June  2012, 5(2): 283-323. doi: 10.3934/krm.2012.5.283

Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension

1. 

Istituto per le Applicazioni del Calcolo (sezione di Bari), via G. Amendola, 122/D, 70126 BARI, Italy

Received  April 2011 Revised  November 2011 Published  April 2012

In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear flow effects follow a scalar integro-differential equation whereas the heat transfer is described by a $2 \times 2$ coupled system. This simplification allows to set up the well-balanced method, involving non-conservative products regularized by solutions of the stationary equations, in order to produce numerical schemes which do stabilize in large times and deliver accurate approximations at numerical steady-state. Boundary-value problems for the stationary equations are solved by the technique of "elementary solutions" at the continuous level and by means of the "analytical discrete ordinates" method at the numerical one. Practically, a comparison with a standard time-splitting method is displayed for a Couette flow by inspecting the shear stress which must be a constant at steady-state. Other test-cases are treated, like heat transfer between two unequally heated walls and also the propagation of a sound disturbance in a gas at rest. Other numerical experiments deal with the behavior of these kinetic models when the Knudsen number becomes small. In particular, a test-case involving a computational domain containing both rarefied and fluid regions characterized by mean free paths of different magnitudes is presented: stabilization onto a physically correct steady-state free from spurious oscillations is observed.
Citation: Laurent Gosse. Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension. Kinetic and Related Models, 2012, 5 (2) : 283-323. doi: 10.3934/krm.2012.5.283
References:
[1]

D. Amadori, L. Gosse and G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws, Arch. Rational Mech. Anal., 162 (2002), 327-366. doi: 10.1007/s002050200198.

[2]

K. Aoki and C. Cercignani, A technique for time-dependent boundary value problems in the kinetic theory of gases. I. Basic analysis, Z. Angew. Math. Phys., 35 (1984), 127-143. doi: 10.1007/BF00947927.

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J. Appell, A. S. Kalitvin and P. P. Zabrejko, Boundary value problems for integro-differential equations of Barbashin type, J. Integral Equ. Applic., 6 (1994), 1-30. doi: 10.1216/jiea/1181075787.

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A. Arnold, J. A. Carrillo and M. D. Tidriri, Large-time behavior of discrete equations with non-symmetric interactions, Math. Mod. Meth. in Appl. Sci., 12 (2002), 1555-1564.

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C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Stat. Phys., 63 (1991), 323-344. doi: 10.1007/BF01026608.

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L. B. Barichello, M. Camargo, P. Rodrigues and C. E. Siewert, Unified solutions to classical flow problems based on the BGK model, Z. Angew. Math. Phys., 52 (2001), 517-534. doi: 10.1007/PL00001559.

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L. B. Barichello and C. E. Siewert, A discrete-ordinates solution for a non-grey model with complete frequency redistribution, JQSRT, 62 (1999), 665-675

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G. R. Bart and R. L. Warnock, Linear integral equations of the third kind, SIAM J. Math. Anal., 4 (1973), 609-622. doi: 10.1137/0504053.

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P. Bassanini, C. Cercignani and C. D. Pagani, Comparison of kinetic theory analyses of linearized heat transfer between parallel plates, Int. J. Heat Mass Transfer, 10 (1967), 447-460. doi: 10.1016/0017-9310(67)90165-2.

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P. Bassanini, C. Cercignani and C. D. Pagani, Influence of the accommodation coefficient on the heat transfer in a rarefied gas, Int. J. Heat Mass Transfer, 11 (1968), 1359-1368. doi: 10.1016/0017-9310(68)90181-6.

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R. Beals and V. Protopopescu, Half-range completeness for the Fokker-Planck equation, J. Stat. Phys., 32 (1983), 565-584. doi: 10.1007/BF01008957.

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M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comp. Phys., 227 (2008), 3781-3803. doi: 10.1016/j.jcp.2007.11.032.

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A. Biryuk, W. Craig and V. Panferov, Strong solutions of the Boltzmann equation in one spatial dimension, C. R. Acad. Sci. Paris, 342 (2006), 843-848. doi: 10.1016/j.crma.2006.04.005.

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C. Cercignani and F. Sernagiotto, The method of elementary solutions for time-dependent problems in linearized kinetic theory, Ann. Physics, 30 (1964), 154-167. doi: 10.1016/0003-4916(64)90308-2.

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Ch. Dalitz, Half-space problem of the Boltzmann equation for charged particles, J. Stat. Phys., 88 (1997), 129-144. doi: 10.1007/BF02508467.

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E. de Groot and Ch. Dalitz, Exact solution for a boundary value problem in semiconductor kinetic theory, J. Math. Phys., 38 (1997), 4629-4643. doi: 10.1063/1.532111.

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L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations, Arch. Rat. Mech. Anal., 110 (1990), 73-91. doi: 10.1007/BF00375163.

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L. Desvillettes and F. Salvarani, Asymptotic behavior of degenerate linear transport equations, Bull. Sci. Math., 133 (2009), 848-858. doi: 10.1016/j.bulsci.2008.09.001.

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A. Frangi, A. Frezzotti and S. Lorenzani, On the application of the BGK kinetic model to the analysis of gas-structure interaction in MEMS, Computers and Structures, 85 (2007), 810-817. doi: 10.1016/j.compstruc.2007.01.011.

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F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comp. Phys., 229 (2010), 7625-7648. doi: 10.1016/j.jcp.2010.06.017.

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L. Gosse, Transient radiative transfer in the grey case: Well-balanced and asymptotic-preserving schemes built on Case's elementary solutions, Journal of Quantitative Spectroscopy & Radiative Transfer, 112 (2011), 1995-2012. doi: 10.1016/j.jqsrt.2011.04.003.

[32]

L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337-342. doi: 10.1016/S1631-073X(02)02257-4.

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L. Gosse and G. Toscani, Space localization and well-balanced scheme for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2003), 641-658. doi: 10.1137/S0036142901399392.

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N. Hadjiconstantinou and A. Garcia, Molecular simulations of sound wave propagation in simple gases, Physics of Fluids, 13 (2001), 1040-1046. doi: 10.1063/1.1352630.

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E. Isaacson and B. Temple, Convergence of the $2 \times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math., 55 (1995), 625-640 doi: 10.1137/S0036139992240711.

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Shi Jin and David Levermore, The discrete-ordinate method in diffusive regimes, Transp. Theor. Stat. Phys., 20 (1991), 413-439. doi: 10.1080/00411459108203913.

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A. Kadir Aziz, D. A. French, S. Jensen and R. B. Kellogg, Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem, M2AN Math. Model. Numer. Anal., 33 (1999), 895-922. doi: 10.1051/m2an:1999125.

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G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases," Interaction of Mechanics and Mathematics, Springer, 2010.

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J. T. Kriese, T. S. Chang and C. E. Siewert, Elementary solutions of coupled model equations in the kinetic theory of gases, Int. J. Eng. Sci., 12 (1974), 441-470. doi: 10.1016/0020-7225(74)90064-0.

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A. V. Latyshev and A. A. Yushkanov, An analytic solution of the problem of temperature and density jumps of a vapor over a surface in the presence of a temperature gradient, J. Applied Math. Mech., 58 (1994), 259-265. doi: 10.1016/0021-8928(94)90054-X.

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T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decomposition and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.

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show all references

References:
[1]

D. Amadori, L. Gosse and G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws, Arch. Rational Mech. Anal., 162 (2002), 327-366. doi: 10.1007/s002050200198.

[2]

K. Aoki and C. Cercignani, A technique for time-dependent boundary value problems in the kinetic theory of gases. I. Basic analysis, Z. Angew. Math. Phys., 35 (1984), 127-143. doi: 10.1007/BF00947927.

[3]

J. Appell, A. S. Kalitvin and P. P. Zabrejko, Boundary value problems for integro-differential equations of Barbashin type, J. Integral Equ. Applic., 6 (1994), 1-30. doi: 10.1216/jiea/1181075787.

[4]

A. Arnold, J. A. Carrillo and M. D. Tidriri, Large-time behavior of discrete equations with non-symmetric interactions, Math. Mod. Meth. in Appl. Sci., 12 (2002), 1555-1564.

[5]

C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations, J. Stat. Phys., 63 (1991), 323-344. doi: 10.1007/BF01026608.

[6]

L. B. Barichello, M. Camargo, P. Rodrigues and C. E. Siewert, Unified solutions to classical flow problems based on the BGK model, Z. Angew. Math. Phys., 52 (2001), 517-534. doi: 10.1007/PL00001559.

[7]

L. B. Barichello and C. E. Siewert, A discrete-ordinates solution for a non-grey model with complete frequency redistribution, JQSRT, 62 (1999), 665-675

[8]

G. R. Bart and R. L. Warnock, Linear integral equations of the third kind, SIAM J. Math. Anal., 4 (1973), 609-622. doi: 10.1137/0504053.

[9]

P. Bassanini, C. Cercignani and C. D. Pagani, Comparison of kinetic theory analyses of linearized heat transfer between parallel plates, Int. J. Heat Mass Transfer, 10 (1967), 447-460. doi: 10.1016/0017-9310(67)90165-2.

[10]

P. Bassanini, C. Cercignani and C. D. Pagani, Influence of the accommodation coefficient on the heat transfer in a rarefied gas, Int. J. Heat Mass Transfer, 11 (1968), 1359-1368. doi: 10.1016/0017-9310(68)90181-6.

[11]

R. Beals, An abstract treatment of some forward-backward problems of transport and scattering, J. Funct. Anal., 34 (1979), 1-20. doi: 10.1016/0022-1236(79)90021-1.

[12]

R. Beals and V. Protopopescu, Half-range completeness for the Fokker-Planck equation, J. Stat. Phys., 32 (1983), 565-584. doi: 10.1007/BF01008957.

[13]

M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics, J. Comp. Phys., 227 (2008), 3781-3803. doi: 10.1016/j.jcp.2007.11.032.

[14]

A. Biryuk, W. Craig and V. Panferov, Strong solutions of the Boltzmann equation in one spatial dimension, C. R. Acad. Sci. Paris, 342 (2006), 843-848. doi: 10.1016/j.crma.2006.04.005.

[15]

Kenneth M. Case, Elementary solutions of the transport equation and their applications, Ann. Physics, 9 (1960), 1-23. doi: 10.1016/0003-4916(60)90060-9.

[16]

K. M. Case and P. F. Zweifel, "Linear Transport Theory," Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967.

[17]

C. Cercignani, Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem, Ann. Physics, 20 (1962), 219-233. doi: 10.1016/0003-4916(62)90199-9.

[18]

C. Cercignani, Plane Couette flow according to the method of elementary solutions, J. Math. Anal. Applic., 11 (1965), 93-101. doi: 10.1016/0022-247X(65)90071-5.

[19]

C. Cercignani, Methods of solution of the linearized Boltzmann equation for rarefied gas dynamics, J. Quant. Spectrosc. Radiat. Transfer, 11 (1971), 973-985. doi: 10.1016/0022-4073(71)90068-9.

[20]

C. Cercignani, Analytic solution of the temperature jump problem for the BGK model, TTSP, 6 (1977), 29-56.

[21]

C. Cercignani, Solution of a linearized kinetic model for an ultrarelativistic gas, J. Stat. Phys., 42 (1986), 601-620. doi: 10.1007/BF01127731.

[22]

C. Cercignani, "Mathematical Methods in Kinetic Theory," Plenum Press, New York, 1969.

[23]

C. Cercignani, "Slow Rarefied Flows. Theory and Application to Micro-Electro-Mechanical Systems," Progress in Mathematical Physics, 41, Birkhäuser Verlag, Basel, 2006.

[24]

C. Cercignani and F. Sernagiotto, The method of elementary solutions for time-dependent problems in linearized kinetic theory, Ann. Physics, 30 (1964), 154-167. doi: 10.1016/0003-4916(64)90308-2.

[25]

Ch. Dalitz, Half-space problem of the Boltzmann equation for charged particles, J. Stat. Phys., 88 (1997), 129-144. doi: 10.1007/BF02508467.

[26]

E. de Groot and Ch. Dalitz, Exact solution for a boundary value problem in semiconductor kinetic theory, J. Math. Phys., 38 (1997), 4629-4643. doi: 10.1063/1.532111.

[27]

L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations, Arch. Rat. Mech. Anal., 110 (1990), 73-91. doi: 10.1007/BF00375163.

[28]

L. Desvillettes and F. Salvarani, Asymptotic behavior of degenerate linear transport equations, Bull. Sci. Math., 133 (2009), 848-858. doi: 10.1016/j.bulsci.2008.09.001.

[29]

A. Frangi, A. Frezzotti and S. Lorenzani, On the application of the BGK kinetic model to the analysis of gas-structure interaction in MEMS, Computers and Structures, 85 (2007), 810-817. doi: 10.1016/j.compstruc.2007.01.011.

[30]

F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comp. Phys., 229 (2010), 7625-7648. doi: 10.1016/j.jcp.2010.06.017.

[31]

L. Gosse, Transient radiative transfer in the grey case: Well-balanced and asymptotic-preserving schemes built on Case's elementary solutions, Journal of Quantitative Spectroscopy & Radiative Transfer, 112 (2011), 1995-2012. doi: 10.1016/j.jqsrt.2011.04.003.

[32]

L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334 (2002), 337-342. doi: 10.1016/S1631-073X(02)02257-4.

[33]

L. Gosse and G. Toscani, Space localization and well-balanced scheme for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2003), 641-658. doi: 10.1137/S0036142901399392.

[34]

J. Greenberg and A. Y. Leroux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16. doi: 10.1137/0733001.

[35]

W. Greenberg, C. V. M. van der Meea and P. F. Zweifel, Generalized kinetic equations, Integr. Equa. Oper. Theory, 7 (1984), 60-95. doi: 10.1007/BF01204914.

[36]

N. Hadjiconstantinou and A. Garcia, Molecular simulations of sound wave propagation in simple gases, Physics of Fluids, 13 (2001), 1040-1046. doi: 10.1063/1.1352630.

[37]

E. Isaacson and B. Temple, Convergence of the $2 \times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math., 55 (1995), 625-640 doi: 10.1137/S0036139992240711.

[38]

Shi Jin and David Levermore, The discrete-ordinate method in diffusive regimes, Transp. Theor. Stat. Phys., 20 (1991), 413-439. doi: 10.1080/00411459108203913.

[39]

A. Kadir Aziz, D. A. French, S. Jensen and R. B. Kellogg, Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem, M2AN Math. Model. Numer. Anal., 33 (1999), 895-922. doi: 10.1051/m2an:1999125.

[40]

H. Kaper, A constructive approach to the solution of a class of boundary value problems of mixed type, J. Math. Anal. Applic., 63 (1978), 691-718. doi: 10.1016/0022-247X(78)90066-5.

[41]

H. Kaper, Boundary value problems of mixed type arising in the kinetic theory of gases, SIAM J. Math. Anal., 10 (1979), 161-178. doi: 10.1137/0510017.

[42]

H. Kaper, Spectral representation of an unbounded linear transformation arising in the kinetic theory of gases, SIAM J. Math. Anal., 10 (1979), 179-191. doi: 10.1137/0510018.

[43]

Tomaž Klinc, On completeness of eigenfunctions of the one-speed transport equation, Commun. Math. Phys., 41 (1975), 273-279. doi: 10.1007/BF01608991.

[44]

G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases," Interaction of Mechanics and Mathematics, Springer, 2010.

[45]

J. T. Kriese, T. S. Chang and C. E. Siewert, Elementary solutions of coupled model equations in the kinetic theory of gases, Int. J. Eng. Sci., 12 (1974), 441-470. doi: 10.1016/0020-7225(74)90064-0.

[46]

A. V. Latyshev, The use of Case's method to solve the linearized BGK equations for the temperature-jump problem, J. Appl. Math. Mech., 54 (1990), 480-484. doi: 10.1016/0021-8928(90)90059-J.

[47]

A. V. Latyshev and A. A. Yushkanov, An analytic solution of the problem of temperature and density jumps of a vapor over a surface in the presence of a temperature gradient, J. Applied Math. Mech., 58 (1994), 259-265. doi: 10.1016/0021-8928(94)90054-X.

[48]

Ph. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of nonconservative products, SIAM J. Math. Anal., 30 (1999), 1309-1342. doi: 10.1137/S0036141098341794.

[49]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation, Physica D, 188 (2004), 178-192. doi: 10.1016/j.physd.2003.07.011.

[50]

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