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

Unique moment set from the order of magnitude method

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.
Citation: Henning Struchtrup. Unique moment set from the order of magnitude method. Kinetic & 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.

[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, (1970).

[3]

H. Grad, Principles of the Kinetic Theory of Gases,, in, (1958), 205.

[4]

P. Kauf, M. Torrilhon and M. Junk, Scale-induced closure for approximations of kinetic equations,, J. Stat. Phys., 141 (2010), 848. 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. doi: 10.3934/krm.2011.4.717.

[6]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'', Modeling and Simulation in Science, (2002).

[7]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis,, Phys. Fluids, 15 (2003), 2668. 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. 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 (): 211.

[10]

H. Struchtrup, Failures of the Burnett and super-Burnett equations in steady state processes,, Cont. Mech. Thermodyn., 17 (2005), 43. 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, (2005).

[12]

H. Struchtrup, Linear kinetic heat transfer: Moment equations, boundary conditions, and Knudsen layers,, Physica A, 387 (2008), 1750. 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. 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. doi: 10.1137/090764542.

[15]

Y. Zheng and H. Struchtrup, Burnett equations for the ellipsoidal statistical BGK Model,, Cont. Mech. Thermodyn., 16 (2004), 97. 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.

[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, (1970).

[3]

H. Grad, Principles of the Kinetic Theory of Gases,, in, (1958), 205.

[4]

P. Kauf, M. Torrilhon and M. Junk, Scale-induced closure for approximations of kinetic equations,, J. Stat. Phys., 141 (2010), 848. 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. doi: 10.3934/krm.2011.4.717.

[6]

Y. Sone, "Kinetic Theory and Fluid Dynamics,'', Modeling and Simulation in Science, (2002).

[7]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis,, Phys. Fluids, 15 (2003), 2668. 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. 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 (): 211.

[10]

H. Struchtrup, Failures of the Burnett and super-Burnett equations in steady state processes,, Cont. Mech. Thermodyn., 17 (2005), 43. 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, (2005).

[12]

H. Struchtrup, Linear kinetic heat transfer: Moment equations, boundary conditions, and Knudsen layers,, Physica A, 387 (2008), 1750. 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. 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. doi: 10.1137/090764542.

[15]

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

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