American Institute of Mathematical Sciences

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March  2015, 8(1): 1-27. doi: 10.3934/krm.2015.8.1

Convergence rate for the method of moments with linear closure relations

Yves Bourgault 1, , Damien Broizat 2, and  Pierre-Emmanuel Jabin 3,

1. 

Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Ave, Ottawa, Ontario, Canada
2. 

Laboratoire J.-A. Dieudonné, Université de Nice - Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
3. 

CSCAMM and Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States

Received  October 2014 Revised  October 2014 Published  December 2014

We study linear closure relations for the moments' method applied to simple kinetic equations. The equations are linear collisional models (velocity jump processes) which are well suited to this type of approximation. In this simplified, 1 dimensional setting, we are able to prove stability estimates for the method (with a kinetic interpretation by a BGK model). Moreover we are also able to obtain convergence rates which automatically increase with the smoothness of the initial data.
Keywords: stability analysis., hyperbolicity, method of moments, Kinetic theory, spectral type convergence, closure relation.
Mathematics Subject Classification: Primary: 35L40, 35Q20, 65M12; Secondary: 65M1.
Citation: Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1
References:
[1]

P. L. Bhatnagar, E. 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. doi: 10.1103/PhysRev.94.511.  Google Scholar

[2]

G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flow, Oxford Engineering Science Series, 42. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.  Google Scholar

[3]

A. V. Bobylev, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262 (1982), 71-75.  Google Scholar

[4]

F. Brini, Hyperbolicity region in extended thermodynamics with 14 moments, Continuum Mech. Thermodyn., 13 (2001), 1-8. doi: 10.1007/s001610100036.  Google Scholar

[5]

Z. Cai, Y. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system in one dimensional space, Commun. Math. Sci., 11 (2013), 547-571. arXiv:1111.3409, 2012. doi: 10.4310/CMS.2013.v11.n2.a12.  Google Scholar

[6]

Z. Cai and R. Li, Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation, SIAM J. Sci. Comput., 32 (2010), 2875-2907. doi: 10.1137/100785466.  Google Scholar

[7]

Z. Cai, R. Li and Y. Wang, An efficient NRxx method for Boltzmann-BGK equation, J. Sci. Comput., 50 (2012), 103-119. doi: 10.1007/s10915-011-9475-5.  Google Scholar

[8]

S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1.  Google Scholar

[9]

L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Archive Rat. Mech. Anal. 123 (1993), 387-404. doi: 10.1007/BF00375586.  Google Scholar

[10]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.  Google Scholar

[11]

P. Le Tallec and J. P. Perlat, Numerical Analysis of Levermore's Moment System, Rapport de recherche 3124, INRIA Rocquencourt, March 1997. Google Scholar

[12]

C. D. Levermore, Moment closure hierarchy for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[13]

C. D. Levermore and W. J. Morokoff, The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1999), 72-96. doi: 10.1137/S0036139996299236.  Google Scholar

[14]

H. G. Othmer and S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. doi: 10.1007/BF00277392.  Google Scholar

[15]

B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property, SIAM J. Numer. Anal., 27 (1990), 1405-1421. doi: 10.1137/0727081.  Google Scholar

[16]

J. Shen and T. Tang, Spectral and High-Order Methods with Applications, volume 3 of Mathematics Monograph Series, Science Press, Beijing, P. R. China, 2006.  Google Scholar

[17]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Springer, 2005.  Google Scholar

[18]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680. doi: 10.1063/1.1597472.  Google Scholar

[19]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, 3rd edition, Springer, 2009. doi: 10.1007/b79761.  Google Scholar

[20]

M. Torrilhon, Two dimensional bulk microflow simulations based on regularized Grad's 13-moment equations, SIAM Multiscale Model. Simul., 5 (2006), 695-728. doi: 10.1137/050635444.  Google Scholar

[21]

M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multivariate Pearson-IV-distributions, Commun. Comput. Phys., 7 (2010), 639-673. doi: 10.4208/cicp.2009.09.049.  Google Scholar

[22]

D. Vernon Widder, The Laplace Transform, Princeton University Press, 1941.  Google Scholar

[23]

G. M. Wing, An Introduction to Transport Theory, New-York, Wiley, 1962.  Google Scholar

show all references

References:
[1]

P. L. Bhatnagar, E. 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. doi: 10.1103/PhysRev.94.511.  Google Scholar

[2]

G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flow, Oxford Engineering Science Series, 42. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.  Google Scholar

[3]

A. V. Bobylev, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262 (1982), 71-75.  Google Scholar

[4]

F. Brini, Hyperbolicity region in extended thermodynamics with 14 moments, Continuum Mech. Thermodyn., 13 (2001), 1-8. doi: 10.1007/s001610100036.  Google Scholar

[5]

Z. Cai, Y. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system in one dimensional space, Commun. Math. Sci., 11 (2013), 547-571. arXiv:1111.3409, 2012. doi: 10.4310/CMS.2013.v11.n2.a12.  Google Scholar

[6]

Z. Cai and R. Li, Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation, SIAM J. Sci. Comput., 32 (2010), 2875-2907. doi: 10.1137/100785466.  Google Scholar

[7]

Z. Cai, R. Li and Y. Wang, An efficient NRxx method for Boltzmann-BGK equation, J. Sci. Comput., 50 (2012), 103-119. doi: 10.1007/s10915-011-9475-5.  Google Scholar

[8]

S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1.  Google Scholar

[9]

L. Desvillettes, Some applications of the method of moments for the homogeneous Boltzmann and Kac equations, Archive Rat. Mech. Anal. 123 (1993), 387-404. doi: 10.1007/BF00375586.  Google Scholar

[10]

H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.  Google Scholar

[11]

P. Le Tallec and J. P. Perlat, Numerical Analysis of Levermore's Moment System, Rapport de recherche 3124, INRIA Rocquencourt, March 1997. Google Scholar

[12]

C. D. Levermore, Moment closure hierarchy for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.  Google Scholar

[13]

C. D. Levermore and W. J. Morokoff, The Gaussian moment closure for gas dynamics, SIAM J. Appl. Math., 59 (1999), 72-96. doi: 10.1137/S0036139996299236.  Google Scholar

[14]

H. G. Othmer and S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. doi: 10.1007/BF00277392.  Google Scholar

[15]

B. Perthame, Boltzmann type schemes for gas dynamics and the entropy property, SIAM J. Numer. Anal., 27 (1990), 1405-1421. doi: 10.1137/0727081.  Google Scholar

[16]

J. Shen and T. Tang, Spectral and High-Order Methods with Applications, volume 3 of Mathematics Monograph Series, Science Press, Beijing, P. R. China, 2006.  Google Scholar

[17]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Springer, 2005.  Google Scholar

[18]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680. doi: 10.1063/1.1597472.  Google Scholar

[19]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, 3rd edition, Springer, 2009. doi: 10.1007/b79761.  Google Scholar

[20]

M. Torrilhon, Two dimensional bulk microflow simulations based on regularized Grad's 13-moment equations, SIAM Multiscale Model. Simul., 5 (2006), 695-728. doi: 10.1137/050635444.  Google Scholar

[21]

M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multivariate Pearson-IV-distributions, Commun. Comput. Phys., 7 (2010), 639-673. doi: 10.4208/cicp.2009.09.049.  Google Scholar

[22]

D. Vernon Widder, The Laplace Transform, Princeton University Press, 1941.  Google Scholar

[23]

G. M. Wing, An Introduction to Transport Theory, New-York, Wiley, 1962.  Google Scholar

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