Abstract
In this paper, we have applied semi-Lagrangian schemes with meshfree interpolation, based on a moving least squares method, to solve the BGK model for rarefied gas dynamics. Sod’s shock tube problems are presented for a large range of mean free paths in one-dimensional physical space and three-dimensional velocity space. In order to validate the solutions obtained from the meshfree method, we have used the piecewise linear spline interpolation. Furthermore, we have compared the solutions of the BGK model with the solutions obtained from direct simulation Monte Carlo method. In the case of a very small mean free path, the numerical solutions are compared with the exact solutions of the compressible Euler equations. Overall, we found that the meshfree interpolation gives better approximation than the piecewise linear spline interpolation.
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References
Cercignani, C., Illner, R., Pulvirenti, M.: The Mathematical Theory of Dilute Gases. Springer, Berlin (1994)
Bird, G.A.: Molecular Gas Dynamics and Direct Simulation of Gas Flows. Oxford University Press, New York (1994)
Babovsky, H.: A convergence proof for Nanbu’s Boltzmann simulation scheme. Eur. J. Mech. 8, 41 (1989)
Neunzert, H., Struckmeier, J.: Particle methods for the Boltzmann equation. Acta Numer. 4, 417 (1995)
Gad-el Hak, M.: The fluid mechanics of microdevices—the Freeman scholar lecture. ASME J. Fluids Eng. 121(403), 5–33 (1999)
Kerniadakis, G.E., Beskok, A., Aluru, N.R.: Microflows and Nanoflows, Fundamentals and Simulation. Springer, New York (2006)
Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. Phys. Rev. 94, 511 (1954)
Mieussens, L.: Discrete velocity model and implicit scheme for the BGK equation of rarefied gas dynamics. Math. Models Methods Appl. Sci. 10, 1121–1149 (2000)
Russo, G., Filbet, F.: Semi-lagrangian schemes applied to moving boundary problems for the BGK model of rarefied gas dynamics, Kinetic and Related Models. Am. Inst. Math. Sci. 2(1), 231–250 (2009)
Tiwari, S., Kuhnert, J.: Modelling of two-phase flow with surface tension by Finite Point-set method (FPM). J. Comp. Appl. Math. 203, 376–386 (2007)
Tiwari, S., Klar, A., Hardt, S.: A particle−particle hybrid method for kinetic and continuum equations. J. Comp. Phys. 228, 7109–7124 (2009)
Deshpande, S.M., Ramesh, V., Malagi, K., Arora, K.: Least squares kinetic upwind mesh-free method. Def. Sci. J. 60(6), 583–597 (2010)
Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comp. Phys. 27, 1–31 (1978)
Chapman, S., Cowling, T.W.: The Mathematical Theory of Non-Uniform Gases. Cambridge University Press, Cambridge (1970)
Carrillo, J.A., Vecil, F.: Non oscillatory interpolation methods applied to Vlasov-based models. SIAM J. Sci. Comput. 27, 1071–1091 (2005)
Groppi, M., Russo, G., Stracquadanio, G.: High order semi-Lagrangian methods for the BGK equation. Commun. Math. Sci. 14(2), 389–414 (2016)
Holway, L.H.: Kinetic theory of shock structure using an ellipsoidal distribution function, In: Rarefied Gas Dynamics, Proceedings of the Fourth International Symposium, vol. 1, pp. 193–215. University of Toronto, Academic Press, New York (1964)
Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Upper Saddle River (1983)
Boscarino, S., Chen, S.-Y., Russo, G., Yun, S.-B.: High order conservative Semi-Lagrangian scheme for the BGK model of the Boltzmann equation, arXiv:1905.03660 (2019)
Acknowledgements
This work is supported by the DFG (German research foundation) under Grant No. KL 1105/30-1 and by the ITN-ETN Marie-Curie Horizon 2020 program ModCompShock, Modeling and computation of shocks and interfaces, Project ID: 642768.
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Tiwari, S., Klar, A. & Russo, G. A meshfree method for the BGK model for rarefied gas dynamics. Int J Adv Eng Sci Appl Math 11, 187–197 (2019). https://doi.org/10.1007/s12572-019-00254-5
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DOI: https://doi.org/10.1007/s12572-019-00254-5