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Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations
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 |
  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.
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. |
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