A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods

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September 2014, 7(3): 531-549. doi: 10.3934/krm.2014.7.531

A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods

Julian Koellermeier ^1, , Roman Pascal Schaerer ^1, and Manuel Torrilhon ^1,

1.	Center for Computational Engineering Science, RWTH Aachen University, Schinkelstr.2, 52062 Aachen, Germany, Germany

Received January 2014 Revised March 2014 Published July 2014

Abstract
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We derive hyperbolic PDE systems for the solution of the Boltzmann Equation. First, the velocity is transformed in a non-linear way to obtain a Lagrangian velocity phase space description that allows for physical adaptivity. The unknown distribution function is then approximated by a series of basis functions.
Standard continuous projection methods for this approach yield PDE systems for the basis coefficients that are in general not hyperbolic. To overcome this problem, we apply quadrature-based projection methods which modify the structure of the system in the desired way so that we end up with a hyperbolic system of equations.
With the help of a new abstract framework, we derive conditions such that the emerging system is hyperbolic and give a proof of hyperbolicity for Hermite ansatz functions in one dimension together with Gauss-Hermite quadrature.

Keywords: Boltzmann equation, quadrature, hyperbolicity., kinetic equation.

Mathematics Subject Classification: Primary: 35Q20; Secondary: 35Q3.

Citation: Julian Koellermeier, Roman Pascal Schaerer, Manuel Torrilhon. A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods. Kinetic and Related Models, 2014, 7 (3) : 531-549. doi: 10.3934/krm.2014.7.531

References:

[1]	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, 1992. doi: 10.1119/1.15378.
[2]	P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-525.
[3]	G. A. Bird, Direct simulation and the Boltzmann equation, Physical Fluids, 13 (1970), 2676-2687. doi: 10.1063/1.1692849.
[4]	G. A. Bird, Monte Carlo simulation in an engineering context, Rarefied Gas Dynamics, 1 (1981), 239-255. doi: 10.2514/5.9781600865480.0239.0255.
[5]	H. Brass and K. Petras, Quadrature Theory - The Theory of Numerical Integration on a Compact Interval, American Mathematical Society, 2011.
[6]	F. Brini, Hyperbolicity region in extended thermodynamics with 14 moments, Contin. Mech. Thermodyn., 13 (2001), 1-8. doi: 10.1007/s001610100036.
[7]	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. doi: 10.4310/CMS.2013.v11.n2.a12.
[8]	Z. Cai, Y. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), 464-518. doi: 10.1002/cpa.21472.
[9]	C. Cercignani, The Boltzmann Equation and its Application, Springer, 1988. doi: 10.1007/978-1-4612-1039-9.
[10]	G.-Q. Chen, Multidimensional conservation laws: Overview, problems, and perspective, IMA Vol. Math. Appl., 153 (2011), 23-72. doi: 10.1007/978-1-4419-9554-4_2.
[11]	R. Duclous, B. Dubroca and M. Frank, A deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843-3857. doi: 10.1088/0031-9155/55/13/018.
[12]	F. Filbet and T. Rey, A rescaling velocity method for dissipative kinetic equations - applications to granular media, J. Comput. Phys., 248 (2013), 177-199. doi: 10.1016/j.jcp.2013.04.023.
[13]	H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.
[14]	E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer Verlag, 2010.
[15]	S. Heinz, Statistical Mechanics of Turbulent Flows, Springer, 2003.
[16]	T. Kataoka, M. Tsutahara, K. Ogawa, Y. Yamamoto, M. Shoji and Y. Sakai, Knudsen pump and its possibility of application to satellite control, Theoretical and Applied Mechanics, 53 (2004), 155-162.
[17]	P. Kauf, Multi-Scale Approximation Models for the Boltzmann Equation, Ph.D thesis, ETH Zürich, 2011. doi: 10.3929/ethz-a-006706585.
[18]	J. Koellermeier, Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection Methods, Master's thesis, RWTH Aachen University, 2013.
[19]	C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.
[20]	G. Metivier, Remarks on the well-posedness of the nonlinear cauchy problem, Contemp. Math., 368 (2005), 337-356. doi: 10.1090/conm/368/06790.
[21]	L. Mieussens, C. Baranger, J. Claudel and N. Hérouard, Locally refined discrete velocity grids for deterministic rarefied flow simulations, Journal of Computational Physics, 257 (2014), 572-593. doi: 10.1016/j.jcp.2013.10.014.
[22]	G. V. Milovanovic and A. S. Cvetkovic, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balkanica, 26 (2012), 169-184.
[23]	X. Shan and X. He, Discretization of the velocity space in the solution of the Boltzmann equation, Phys. Rev. Lett., 80 (1998), 65-68. doi: 10.1103/PhysRevLett.80.65.
[24]	A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Englewood Cliffs, 1966.
[25]	H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.
[26]	H. Struchtrup and M. Torrilhon, H-theorem, regularization, and boundary conditions for linearized 13-moment-equations, Phys. Rev. Lett., 99 (2007), 014502. doi: 10.1103/PhysRevLett.99.014502.
[27]	M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Comm. Comput. Phys., 7 (2010), 639-673. doi: 10.4208/cicp.2009.09.049.
[28]	M. Torrilhon, H-theorem for nonlinear regularized 13-moment equations in kinetic gas theory, Kinet. Relat. Models, 5 (2012), 185-201. doi: 10.3934/krm.2012.5.185.

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References:

[1]	M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York, 1992. doi: 10.1119/1.15378.
[2]	P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-525.
[3]	G. A. Bird, Direct simulation and the Boltzmann equation, Physical Fluids, 13 (1970), 2676-2687. doi: 10.1063/1.1692849.
[4]	G. A. Bird, Monte Carlo simulation in an engineering context, Rarefied Gas Dynamics, 1 (1981), 239-255. doi: 10.2514/5.9781600865480.0239.0255.
[5]	H. Brass and K. Petras, Quadrature Theory - The Theory of Numerical Integration on a Compact Interval, American Mathematical Society, 2011.
[6]	F. Brini, Hyperbolicity region in extended thermodynamics with 14 moments, Contin. Mech. Thermodyn., 13 (2001), 1-8. doi: 10.1007/s001610100036.
[7]	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. doi: 10.4310/CMS.2013.v11.n2.a12.
[8]	Z. Cai, Y. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), 464-518. doi: 10.1002/cpa.21472.
[9]	C. Cercignani, The Boltzmann Equation and its Application, Springer, 1988. doi: 10.1007/978-1-4612-1039-9.
[10]	G.-Q. Chen, Multidimensional conservation laws: Overview, problems, and perspective, IMA Vol. Math. Appl., 153 (2011), 23-72. doi: 10.1007/978-1-4419-9554-4_2.
[11]	R. Duclous, B. Dubroca and M. Frank, A deterministic partial differential equation model for dose calculation in electron radiotherapy, Phys. Med. Biol., 55 (2010), 3843-3857. doi: 10.1088/0031-9155/55/13/018.
[12]	F. Filbet and T. Rey, A rescaling velocity method for dissipative kinetic equations - applications to granular media, J. Comput. Phys., 248 (2013), 177-199. doi: 10.1016/j.jcp.2013.04.023.
[13]	H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407. doi: 10.1002/cpa.3160020403.
[14]	E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer Verlag, 2010.
[15]	S. Heinz, Statistical Mechanics of Turbulent Flows, Springer, 2003.
[16]	T. Kataoka, M. Tsutahara, K. Ogawa, Y. Yamamoto, M. Shoji and Y. Sakai, Knudsen pump and its possibility of application to satellite control, Theoretical and Applied Mechanics, 53 (2004), 155-162.
[17]	P. Kauf, Multi-Scale Approximation Models for the Boltzmann Equation, Ph.D thesis, ETH Zürich, 2011. doi: 10.3929/ethz-a-006706585.
[18]	J. Koellermeier, Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection Methods, Master's thesis, RWTH Aachen University, 2013.
[19]	C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065. doi: 10.1007/BF02179552.
[20]	G. Metivier, Remarks on the well-posedness of the nonlinear cauchy problem, Contemp. Math., 368 (2005), 337-356. doi: 10.1090/conm/368/06790.
[21]	L. Mieussens, C. Baranger, J. Claudel and N. Hérouard, Locally refined discrete velocity grids for deterministic rarefied flow simulations, Journal of Computational Physics, 257 (2014), 572-593. doi: 10.1016/j.jcp.2013.10.014.
[22]	G. V. Milovanovic and A. S. Cvetkovic, Special classes of orthogonal polynomials and corresponding quadratures of Gaussian type, Math. Balkanica, 26 (2012), 169-184.
[23]	X. Shan and X. He, Discretization of the velocity space in the solution of the Boltzmann equation, Phys. Rev. Lett., 80 (1998), 65-68. doi: 10.1103/PhysRevLett.80.65.
[24]	A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Englewood Cliffs, 1966.
[25]	H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.
[26]	H. Struchtrup and M. Torrilhon, H-theorem, regularization, and boundary conditions for linearized 13-moment-equations, Phys. Rev. Lett., 99 (2007), 014502. doi: 10.1103/PhysRevLett.99.014502.
[27]	M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson-IV-distributions, Comm. Comput. Phys., 7 (2010), 639-673. doi: 10.4208/cicp.2009.09.049.
[28]	M. Torrilhon, H-theorem for nonlinear regularized 13-moment equations in kinetic gas theory, Kinet. Relat. Models, 5 (2012), 185-201. doi: 10.3934/krm.2012.5.185.

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