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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

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 & 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.  Google Scholar

[2]

P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-525. Google Scholar

[3]

G. A. Bird, Direct simulation and the Boltzmann equation, Physical Fluids, 13 (1970), 2676-2687. doi: 10.1063/1.1692849.  Google Scholar

[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.  Google Scholar

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H. Brass and K. Petras, Quadrature Theory - The Theory of Numerical Integration on a Compact Interval, American Mathematical Society, 2011.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Cercignani, The Boltzmann Equation and its Application, Springer, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[13]

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

[14]

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer Verlag, 2010.  Google Scholar

[15]

S. Heinz, Statistical Mechanics of Turbulent Flows, Springer, 2003. Google Scholar

[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. Google Scholar

[17]

P. Kauf, Multi-Scale Approximation Models for the Boltzmann Equation, Ph.D thesis, ETH Zürich, 2011. doi: 10.3929/ethz-a-006706585.  Google Scholar

[18]

J. Koellermeier, Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection Methods, Master's thesis, RWTH Aachen University, 2013. Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Englewood Cliffs, 1966.  Google Scholar

[25]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases, Phys. Rev., 94 (1954), 511-525. Google Scholar

[3]

G. A. Bird, Direct simulation and the Boltzmann equation, Physical Fluids, 13 (1970), 2676-2687. doi: 10.1063/1.1692849.  Google Scholar

[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.  Google Scholar

[5]

H. Brass and K. Petras, Quadrature Theory - The Theory of Numerical Integration on a Compact Interval, American Mathematical Society, 2011.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Cercignani, The Boltzmann Equation and its Application, Springer, 1988. doi: 10.1007/978-1-4612-1039-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[14]

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer Verlag, 2010.  Google Scholar

[15]

S. Heinz, Statistical Mechanics of Turbulent Flows, Springer, 2003. Google Scholar

[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. Google Scholar

[17]

P. Kauf, Multi-Scale Approximation Models for the Boltzmann Equation, Ph.D thesis, ETH Zürich, 2011. doi: 10.3929/ethz-a-006706585.  Google Scholar

[18]

J. Koellermeier, Hyperbolic Approximation of Kinetic Equations Using Quadrature-Based Projection Methods, Master's thesis, RWTH Aachen University, 2013. Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Englewood Cliffs, 1966.  Google Scholar

[25]

H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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