# American Institute of Mathematical Sciences

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

## Convergence rate for the method of moments with linear closure relations

 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.
Citation: Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic and 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. [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. [3] A. V. Bobylev, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262 (1982), 71-75. [4] F. Brini, Hyperbolicity region in extended thermodynamics with 14 moments, Continuum Mech. Thermodyn., 13 (2001), 1-8. doi: 10.1007/s001610100036. [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. [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. [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. [8] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1. [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. [10] H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403. [11] P. Le Tallec and J. P. Perlat, Numerical Analysis of Levermore's Moment System, Rapport de recherche 3124, INRIA Rocquencourt, March 1997. [12] C. D. Levermore, Moment closure hierarchy for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. [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. [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. [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. [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. [17] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Springer, 2005. [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. [19] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, 3rd edition, Springer, 2009. doi: 10.1007/b79761. [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. [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. [22] D. Vernon Widder, The Laplace Transform, Princeton University Press, 1941. [23] G. M. Wing, An Introduction to Transport Theory, New-York, Wiley, 1962.

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##### 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. [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. [3] A. V. Bobylev, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Dokl. Akad. Nauk SSSR, 262 (1982), 71-75. [4] F. Brini, Hyperbolicity region in extended thermodynamics with 14 moments, Continuum Mech. Thermodyn., 13 (2001), 1-8. doi: 10.1007/s001610100036. [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. [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. [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. [8] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), 1-89. doi: 10.1103/RevModPhys.15.1. [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. [10] H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403. [11] P. Le Tallec and J. P. Perlat, Numerical Analysis of Levermore's Moment System, Rapport de recherche 3124, INRIA Rocquencourt, March 1997. [12] C. D. Levermore, Moment closure hierarchy for kinetic theories, J. Statist. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552. [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. [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. [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. [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. [17] H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows: Approximation Methods in Kinetic Theory, Springer, 2005. [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. [19] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, A Practical Introduction, 3rd edition, Springer, 2009. doi: 10.1007/b79761. [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. [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. [22] D. Vernon Widder, The Laplace Transform, Princeton University Press, 1941. [23] G. M. Wing, An Introduction to Transport Theory, New-York, Wiley, 1962.
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