperbolic cross approximation for the spatially homogeneous Boltzmann equation, IMA Journal of Numerical Analysis, 35 (2015), 1533-1567. doi: 10.1093/imanum/dru042. Google Scholar

[23]

R. O. Fox and P. Vedula, Quadrature-based moment model for moderately dense polydisperse gas-particle flows, Industrial and Engineering Chemistry Research, 49 (2010), 5174-5187. doi: 10.1021/ie9013138. Google Scholar

[24]

I. M. Gamba and S. H. Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., 228 (2009), 2012-2036. doi: 10.1016/j.jcp.2008.09.033. Google Scholar

[25]

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

[26]

I. M. Gamba and C. Zhang, A conservative discontinuous Galerkin scheme with $O(n^2)$ 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, 8pp.Google Scholar

[27]

B. I. Green and P. Vedula, Validation of a collisional lattice Boltzmann method, in 20th AIAA Computational Fluid Dynamics Conference, 27-30 June 2011, Honolulu Hawaii, AIP Conference Proceedings, American Institute of Physics, 2011, 14pp.Google Scholar

[28]

L. H. Holway, New statistical models for kinetic theory: Methods of construction, Phys. Fluids, 9 (1966), 1658-1673. doi: 10.1063/1.1761920. Google Scholar

[29]

M. Ivanov, A. Kashkovsky, S. Gimelshein, G. Markelov, A. Alexeenko, Y. Bondar, G. Zhukova and S. Nikiforov, SMILE system for 2D/3D DSMC computations, in 25th International Symposium on Rarefied Gas Dynamics, 21–28 July 2006, St. Petersburg, Russia, AIP Conference Proceedings, Publishing House of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia, 2007, 8pp.Google Scholar

[30]

D. Kalman, A singularly valuable decomposition: The SVD of a matrix, College Math Journal, 27 (1996), 2-23. Google Scholar

[31]

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. doi: 10.1007/s10955-007-9374-1. Google Scholar

[32]

Y. Y. KlossF. G. Tcheremissine and P. V. Shuvalov, Solution of the Boltzmann equation for unsteady flows with shock waves in narrow channels, Computational Mathematics and Mathematical Physics, 50 (2010), 1093-1103. doi: 10.1134/S096554251006014X. Google Scholar

[33]

R. Larsen, PROPACK: Computing the singular value decomposition of large and sparse or structured matrices. Computer software, 2005.Google Scholar

[34]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065. doi: 10.1007/BF02179552. Google Scholar

[35]

A. Majorana, A numerical model of the Boltzmann equation related to the discontinuous Galerkin method, Kinetic and Related Models, 4 (2011), 139-151. doi: 10.3934/krm.2011.4.139. Google Scholar

[36]

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

[37]

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

[38]

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

C. R. Schrock and A. W. Wood, Convergence of a distributional Monte Carlo method for the Boltzmann equation, Advances in Applied Mathematics and Mechanics, 4 (2012), 102-121. doi: 10.4208/aamm.10-m11113. Google Scholar

[42]

C. R. Schrock and A. W. Wood, Distributional Monte Carlo solution technique for rarefied gasdynamics, Journal of Thermophysics and Heat Transfer, 26 (2012), 185-189. Google Scholar

[43]

N. Selden, C. Ngalande, N. Gimelshein, S. Gimelshein and A. Ketsdever, Origins of radiometric forces on a circular vane with a temperature gradient, Journal of Fluid Mechanics, 634 (2009), 419–431, URL http://journals.cambridge.org/article_S0022112009007976. doi: 10.1017/S0022112009007976. Google Scholar

[44]

E. M. Shakhov, Approximate kinetic equations in rarefied gas theory, Fluid Dynamics, 3 (1968), 112-115. doi: 10.1007/BF01016254. Google Scholar

[45]

E. M. Shakhov, Generalization of the Krook kinetic relaxation equation, Fluid Dynamics, 3 (1968), 95-96. doi: 10.1007/BF01029546. Google Scholar

[46]

K. Stephani, D. Goldstein and P. Varghese, Generation of a hybrid DSMC/CFD solution for gas mixtures with internal degrees of freedom, in 50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, Aerospace Sciences Meetings, American Institute of Aeronautics and Astronautics, 2012, p648. doi: 10.2514/6.2012-648. Google Scholar

[47]

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

[48]

S. Succi, The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Numerical Mathematics and Scientific Computation, Clarendon Press, Oxford, 2013, URL https://books.google.com/books?id=OC0Sj_xgnhAC. Google Scholar

[49]

F. G. Tcheremissine, Solution to the Boltzmann kinetic equation for high-speed flows, Computational Mathematics and Mathematical Physics, 46 (2006), 315-329. Google Scholar

[50]

F. G. Tcheremissine, Method for solving the Boltzmann kinetic equation for polyatomic gases, Computational Mathematics and Mathematical Physics, 52 (2012), 252-268. doi: 10.1134/S0965542512020054. Google Scholar

[51]

V. A. Titarev, Efficient deterministic modelling of three-dimensional rarefied gas flows, Communications in Computational Physics, 12 (2012), 162-192. doi: 10.4208/cicp.220111.140711a. Google Scholar

[52]

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. Google Scholar

[53]

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

Figure 1.  Magnitudes of eigenvalues of $A_{\alpha\beta}$ for the cases $s_u = s_v = s_w = 1$, $N = 15^3$ (a) and $s = 1$, $N = 33^3$ (b)
Figure 2.  Directional temperatures in the solutions to the problem of spatially homogeneous relaxation for the initial data corresponding to Mach 1.55 (a), (b) and Mach 3.0 (c), (d) shock waves. The solutions shown in Figures (a) and (c) are obtained by using the truncated kernel method for evaluating the collision integral. The solid line corresponds to the solution using the full Boltzmann integral. The new model looses accuracy at about three mean free times for both Mach 1.55 and Mach 3.0 solutions. The solutions shown in Figures (b) and (d) are obtained by using the truncated kernel model for about three mean free times and by using the ellipsoidal-statistical Bhatnagar-Gross-Krook model after
Table 1.  Convergence of errors in the approximation of $\mathcal{A}(f, f)$ by $\hat{\mathcal{A}}_M(f, f)$ for the case $N = 15^3$ with respect to the number $M$ of retained eigenvectors: $\vert\lambda_{M}\vert$ is the magnitude of the $M^{\mathrm{th}}$ eigenvalue; $\varepsilon_1$ is the absolute error between $\mathcal{A}(f, f)$ and $\hat{\mathcal{A}}_M(f, f)$ when $f(t, \vec{x}, \vec{v})$ is a single Maxwellian stream with density $n = 1$, bulk velocity $\bar{\vec{u}} = (0, 0, 0)$ and temperature $T = 0.59$; $\varepsilon_2$ is the relative error between $\mathcal{A}(f, f)$ and $\hat{\mathcal{A}}_M(f, f)$ when $f(t, \vec{x}, \vec{v})$ is a sum of two Maxwellian streams with densities $n_1 = n_2 = 1$, bulk velocities $\bar{\vec{u}}_1 = (0.4, 0, 0)$ and $\bar{\vec{u}}_2 = (-0.4, 0, 0)$ and temperatures $T_1 = T_2 = 0.89$; $\varepsilon_3$ is the relative error between $\mathcal{A}(f, f)$ and $\hat{\mathcal{A}}_M(f, f)$ when $f(t, \vec{x}, \vec{v})$ is a sum of two Maxwellian streams with densities $n_1 = n_2 = 1$, bulk velocities $\bar{\vec{u}}_1 = (0.4, 0, 0)$ and $\bar{\vec{u}}_2 = (-0.4, 0, 0)$ and temperatures $T_1 = T_2 = 0.31$
$M$$\vert\lambda_{M}\vert$$\varepsilon_{1}$$\varepsilon_{2}$$\varepsilon_{3}$
104.1E+01.7E-13.1E-12.1E-1
801.0E+03.7E-25.8E-15.3E-2
1605.7E-11.6E-22.9E-22.0E-2
3203.6E-11.5E-22.5E-21.8E-2
6402.2E-13.3E-38.4E-34.4E-3
9601.1E-11.3E-35.7E-32.2E-3
11205.8E-22.3E-35.9E-32.3E-3
12804.1E-31.7E-52.1E-33.1E-5
14004.6E-52.2E-72.0E-33.9E-7
$M$$\vert\lambda_{M}\vert$$\varepsilon_{1}$$\varepsilon_{2}$$\varepsilon_{3}$
104.1E+01.7E-13.1E-12.1E-1
801.0E+03.7E-25.8E-15.3E-2
1605.7E-11.6E-22.9E-22.0E-2
3203.6E-11.5E-22.5E-21.8E-2
6402.2E-13.3E-38.4E-34.4E-3
9601.1E-11.3E-35.7E-32.2E-3
11205.8E-22.3E-35.9E-32.3E-3
12804.1E-31.7E-52.1E-33.1E-5
14004.6E-52.2E-72.0E-33.9E-7
Table 2.  Convergence of errors in the approximation of $\mathcal{A}(f,f)$ by $\hat{\mathcal{A}}_M(f,f)$ for the case $N=33^3$ with respect to the number $M$ of retained eigenvectors: $\vert\lambda_{M}\vert$ is the magnitude of the $M^{\mathrm{th}}$ eigenvalue; $\varepsilon_1$ is the absolute error between $\mathcal{A}(f,f)$ and $\hat{\mathcal{A}}_M(f,f)$ when $f(t,\vec{x},\vec{v})$ is a single Maxwellian stream with density $n=1$, bulk velocity $\bar{\vec{u}}=(0,0,0)$ and temperature $T=0.89$. $\varepsilon_2$ is the relative error between values of $\mathcal{A}(f,f)$ and values $\hat{\mathcal{A}}_M(f,f)$ when $f(t,\vec{x},\vec{v})$ is a sum of two Maxwellian streams with densities $n_1=n_2=1$, bulk velocities $\bar{\vec{u}}_1=(0.\overline{36},0,0)$ and $\bar{\vec{u}}_2=(-0.\overline{36},0,0)$ and temperatures $T_1=T_2=0.89$. The quantity $\varepsilon_3$ is the relative error between values of $\mathcal{A}(f,f)$ and values $\hat{\mathcal{A}}_M(f,f)$ when $f(t,\vec{x},\vec{v})$ is a sum of two Maxwellian streams with densities $n_1=n_2=1$, bulk velocities $\bar{\vec{u}}_1=(0.\overline{36},0,0)$ and $\bar{\vec{u}}_2=(-0.\overline{36},0,0)$ and temperatures $T_1=T_2=0.31$
$M$ $\vert\lambda_{M}\vert$ $\varepsilon_1$ $\varepsilon_2$ $\varepsilon_3$
105.0E+01.0E-15.2E-13.7E-1
1001.7E+04.1E-22.0E-13.5E-1
5005.4E-15.0E-32.3E-22.5E-2
10002.9E-13.3E-31.6E-21.5E-2
20001.7E-11.6E-37.9E-34.7E-3
29001.3E-11.5E-37.5E-33.9E-3
48009.6E-29.2E-44.6E-32.3E-3
99004.3E-21.6E-43.5E-35.3E-4
144002.8E-122.8E-53.5E-39.2E-7
$M$ $\vert\lambda_{M}\vert$ $\varepsilon_1$ $\varepsilon_2$ $\varepsilon_3$
105.0E+01.0E-15.2E-13.7E-1
1001.7E+04.1E-22.0E-13.5E-1
5005.4E-15.0E-32.3E-22.5E-2
10002.9E-13.3E-31.6E-21.5E-2
20001.7E-11.6E-37.9E-34.7E-3
29001.3E-11.5E-37.5E-33.9E-3
48009.6E-29.2E-44.6E-32.3E-3
99004.3E-21.6E-43.5E-35.3E-4
144002.8E-122.8E-53.5E-39.2E-7
Table 3.  CPU time in seconds for computing one time step of the spatially homogeneous solution on a 2.4 GHz processor. Results of simulations for the $N = 15^3$ and $N = 33^3$ grids are presented. $M$ is the number of eigenvectors used in (17). The bottom row gives the time for computing one time step in the method of [3]. Speedup is the ratio of the time used in the new method and the time used by the algorithm of [3]
$n=15$, $s_u=s_v=s_w=1$ $n=33$, $s_u=s_v=s_w=1$
$M$CPU Time, sSpeedup $M$CPU Time, sSpeedup
1602.016.44000444.140.0
3203.88.76000668.1026.6
4805.46.18000890.219.9
6407.04.8100001113.415.9
96010.33.2120001333.613.3
140016.02.1140001559.411.4
DG Boltzmann CPU Time: 33.18 sDG Boltzmann CPU Time: 17758 s
$n=15$, $s_u=s_v=s_w=1$ $n=33$, $s_u=s_v=s_w=1$
$M$CPU Time, sSpeedup $M$CPU Time, sSpeedup
1602.016.44000444.140.0
3203.88.76000668.1026.6
4805.46.18000890.219.9
6407.04.8100001113.415.9
96010.33.2120001333.613.3
140016.02.1140001559.411.4
DG Boltzmann CPU Time: 33.18 sDG Boltzmann CPU Time: 17758 s
Table 4.  Results of approximating the solution to the problem of spatially homogeneous relaxation [3] by a sum of two Maxwellian distributions at different stages of the relaxation process. The values of the time are normalized to the mean free time. The initial data for the problem is a sum of two Maxwellians with dimensionless densities, bulk velocities and temperatures, $n_1 = 1.609$, $\bar{\vec{u}}_1 = (0.775, 0, 0)$, $T_1 = 0.3$ and $n_2 = 2.863$, $\bar{\vec{u}}_2 = (0.436, 0, 0)$, $T_2$ = 0.464, respectively. Columns $n_i$, $T_i$, $\bar{u}_i$, $\bar{v}_i$, and $\bar{w}_i$, $i = 1, 2$ show the density, temperature, and the components of the bulk velocity of the approximating Maxwellians, respectively; $\varepsilon_{L^1}$ is the $L^1$ norm error given by (21)
$t/\tau$ $n_1$ $T_1$ $\bar{u}_{1}$ $\bar{v}_{1}$ $\bar{w}_{1}$ $n_2$ $T_2$ $\bar{u}_{2}$ $\bar{v}_{2}$ $\bar{w}_{2}$ $\varepsilon_{L_1}$
0.01.540.300.781.1E-23.6E-42.940.460.44-7.4E-3-2.9E-38.5E-3
0.041.660.300.771.9E-3-7.4E-32.810.470.43-2.5E-49.7E-48.3E-3
0.453.310.460.485.8E-41.2E-31.160.290.786.3E-34.7E-47.8E-3
0.871.190.300.761.9E-31.0E-23.280.450.483.3E-4-2.0E-35.8E-3
1.71.300.330.696.6E-41.5E-23.170.460.506.6E-4-8.9E-37.8E-3
2.92.930.450.51-3.7E-37.8E-31.540.370.654.5E-3-9.3E-37.1E-3
3.81.290.360.65-1.6E-2-1.2E-23.180.440.529.0E-35.2E-36.9E-3
4.23.530.440.53-4.0E-3-3.2E-30.940.360.661.2E-21.3E-25.9E-3
4.52.210.440.52-1.2E-2-2.7E-22.260.400.601.3E-22.8E-27.0E-3
6.22.340.420.532.6E-2-8.2E-32.130.420.59-2.7E-29.9E-36.1E-3
7.91.580.420.534.4E-2-4.3E-22.890.420.57-2.4E-22.3E-24.6E-3
$t/\tau$ $n_1$ $T_1$ $\bar{u}_{1}$ $\bar{v}_{1}$ $\bar{w}_{1}$ $n_2$ $T_2$ $\bar{u}_{2}$ $\bar{v}_{2}$ $\bar{w}_{2}$ $\varepsilon_{L_1}$
0.01.540.300.781.1E-23.6E-42.940.460.44-7.4E-3-2.9E-38.5E-3
0.041.660.300.771.9E-3-7.4E-32.810.470.43-2.5E-49.7E-48.3E-3
0.453.310.460.485.8E-41.2E-31.160.290.786.3E-34.7E-47.8E-3
0.871.190.300.761.9E-31.0E-23.280.450.483.3E-4-2.0E-35.8E-3
1.71.300.330.696.6E-41.5E-23.170.460.506.6E-4-8.9E-37.8E-3
2.92.930.450.51-3.7E-37.8E-31.540.370.654.5E-3-9.3E-37.1E-3
3.81.290.360.65-1.6E-2-1.2E-23.180.440.529.0E-35.2E-36.9E-3
4.23.530.440.53-4.0E-3-3.2E-30.940.360.661.2E-21.3E-25.9E-3
4.52.210.440.52-1.2E-2-2.7E-22.260.400.601.3E-22.8E-27.0E-3
6.22.340.420.532.6E-2-8.2E-32.130.420.59-2.7E-29.9E-36.1E-3
7.91.580.420.534.4E-2-4.3E-22.890.420.57-2.4E-22.3E-24.6E-3
Table 5.  Results of approximating the solution to the problem of spatially homogeneous relaxation by a sum of three Maxwellian distributions at different stages of the relaxation process. The initial data for the problem is a sum of two Maxwellians with dimensionless densities, bulk velocities and temperatures, $n_1 = 1.609$, $\bar{\vec{u}}_1 = (0.775, 0, 0)$, $T_1 = 0.3$ and $n_2 = 2.863$, $\bar{\vec{u}}_2 = (0.436, 0, 0)$, $T_2$ = 0.464, respectively
$t/\tau$ $n_1$ $T_1$ $\bar{u}_1$ $\bar{v}_1$ $n_2$ $T_2$ $\bar{u}_2$ $\bar{v}_2$ $n_3$ $T_3$ $\bar{u}_3$ $\bar{v}_3$ $\varepsilon_{L_1}$
0.01.350.480.38-2.8E-21.510.440.483.0E-31.610.300.78-5.9E-36.6E-3
.041.970.430.511.0E-21.170.480.38-1.4E-21.330.290.79-9.6E-36.2E-3
.451.860.330.733.6E-30.680.470.45-4.9E-21.930.470.431.3E-27.9E-3
.870.890.460.40-9.0E-22.470.440.533.9E-21.110.310.75-1.1E-28.9E-3
1.70.920.290.735.5E-30.680.440.521.0E-12.870.450.51-2.7E-27.0E-3
2.91.590.420.581.4E-20.920.340.67-1.9E-21.960.450.495.7E-27.0E-3
3.82.010.390.62-3.0E-31.030.430.525.2E-21.430.450.50-3.0E-26.2E-3
4.21.640.430.54-2.5E-21.030.380.64-2.7E-21.800.430.52-6.7E-37.6E-3
4.51.500.450.49-4.1E-20.840.370.65-4.0E-22.130.420.574.2E-25.4E-3
6.21.900.410.56-2.3E-21.810.440.534.3E-20.760.390.63-3.0E-28.0E-3
7.91.340.420.602.9E-42.270.420.57-2.3E-20.840.410.476.1E-25.9E-3
$t/\tau$ $n_1$ $T_1$ $\bar{u}_1$ $\bar{v}_1$ $n_2$ $T_2$ $\bar{u}_2$ $\bar{v}_2$ $n_3$ $T_3$ $\bar{u}_3$ $\bar{v}_3$ $\varepsilon_{L_1}$
0.01.350.480.38-2.8E-21.510.440.483.0E-31.610.300.78-5.9E-36.6E-3
.041.970.430.511.0E-21.170.480.38-1.4E-21.330.290.79-9.6E-36.2E-3
.451.860.330.733.6E-30.680.470.45-4.9E-21.930.470.431.3E-27.9E-3
.870.890.460.40-9.0E-22.470.440.533.9E-21.110.310.75-1.1E-28.9E-3
1.70.920.290.735.5E-30.680.440.521.0E-12.870.450.51-2.7E-27.0E-3
2.91.590.420.581.4E-20.920.340.67-1.9E-21.960.450.495.7E-27.0E-3
3.82.010.390.62-3.0E-31.030.430.525.2E-21.430.450.50-3.0E-26.2E-3
4.21.640.430.54-2.5E-21.030.380.64-2.7E-21.800.430.52-6.7E-37.6E-3
4.51.500.450.49-4.1E-20.840.370.65-4.0E-22.130.420.574.2E-25.4E-3
6.21.900.410.56-2.3E-21.810.440.534.3E-20.760.390.63-3.0E-28.0E-3
7.91.340.420.602.9E-42.270.420.57-2.3E-20.840.410.476.1E-25.9E-3
[1]

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

[2]

Carlota M. Cuesta, Sabine Hittmeir, Christian Schmeiser. Weak shocks of a BGK kinetic model for isentropic gas dynamics. Kinetic & Related Models, 2010, 3 (2) : 255-279. doi: 10.3934/krm.2010.3.255

[3]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[4]

Bernard Ducomet, Alexander Zlotnik. On a regularization of the magnetic gas dynamics system of equations. Kinetic & Related Models, 2013, 6 (3) : 533-543. doi: 10.3934/krm.2013.6.533

[5]

Mirosław Lachowicz, Andrea Quartarone, Tatiana V. Ryabukha. Stability of solutions of kinetic equations corresponding to the replicator dynamics. Kinetic & Related Models, 2014, 7 (1) : 109-119. doi: 10.3934/krm.2014.7.109

[6]

Adriano Festa, Andrea Tosin, Marie-Therese Wolfram. Kinetic description of collision avoidance in pedestrian crowds by sidestepping. Kinetic & Related Models, 2018, 11 (3) : 491-520. doi: 10.3934/krm.2018022

[7]

Manuel Torrilhon. H-Theorem for nonlinear regularized 13-moment equations in kinetic gas theory. Kinetic & Related Models, 2012, 5 (1) : 185-201. doi: 10.3934/krm.2012.5.185

[8]

Yin Yang, Yunqing Huang. Spectral Jacobi-Galerkin methods and iterated methods for Fredholm integral equations of the second kind with weakly singular kernel. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 685-702. doi: 10.3934/dcdss.2019043

[9]

Mehdi Badsi, Martin Campos Pinto, Bruno Després. A minimization formulation of a bi-kinetic sheath. Kinetic & Related Models, 2016, 9 (4) : 621-656. doi: 10.3934/krm.2016010

[10]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817

[11]

Yinhua Xia, Yan Xu, Chi-Wang Shu. Efficient time discretization for local discontinuous Galerkin methods. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 677-693. doi: 10.3934/dcdsb.2007.8.677

[12]

André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135

[13]

Kunquan Lan, Wei Lin. Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1943-1960. doi: 10.3934/dcdsb.2018256

[14]

Sami Injrou, Morgan Pierre. Stable discretizations of the Cahn-Hilliard-Gurtin equations. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 1065-1080. doi: 10.3934/dcds.2008.22.1065

[15]

Marion Acheritogaray, Pierre Degond, Amic Frouvelle, Jian-Guo Liu. Kinetic formulation and global existence for the Hall-Magneto-hydrodynamics system. Kinetic & Related Models, 2011, 4 (4) : 901-918. doi: 10.3934/krm.2011.4.901

[16]

Dmitri Finkelshtein, Yuri Kondratiev, Yuri Kozitsky. Glauber dynamics in continuum: A constructive approach to evolution of states. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1431-1450. doi: 10.3934/dcds.2013.33.1431

[17]

Gabriella Bretti, Ciro D’Apice, Rosanna Manzo, Benedetto Piccoli. A continuum-discrete model for supply chains dynamics. Networks & Heterogeneous Media, 2007, 2 (4) : 661-694. doi: 10.3934/nhm.2007.2.661

[18]

Gilles Pijaudier-Cabot, David Grégoire. A review of non local continuum damage: Modelling of failure?. Networks & Heterogeneous Media, 2014, 9 (4) : 575-597. doi: 10.3934/nhm.2014.9.575

[19]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955

[20]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic & Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (81)
  • HTML views (824)
  • Cited by (0)

Other articles
by authors

[Back to Top]