Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T06:22:04.463Z Has data issue: false hasContentIssue false
  • English
  • Français

High-Order Conservative Asymptotic-Preserving Schemes for Modeling Rarefied Gas Dynamical Flows with Boltzmann-BGK Equation

Part of: Equations of mathematical physics and other areas of application Rarefied gas flows, Boltzmann equation

Published online by Cambridge University Press:  15 October 2015

Manuel A. Diaz ,
Min-Hung Chen  and
Jaw-Yen Yang
Manuel A. Diaz
Affiliation:
Institute of Applied Mechanics, National Taiwan University, Taiwan, Taipei 10167
Min-Hung Chen
Affiliation:
Department of Mathematics, National Cheng-Kung University, Taiwan, Tainan 701
Jaw-Yen Yang*
Affiliation:
Institute of Applied Mechanics, National Taiwan University, Taiwan, Taipei 10167 Institute of Advanced Study in Theoretical Science, National Taiwan University, Taiwan, Taipei 10167
*
*Corresponding author. Email addresses: f99543083@ntu.edu.tw (M. A. Diaz), tengch@mail.ncku.edu.tw (M.-H. Chen), yangjy@iam.ntu.edu.tw (J.-Y. Yang)
Get access

Abstract

High-order and conservative phase space direct solvers that preserve the Euler asymptotic limit of the Boltzmann-BGK equation for modelling rarefied gas flows are explored and studied. The approach is based on the conservative discrete ordinate method for velocity space by using Gauss Hermite or Simpsons quadrature rule and conservation of macroscopic properties are enforced on the BGK collision operator. High-order asymptotic-preserving time integration is adopted and the spatial evolution is performed by high-order schemes including a finite difference weighted essentially non-oscillatory method and correction procedure via reconstruction schemes. An artificial viscosity dissipative model is introduced into the Boltzmann-BGK equation when the correction procedure via reconstruction scheme is used. The effects of the discrete velocity conservative property and accuracy of high-order formulations of kinetic schemes based on BGK model methods are provided. Extensive comparative tests with one-dimensional and two-dimensional problems in rarefied gas flows have been carried out to validate and illustrate the schemes presented. Potentially advantageous schemes in terms of stable large time step allowed and higher-order of accuracy are suggested.

Keywords

Rarefied gas dynamicsBoltzmann-BGK equationasymptotic preservingconservative discrete ordinate methodweighted essentially non-oscillatorycorrection procedure via reconstructionartificial viscosity

MSC classification

Secondary: 35Q20: Boltzmann equations 76P05: Rarefied gas flows, Boltzmann equation
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 4 , October 2015 , pp. 1012 - 1049
Copyright
Copyright © Global-Science Press 2015 

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]Ascher, U.M., Ruuth, S.J., Spiteri, R.J.. Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics 1997;25(2):151–167.Google Scholar
[2]Atkins, H.L., Pampell, A.. Robust and accurate shock capturing method for high-order discontinuous Galerkin methods. In: 20th AIAA Computaional Fluid Dynamics Conference. 2011.Google Scholar
[3]Barter, G.E.. Shock capturing with PDE-based artificial viscosity for an adaptive, higher-order discontinuous Galerkin finite element method. Technical Report; DTIC Document; 2008.Google Scholar
[4]Bennoune, M., Lemou, M., Mieussens, L.. Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics. Journal of Computational Physics 2008;227(8):3781–3803.Google Scholar
[5]Bernard, F., Iollo, A., Puppo, G.. Accurate asymptotic preserving boundary conditions for kinetic equations on cartesian grids. Journal of Scientific Computing 2015;:1–32.Google Scholar
[6]Bhatnagar, P.L., Gross, E.P., Krook, M.. A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Phys Rev 1954;94:511–525.Google Scholar
[7]Chu, C.. Kinetic-theoretic description of the formation of a shock wave. Physics of Fluids (1958-1988) 1965;8(1):12–22.Google Scholar
[8]Cockburn, B., Shu, C. W.. TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws II: General Framework. Mathematics of Computation 1989;52(186):411–435.Google Scholar
[9]Dimarco, G., Pareschi, L.. Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations. SIAM Journal on Numerical Analysis 2013;51(2):1064–1087.Google Scholar
[10]Hesthaven, J., Warburton, T.. Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Texts in Applied Mathematics. Springer, 2008.Google Scholar
[11]Holway, L.H.. New statistical models for kinetic theory: Methods of construction. Physics of Fluids (1958-1988) 1966;9(9):1658–1673.Google Scholar
[12]Huang, J.C.. A conservative discrete ordinate method for model boltzmann equations. Computers & Fluids 2011;45(1):261–267. 22nd International Conference on Parallel Computational Fluid Dynamics (ParCFD 2010) ParCFD.Google Scholar
[13]Huynh, H.T.. A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods; American Institute of Aeronautics and Astronautics. Fluid Dynamics and Co-located Conferences.Google Scholar
[14]Kanevsky, A., Carpenter, M.H., Hesthaven, J.S.. Idempotent filtering in spectral and spectral element methods. Journal of Computational Physics 2006;220(1):41–58.Google Scholar
[15]Kennedy, C.A., Carpenter, M.H.. Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Applied Numerical Mathematics 2003;44(1-2):139–181.Google Scholar
[16]Klöckner, A., Warburton, T., Hesthaven, J.S.. Viscous shock capturing in a time-explicit discontinuous Galerkin method. Mathematical Modelling of Natural Phenomena 2011;6(03):57–83.Google Scholar
[17]Lax, P.D., Liu, X.D.. Solution of two-dimensional Riemann problems of gas dynamics by positive schemes. SIAM Journal on Scientific Computing 1998;19(2):319–340.Google Scholar
[18]Mieussens, L.. Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J Comput Phys 2000;162(2):429–466.Google Scholar
[19]Mieussens, L.. A survey of deterministic solvers for rarefied flows. In: Proceedings of the 29th International Symposium on Rarefied Gas Dynamics. AIP Publishing; volume 1628; 2014. p. 943–951.Google Scholar
[20]Persson, P.O., Peraire, J.. Sub-cell shock capturing for discontinuous Galerkin methods. In: Aerospace Sciences Meetings. American Institute of Aeronautics and Astronautics; 2006.Google Scholar
[21]Pieraccini, S., Puppo, G.. Implicit-explicit schemes for BGK kinetic equations. Journal of Scientific Computing 2007;32(1):1–28.Google Scholar
[22]Ren, X., Xu, K., Shyy, W., Gu, C.. A multi-dimensional high-order discontinuous Galerkin method based on gas kinetic theory for viscous flow computations. Journal of Computational Physics 2015;292:176–193.Google Scholar
[23]Shakhov, E.. Generalization of the Krook kinetic relaxation equation. Fluid Dynamics 1968;3(5):95–96.CrossRefGoogle Scholar
[24]Shu, C.W.. High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM review 2009;51(1):82–126.CrossRefGoogle Scholar
[25]Toro, E.F.. Riemann solvers and numerical methods for fluid dynamics: a practical introduction. Springer Science & Business Media, 2009.Google Scholar
[26]Vincent, P.E., Castonguay, P., Jameson, A.. A new class of high-order energy stable flux reconstruction schemes. Journal of Scientific Computing 2011;47(1):50–72.Google Scholar
[27]Wang, Z.J., Gao, H.. A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. Journal of Computational Physics 2009;228:8161–8186.Google Scholar
[28]Welander, P.. On the temperature jump in a rarefied gas. Arkiv fysik 1954;7.Google Scholar
[29]Xiong, T., Jang, J., Li, F., Qiu, J.M.. High order asymptotic preserving nodal discontinuous Galerkin IMEX schemes for the BGK equation. Journal of Computational Physics 2015;284(0):70–94.Google Scholar
[30]Xu, K.. A gas-kinetic {BGK} scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. Journal of Computational Physics 2001;171(1):289–335.Google Scholar
[31]Xu, K.. Discontinuous Galerkin BGK method for viscous flow equations: one-dimensional systems. SIAM journal on scientific computing 2004;25(6):1941–1963.Google Scholar
[32]Yang, J., Huang, J.. Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations. Journal of Computational Physics 1995;120(2):323–339.Google Scholar
[33]Zhang, X., Shu, C.W.. On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. Journal of Computational Physics 2010;229(23):8918–8934.CrossRefGoogle Scholar
[34]Zhu, H., C.Y., J., Q., . A comparison of the performance of limiters for Runge-Kutta discontinuous Galerkin methods. Adv Appl Math Mech 2013;5(3):365–390.CrossRefGoogle Scholar