• Previous Article
    One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space
  • KRM Home
  • This Issue
  • Next Article
    Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations
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

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.
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.

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.

[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.

[1]

Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341

[2]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[3]

Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145

[4]

Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic and Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014

[5]

Marco Berardi, Fabio V. Difonzo. A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards' equation. Journal of Computational Dynamics, 2022, 9 (2) : 69-84. doi: 10.3934/jcd.2022001

[6]

Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic and Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039

[7]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic and Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205

[8]

Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic and Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499

[9]

El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401

[10]

Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic and Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237

[11]

Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic and Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551

[12]

Kamel Hamdache, Djamila Hamroun. Macroscopic limit of the kinetic Bloch equation. Kinetic and Related Models, 2021, 14 (3) : 541-570. doi: 10.3934/krm.2021015

[13]

Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic and Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036

[14]

Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic and Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673

[15]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221

[16]

Kevin Zumbrun. L resolvent bounds for steady Boltzmann's Equation. Kinetic and Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048

[17]

Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579

[18]

Nicolas Fournier. A new regularization possibility for the Boltzmann equation with soft potentials. Kinetic and Related Models, 2008, 1 (3) : 405-414. doi: 10.3934/krm.2008.1.405

[19]

Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939

[20]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic and Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (125)
  • HTML views (0)
  • Cited by (17)

[Back to Top]