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Convergence of A Distributional Monte Carlo Method for the Boltzmann Equation

Published online by Cambridge University Press:  03 June 2015

Christopher R. Schrock  and
Aihua W. Wood
Christopher R. Schrock*
Affiliation:
Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA
Aihua W. Wood*
Affiliation:
Department of Mathematics and Statistics, Air Force Institute of Technology, Wright-Patterson AFB, OH 45433, USA
*
Email: [email protected]
Corresponding author. Email: [email protected]
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Abstract

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and L∞ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.

Keywords

82B4076P0565C3582C80Direct simulation monte carlorarefied gas dynamicsBoltzmann equationconvergence proof
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 1 , February 2012 , pp. 102 - 121
Copyright
Copyright © Global-Science Press 2012

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References

[1] Al-Mohssen, H. A. and Hadjiconstantinou, N. G., Yet another variance reduction method for direct monte carlo simulations of low-signal flows, Proceedings of the 26th International Symposium on Rarefied Gas Dynamics, 1084 (2008), pp. 257–262.CrossRefGoogle Scholar
[2] H. A., Al-Mohssen and Hadjiconstantinou, N. G., Low-variance monte carlo simulations using importance weights, ESAIM: Math. Model. Numer. Anal., 44(5) (2010), pp. 1069–1083.Google Scholar
[3] Al-Mohssen, H. A. and Hadjiconstantinou, N. G., A practical variance reduced dsmc method, Proceedings of the 27th International Symposium on Rarefied Gas Dynamics, 2010.Google Scholar
[4] Aristov, V. V., Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows, Springer-Verlag, 2002.Google Scholar
[5] Babovsky, H., A convergence proof for nanbu’s Boltzmann simulation scheme, Euro. J. Mech. B. Fluids., 8(1) (1989), pp. 41–55.Google Scholar
[6] Babovsky, H. and Illner, R., A convergence proof for Nanbu’s simulation method for the full Boltzmann equation, SIAM J. Numer. Anal., 26(1) (1989), pp. 45–65.CrossRefGoogle Scholar
[7] Bhatnagar, P. L., Gross, E. P. and Krook, M., A model for collision processes in gases I: small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), pp. 511–525.CrossRefGoogle Scholar
[8] Bird, G. A., Approach to translational equilibrium in a rigid sphere gas, Phys. Fluids., 6 (1963), 1518.CrossRefGoogle Scholar
[9] Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, 1994.CrossRefGoogle Scholar
[10] Cercignani, C., Rarefied Gas Dynamics from Basic Concepts to Actual Calculations, Cambridge University Press, 2000.Google Scholar
[11] Cercignani, C., Illner, R. and Pulvirenti, M., The Mathematical Theory of Dilute Gases, Springer-Verlag, 1994.CrossRefGoogle Scholar
[12] Chigullapalli, S., Venkattraman, A. and Alexeenko, A. A., Modeling of viscous shock tube using ES-BGK model kinetic equations, AIAA Paper, (2009), 2009–1317.Google Scholar
[13] Gamba, I. and Tharkabhushanam, S., Spectral-lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys., 228 (2009), pp. 2012–2036.CrossRefGoogle Scholar
[14] Homolle, T. M. and Hadjiconstantinou, N. G., Low-variance deviational simulation monte carlo, Phys. Fluids., 19 (2007), 041701.CrossRefGoogle Scholar
[15] Landon, C. D. and Hadjiconstantinou, N. G., Variance-reduced direct simulation monte carlo with the bhatnagar-gross-krook collision operator, Proceedings of the 27th International Symposium on Rarefied Gas Dynamics, 2010.Google Scholar
[16] Mieussens, L., Discrete velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries, J. Comput. Phys., 162(2) (2000), pp. 429–466.CrossRefGoogle Scholar
[17] Nanbu, K., Direct simulation scheme derived from the Boltzmann equation I: monocomponent gases, J. Phys. Soc. Jpn., 49 (1980), pp. 2042–2049.CrossRefGoogle Scholar
[18] Pareschi, L. and Perthame, B., A fourier spectral method for homogeneous Boltzmann equations, Trans. Theory. Stat. Phys., 25 (1996), pp. 369–382.CrossRefGoogle Scholar
[19] Platkowski, T. and Illner, R., Discrete velocity models of the Boltzmann equation: a survey on the mathematical aspects of the theory, SIAM Rev., 30(2) (1988), pp. 213–255.CrossRefGoogle Scholar
[20] Rjasanow, S. and Wagner, W., A stochastic weighted particle method for the Boltzmann equation, J. Comput. Phys., 124 (1996), pp. 243–253.CrossRefGoogle Scholar
[21] Stakgold, I., Green’s Functions and Boundary Value Problems, John Wiley & Sons, 1998.Google Scholar
[22] Wagner, W., A convergence proof for bird’s direct simulation monte carlo method for the Boltz-mann equation, J. Stat. Phys., 66 (1992), pp. 1011–1044.CrossRefGoogle Scholar
[23] Wagner, W., Deviational particle monte carlo for the Boltzmann equation, Monte. Carlo. Methods. Appl., 14(3) (2008), pp. 191–268.CrossRefGoogle Scholar
[24] Wand, M. P. and Jones, M. C., Kernel Smoothing, Chapman & Hall, 1995.CrossRefGoogle Scholar
[25] Xu, K. and Huang, J., A unified gas-kinetic scheme for continuum and rarefied flows, J. Com-put. Phys., 229 (2010), pp. 7747–7764.CrossRefGoogle Scholar