American Institute of Mathematical Sciences

Advanced
June  2012, 5(2): 417-440. doi: 10.3934/krm.2012.5.417

Unique moment set from the order of magnitude method

Henning Struchtrup 1,

1. 

Department of Mechanical Engineering, University of Victoria, Victoria BC V8W 3P6, Canada

Received  November 2011 Revised  February 2012 Published  April 2012

The order of magnitude method [Struchtrup, Phys. Fluids 16, 3921-3934 (2004)] is used to construct a unique moment set for 1-D transport with scattering. Simply speaking, the method uses a series of leading order Chapman-Enskog expansions in the Knudsen number to construct the moments such that the number of moments at a given Chapman-Enskog order is minimal. For isotropic scattering, when one begins with monomials for the moments, the method constructs step by step moments of the Legendre polynomials. For anisotropic scattering, however, it constructs moments of new polynomials relevant for the particular scattering mechanism. All terms in the final moment equations are scaled by powers of the Knudsen number, which gives an easy handle to model reduction.
Keywords: order of magnitude method., moment method, Knudsen number approximation, Chapman-Enskog method, Kinetic theory of gases.
Mathematics Subject Classification: Primary: 76P05, 82B40; Secondary: 53C3.
Citation: Henning Struchtrup. Unique moment set from the order of magnitude method. Kinetic and Related Models, 2012, 5 (2) : 417-440. doi: 10.3934/krm.2012.5.417
References:
[1]

A. V. Bobylëv, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27 (1982), 29-31.

[2]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases,'' Third edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.

[3]

H. Grad, Principles of the Kinetic Theory of Gases, in "1958 Handbuch der Physik" (ed. S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205-294.

[4]

P. Kauf, M. Torrilhon and M. Junk, Scale-induced closure for approximations of kinetic equations, J. Stat. Phys., 141 (2010), 848-888. doi: 10.1007/s10955-010-0073-y.

[5]

M. Frank and B. Seibold, Optimal prediction for radiative transfer: A new perspective on moment closure, Kinetic and Related Models, 4 (2011), 717-733. doi: 10.3934/krm.2011.4.717.

[6]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002.

[7]

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.

[8]

H. Struchtrup, Stable transport equations for rarefied gases at high orders in the Knudsen number, Phys. Fluids, 16 (2004), 3921-3934. doi: 10.1063/1.1782751.

[9]

H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials, Multiscale Model. Simul., 3 (2004/05), 211-243.

[10]

H. Struchtrup, Failures of the Burnett and super-Burnett equations in steady state processes, Cont. Mech. Thermodyn., 17 (2005), 43-50. doi: 10.1007/s00161-004-0186-0.

[11]

H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory,'' Interaction of Mechanics and Mathematics, Springer, Berlin, 2005.

[12]

H. Struchtrup, Linear kinetic heat transfer: Moment equations, boundary conditions, and Knudsen layers, Physica A, 387 (2008), 1750-1766. doi: 10.1016/j.physa.2007.11.044.

[13]

H. Struchtrup and P. Taheri, Macroscopic transport models for rarefied gas flows: A brief review, IMA J. Appl. Math., 76 (2011), 672-697. doi: 10.1093/imamat/hxr004.

[14]

M. Schäfer, M. Frank and C. D. Levermore, Diffusive correction to $P_N-$ approximations, Multiscale. Model. Simul., 9 (2011), 1-28. doi: 10.1137/090764542.

[15]

Y. Zheng and H. Struchtrup, Burnett equations for the ellipsoidal statistical BGK Model, Cont. Mech. Thermodyn., 16 (2004), 97-108. doi: 10.1007/s00161-003-0143-3.

show all references

References:
[1]

A. V. Bobylëv, The Chapman-Enskog and Grad methods for solving the Boltzmann equation, Sov. Phys. Dokl., 27 (1982), 29-31.

[2]

S. Chapman and T. G. Cowling, "The Mathematical Theory of Non-Uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases,'' Third edition, prepared in co-operation with D. Burnett, Cambridge University Press, London, 1970.

[3]

H. Grad, Principles of the Kinetic Theory of Gases, in "1958 Handbuch der Physik" (ed. S. Flügge), Bd. 12, Thermodynamik der Gase, Springer-Verlag, Berlin-Göttingen-Heidelberg, (1958), 205-294.

[4]

P. Kauf, M. Torrilhon and M. Junk, Scale-induced closure for approximations of kinetic equations, J. Stat. Phys., 141 (2010), 848-888. doi: 10.1007/s10955-010-0073-y.

[5]

M. Frank and B. Seibold, Optimal prediction for radiative transfer: A new perspective on moment closure, Kinetic and Related Models, 4 (2011), 717-733. doi: 10.3934/krm.2011.4.717.

[6]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'' Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002.

[7]

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.

[8]

H. Struchtrup, Stable transport equations for rarefied gases at high orders in the Knudsen number, Phys. Fluids, 16 (2004), 3921-3934. doi: 10.1063/1.1782751.

[9]

H. Struchtrup, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials, Multiscale Model. Simul., 3 (2004/05), 211-243.

[10]

H. Struchtrup, Failures of the Burnett and super-Burnett equations in steady state processes, Cont. Mech. Thermodyn., 17 (2005), 43-50. doi: 10.1007/s00161-004-0186-0.

[11]

H. Struchtrup, "Macroscopic Transport Equations for Rarefied Gas Flows. Approximation Methods in Kinetic Theory,'' Interaction of Mechanics and Mathematics, Springer, Berlin, 2005.

[12]

H. Struchtrup, Linear kinetic heat transfer: Moment equations, boundary conditions, and Knudsen layers, Physica A, 387 (2008), 1750-1766. doi: 10.1016/j.physa.2007.11.044.

[13]

H. Struchtrup and P. Taheri, Macroscopic transport models for rarefied gas flows: A brief review, IMA J. Appl. Math., 76 (2011), 672-697. doi: 10.1093/imamat/hxr004.

[14]

M. Schäfer, M. Frank and C. D. Levermore, Diffusive correction to $P_N-$ approximations, Multiscale. Model. Simul., 9 (2011), 1-28. doi: 10.1137/090764542.

[15]

Y. Zheng and H. Struchtrup, Burnett equations for the ellipsoidal statistical BGK Model, Cont. Mech. Thermodyn., 16 (2004), 97-108. doi: 10.1007/s00161-003-0143-3.

[1]

Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic and Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033
[2]

Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic and Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79
[3]

Pierre Degond, Amic Frouvelle, Jian-Guo Liu. From kinetic to fluid models of liquid crystals by the moment method. Kinetic and Related Models, 2022, 15 (3) : 417-465. doi: 10.3934/krm.2021047
[4]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[5]

Sho Matsumoto, Jonathan Novak. A moment method for invariant ensembles. Electronic Research Announcements, 2018, 25: 60-71. doi: 10.3934/era.2018.25.007
[6]

Vincent Giovangigli, Wen-An Yong. Erratum: ``Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion''. Kinetic and Related Models, 2016, 9 (4) : 813-813. doi: 10.3934/krm.2016018
[7]

Gilberto M. Kremer, Wilson Marques Jr.. Fourteen moment theory for granular gases. Kinetic and Related Models, 2011, 4 (1) : 317-331. doi: 10.3934/krm.2011.4.317
[8]

Pierre Lissy. Construction of gevrey functions with compact support using the bray-mandelbrojt iterative process and applications to the moment method in control theory. Mathematical Control and Related Fields, 2017, 7 (1) : 21-40. doi: 10.3934/mcrf.2017002
[9]

Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic and Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291
[10]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[11]

Jiantao Jiang, Jing An, Jianwei Zhou. A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022066
[12]

Luciano Pandolfi. Riesz systems and moment method in the study of viscoelasticity in one space dimension. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1487-1510. doi: 10.3934/dcdsb.2010.14.1487
[13]

Shigeru Takata, Hitoshi Funagane, Kazuo Aoki. Fluid modeling for the Knudsen compressor: Case of polyatomic gases. Kinetic and Related Models, 2010, 3 (2) : 353-372. doi: 10.3934/krm.2010.3.353
[14]

Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219
[15]

Chen Li, Fajie Wei, Shenghan Zhou. Prediction method based on optimization theory and its application. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1213-1221. doi: 10.3934/dcdss.2015.8.1213
[16]

Palash Sarkar, Shashank Singh. A unified polynomial selection method for the (tower) number field sieve algorithm. Advances in Mathematics of Communications, 2019, 13 (3) : 435-455. doi: 10.3934/amc.2019028
[17]

Feng Ma, Mingfang Ni. A two-phase method for multidimensional number partitioning problem. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 203-206. doi: 10.3934/naco.2013.3.203
[18]

Sergio Grillo, Leandro Salomone, Marcela Zuccalli. Explicit solutions of the kinetic and potential matching conditions of the energy shaping method. Journal of Geometric Mechanics, 2021, 13 (4) : 629-646. doi: 10.3934/jgm.2021022
[19]

Aleksa Srdanov, Radiša Stefanović, Aleksandra Janković, Dragan Milovanović. "Reducing the number of dimensions of the possible solution space" as a method for finding the exact solution of a system with a large number of unknowns. Mathematical Foundations of Computing, 2019, 2 (2) : 83-93. doi: 10.3934/mfc.2019007
[20]

Xiaojun Chen, Guihua Lin. CVaR-based formulation and approximation method for stochastic variational inequalities. Numerical Algebra, Control and Optimization, 2011, 1 (1) : 35-48. doi: 10.3934/naco.2011.1.35

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy