Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T19:41:46.187Z Has data issue: false hasContentIssue false
  • English
  • Français

A Local Velocity Grid Approach for BGK Equation

Published online by Cambridge University Press:  03 June 2015

Florian Bernard ,
Angelo Iollo  and
Gabriella Puppo
Florian Bernard*
Affiliation:
Department of Mechanical and Aerospace Engineering, Politecnico di Torino, 10129 Torino, Italy Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France. INRIA, F-33400 Talence, France
Angelo Iollo*
Affiliation:
Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France. INRIA, F-33400 Talence, France
Gabriella Puppo*
Affiliation:
Dip. di Scienza ed Alta Tecnologia, Università dell’Insubria, Como, Italy
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

The solution of complex rarefied flows with the BGK equation and the Discrete Velocity Method (DVM) requires a large number of velocity grid points leading to significant computational costs. We propose an adaptive velocity grid approach exploiting the fact that locally in space, the distribution function is supported only by a sub-set of the global velocity grid. The velocity grid is adapted thanks to criteria based on local temperature, velocity and on the enforcement of mass conservation. Simulations in 1D and 2D are presented for different Knudsen numbers and compared to a global velocity grid BGK solution, showing the computational gain of the proposed approach.

Keywords

76P05Kinetic modelsBGK modeldiscrete velocity method
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 4 , October 2014 , pp. 956 - 982
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alaia, A., and Puppo, G.A hybrid method for hydrodynamic-kinetic flow - Part II - Coupling of hydrodynamic and kinetic models. Journal of Computational Physics 231,16 (June 2012), 5217–5242.Google Scholar
[2]Andries, P., Bourgat, J.-F., Le Tallec, P., and Perthame, B.Numerical comparison between the Boltzmann and ES-BGK models for rarefied gases. Computer Methods in Applied Mechanics and Engineering 191,31 (May 2002), 3369–3390.Google Scholar
[3]Andries, P., Le Tallec, P., Perlat, J.-P., and Perthame, B.The Gaussian-BGK model of Boltz-mann equation with small Prandtl number. European Journal of Mechanics. B. Fluids 19, 6 (2000), 813–830.CrossRefGoogle Scholar
[4]Baranger, C., Claudel, J., Hérouard, N., and Mieussens, L. AIP Conference Proceedings. In 28th International Symposium on Rarefied Gas Dynamics 2012, AIP, pp. 389–396.Google Scholar
[5]Bernard, F., Iollo, A., and Puppo G., AccurateAsymptotic Preserving Boundary Conditions for Kinetic Equations on Cartesian Grids. Rapport de recherche RR-8471, INRIA, Feb. 2014.Google Scholar
[6]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. Physics Review 94, (May 1954), 511–525.Google Scholar
[7]Bird, G. A.Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford Engineering Science Series. Clarendon Press, 1994.Google Scholar
[8]Buet, C.A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transport Theory and Statistical Physics 25,1 (1996), 33–60.CrossRefGoogle Scholar
[9]Cercignani, C.The Boltzmann Equation and Its Applications. Springer-Verlag GmbH, 1988.Google Scholar
[10]Chapman, S., and Cowling, T.The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases. Cambridge Mathematical Library. Cambridge University Press, 1970.Google Scholar
[11]Chen, S., Xu, K., Lee, C., and Cai, Q.A unified gas kinetic scheme with moving mesh and velocity space adaptation. Journal of Computational Physics 231, 20 (Aug. 2012), 6643–6664.Google Scholar
[12]Chu, C. K.Kinetic-theoretic description of the formation of a shock wave. Physics of Fluids 8, (1965), 12–22.Google Scholar
[13]Coron, F., and Perthame, B.Numerical passage from kinetic to fluid equations. SIAM Journal on Numerical Analysis 28, (1991), 26–42.Google Scholar
[14]Degond, P., Pareschi, L., and Russo, G.Modeling and Computational Methods for Kinetic Equations. Modeling and Simulation in Science, Engineering and Technology. Springer, 2004.Google Scholar
[15]Dimarco, G., and Pareschi, L.Numerical methods for kinetic equations. Acta Numerica 23, (2014), 369–520.Google Scholar
[16]Filbet, F., and Jin, S.A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. Journal of Computational Physics 229, 20 (Oct. 2010), 7625–7648.Google Scholar
[17]Filbet, F., and Jin, S.An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation. Journal of Scientific Computing 46, 2 (June 2010), 204–224.Google Scholar
[18]Jin, S.Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM Journal on Scientific Computing 21, 2 (1999), 441–454.CrossRefGoogle Scholar
[19]Kennedy, C. A., and Carpenter, M. H.Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Applied Numerical Mathematics 44,1–2 (Jan. 2003), 139–181.Google Scholar
[20]Mieussens, L.Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries. Journal of Computational Physics 162, 2 (Aug. 2000), 429–466.CrossRefGoogle Scholar
[21]Pareschi, L., and Russo, G.An introduction to the numerical analysis of the Boltzmann equation. Rivista di Matematica della Universita di Parma. Serie 7 4**, (2005), 145–250.Google Scholar
[22]Pareschi, L., and Russo, G.Implicit-Explicit Runge-Kutta Schemes and Applications to Hyperbolic Systems with Relaxation. Journal of Scientific Computing 25,1 (Oct. 2005), 129–155.Google Scholar
[23]Pieraccini, S., and Puppo, G.Implicit-Explicit schemes for BGK kinetic equations. Journal of Scientific Computing 32,1 (2007), 1–28.Google Scholar
[24]Pieraccini, S., and Puppo, G.Microscopically Implicit-Macroscopically Explicit schemes for the BGK equation. Journal of Computational Physics 231, (2012), 299–327.Google Scholar
[25]Tcheremissine, F.Conservative evaluation of Boltzmann collision integral in discrete ordinates approximation. Computers & Mathematics with Applications 35,1 (1998), 215–221.Google Scholar
[26]Woodward, P., and Colella, P.The numerical simulation of two-dimensional fluid flow with strong shocks. Journal of Computational Physics (ISSN 0021-9991) 54, (Apr. 1984), 115–173.Google Scholar
[27]Xu, K., and Huang, J.-C. A unified gas-kinetic scheme for continuum and rarefied flows. Journal of Computational Physics 229, 20 (Oct. 2010), 7747–7764.Google Scholar