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Exponential Runge-Kutta Methods for the Multispecies Boltzmann Equation

Published online by Cambridge University Press:  03 June 2015

Qin Li  and
Xu Yang
Qin Li*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
Xu Yang*
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106-3080, USA
*
Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

This paper generalizes the exponential Runge-Kutta asymptotic preserving (AP) method developed in [G. Dimarco and L. Pareschi, SIAM Numer. Anal., 49 (2011), pp. 2057-2077] to compute the multi-species Boltzmann equation. Compared to the single species Boltzmann equation that the method was originally applied on, this set of equation presents a new difficulty that comes from the lack of local conservation laws due to the interaction between different species. Hence extra stiff nonlinear source terms need to be treated properly to maintain the accuracy and the AP property. The method we propose does not contain any nonlinear nonlocal implicit solver, and can capture the hydrodynamic limit with time step and mesh size independent of the Knudsen number. We prove the positivity and strong AP properties of the scheme, which are verified by two numerical examples.

Keywords

65C2065M0676D0582C40Multispecies Boltzmann equationexponential Runge-Kutta methodhydrodynamic limitasymptotic preserving propertypositivity preserving
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 996 - 1011
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Andries, P., Aoki, K., and Perthame, B., A consistent BGK-type model for gas mixtures, J. Stat. Phys., 106 (2002), pp. 993–1018.Google Scholar
[2]Bardos, C., Golse, F., and Sone, Y., Half-space problems for the boltzmann equation: A survey, J. Stat. Phys., 124 (2006), pp. 275–300.Google Scholar
[3]Besse, C., Borghol, S., Goudon, T., Lacroix-Violet, I., and Dudon, J.-P., Hydrodynamic regimes, Knudsen layer, numerical schemes: Definition of boundary fluxes, Adv. Appl. Math. Mech., 3 (2011), pp. 519–561.Google Scholar
[4]Bobylev, A., Grzhibovskis, R., and Heintz, A., Entropy inequalities for evaporation/condensation problem in rarefied gas dynamics, J. Stat. Phys., 102 (2001), pp. 1151–1176.Google Scholar
[5]Bourgat, J., Le Tallec, P., Perthame, B., and Qiu, Y., Coupling Boltzmann and Euler equations without overlapping, in Domain Decomposition Methods in Science and Engineering, Quarteroni, A., Périaux, J., Kuznetsov, Y. A., and Widlund, O., eds., Providence, RI, 1994, AMS, pp. 377–398.CrossRefGoogle Scholar
[6]Caflisch, R., Jin, S., and Russo, G.,Uniformly accurate schemes for hyperbolic systems with relaxations, SIAM J. Numer. Anal., 34 (1997), pp. 246–281.Google Scholar
[7]Coron, F. and Perthame, B., Numerical passage from kinetic to fluid equations, SIAM J. Numer. Anal., 28 (1991), pp. 26–42.Google Scholar
[8]Degond, P. and Jin, S., A smooth transition model between kinetic and diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 2671–2687.CrossRefGoogle Scholar
[9]Degond, P., Jin, S., and Mieussens, L., A smooth transition model between kinetic and hydro-dynamic equations, J. Comput. Phys., 209 (2005), pp. 665–694.CrossRefGoogle Scholar
[10]Dimarco, G. and Pareschi, L., Exponential Runge–Kutta methods for stiff kinetic equations, SIAM J. Numer. Anal., 49 (2011), pp. 2057–2077.Google Scholar
[11]Ferziger, J. H. and Kaper, H. G., Mathematical Theory of Transport Processes in Gases, North-Holland Pub. Co, 1972.Google Scholar
[12]Filbet, F. and Jin, S., A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), pp. 7625–7648.Google Scholar
[13]Filbet, F., Mouhot, C., and Pareschi, L., Solving the Boltzmann equation in N log2N, SIAM J. Sci. Comput., 28 (2006), pp. 1029–1053.CrossRefGoogle Scholar
[14]Filbet, F. and Russo, G., High order numerical methods for the space non homogeneous Boltz-mann equation, J. Comput. Phys, 186 (2003), pp. 457–480.Google Scholar
[15]Gabetta, E., Pareschi, L., and Toscani, G., Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal, 34 (1997), pp. 2168–2194.Google Scholar
[16]Golse, F., Jin, S., and Levermore, C. D., A domain decomposition analysis for a two-scale linear transport problem, Math. Model Num. Anal, 37 (2002), pp. 869–892.Google Scholar
[17]Grad, H., in Rarefied Gas Dynamics, F. Devienne, ed., Pergamon, London, England, 1960, pp. 10–138.Google Scholar
[18]Guenther, M., Struckmeier, J., Le Tallec, P., and Perlat, J., Numerical modeling of gas flows in the transition between rarefied and continuum regimes, Notes on Numerical Fluid Mechanics, 66 (1998), pp. 222–241.Google Scholar
[19]Hamel, B. B., Kinetic model for binary gas mixtures, Physics of Fluids, 8 (1965), pp. 418–425.Google Scholar
[20]Jin, S., Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 122 (1995), pp. 51–67.Google Scholar
[21]Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21 (1999), pp. 441–454.Google Scholar
[22]Jin, S., Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review, Lecture Notes for Summer School on “Methods and Models of Kinetic Theory” (M&MKT), Porto Ercole (Grosseto, Italy), June 2010.Google Scholar
[23]Jin, S. and Li, Q., A BGK-penalization-based asymptotic-preserving scheme for the multi-species Boltzmann equation, Numer. Methods Partial Differential Equations, (2012).Google Scholar
[24]Jin, S. and Shi, Y., A micro-macro decomposition-based asymptotic-preserving scheme for the multispecies Boltzmann equation, SIAM J. Sci. Comput., 31 (2010), pp. 4580–4606.Google Scholar
[25]Jin, S., Yang, X., and Yuan, G., A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface, Kinet. Relat. Models, 1 (2008), pp. 65– 84.Google Scholar
[26]Klar, A., Neunzert, H., and Struckmeier, J., Transition from kinetic theory to macroscopic fluid equations: A problem for domain decomposition anda source for new algorithms, Transport Theory Statist. Phys., 29 (2000), pp. 93–106.Google Scholar
[27]Le, P. Tallec and Mallinger, F., Coupling Boltzmann and Navier-Stokes equations by half fluxes, J. Comput. Phys., 136 (1997), pp. 51–67.Google Scholar
[28]Lemou, M. and Mieussens, L., A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), pp. 334–368.Google Scholar
[29]Li, Q. and Pareschi, L., Exponential Runge-Kutta schemes for inhomogeneous Boltzmann equations with high order of accuracy, arXiv:1208.2622.Google Scholar
[30]Pareschi, L. and Russo, G., Numerical solution of the Boltzmann equation I: Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal, 37 (2000), pp. 1217–1245.Google Scholar
[31]Toscani, G. and Villani, C., Probability metrics and uniqueness of the solution to the Boltz-mann equation for a Maxwell gas, J. Stat. Phys., 94 (1999), pp. 619–637.Google Scholar