Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-27T06:25:38.123Z Has data issue: false hasContentIssue false
  • English
  • Français

A Unified Gas Kinetic Scheme for Continuum and Rarefied Flows V: Multiscale and Multi-Component Plasma Transport

Published online by Cambridge University Press:  31 October 2017

Chang Liu  and
Kun Xu
Chang Liu*
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Kun Xu*
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
*
*Corresponding author. Email addresses:[email protected](C. Liu), [email protected](K. Xu)
*Corresponding author. Email addresses:[email protected](C. Liu), [email protected](K. Xu)
Get access

Abstract

As a continuation of developing multiscale method for the transport phenomena, a unified gas kinetic scheme (UGKS) for multi-scale and multi-component plasma simulation is constructed. The current scheme is a direct modeling method, where the time evolution solutions from the Vlasov-BGK equations of electron and ion and the Maxwell equations are used to construct a scale-dependent plasma simulation model. The modeling scale used in the UGKS is the mesh size scale, which can be comparable to or much larger than the local mean free path. As a result, with the variation of modeling scales in space and time through the so-called cell's Knudsen number and normalized Larmor radius, the discretized governing equations can recover a wide range of plasma evolution from the Vlasov equation in the kinetic scale to different-type of magnetohydrodynamic (MHD) equations in the hydrodynamic scale. The UGKS provides a general evolution model, which goes to the Vlasov equation in the kinetic scale and many types of MHD equations in the hydrodynamic scale, such as the two fluids model, the Hall, the resistive, and the ideal MHD equations. All current existing governing equations become the subsets of the UGKS, and the UGKS bridges these distinguishable governing equations seamlessly. The construction of UGKS is based on the implementation of physical conservation laws and the un-splitting treatment of particle collision, acceleration, and transport in the construction of a scale-dependent numerical flux across a cell interface. At the same time, the discretized plasma evolution equations are coupled with the Maxwell equations for electro-magnetic fields, which also cover a scale-dependent transition between the Ampére's law and the Ohm's law for the calculation of electric field. The time step of UGKS is not limited by the relaxation time, the cyclotron period, and the speed of light in the ideal-MHD regime. Our scheme is able to give a physically accurate solution for plasma simulation with a wide range of Knudsen number and normalized Larmor radius. It can be used to study the phenomena from the Vlasov limit to the scale of plasma skin depth for the capturing of two-fluid effect, and the phenomena in the plasma transition regime with a modest Knudsen number and Larmor radius. The UGKS is validated by numerical test cases, such as the Landau damping and two stream instability in the kinetic regime, and the Brio-Wu shock tube problem, and the Orszag-Tang MHD turbulence problem in the hydrodynamic regime. The scheme is also used to study the geospace environment modeling (GEM), such as the challenging magnetic reconnection problem in the transition regime. At the same time, the magnetic reconnection mechanism of the Sweet-Parker model and the Hall effect model can be connected smoothly through the variation of Larmor radius in the UGKS simulations. Overall, the UGKS is a physically reliable multi-scale plasma simulation method, and it provides a powerful and unified approach for the study of plasma physics.

Keywords

Unified gas-kinetic schemeplasmaVlasov equationtwo-fluid equationsMHD equationsmagnetic field reconnection
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 5 , November 2017 , pp. 1175 - 1223
Copyright
Copyright © Global-Science Press 2017 

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] National Research Council, Plasma science: advancing knowledge in the national interest. National Academies Press (2008).Google Scholar
[2] Chen, F. F., Introduction to plasma physics and controlled fusion. Plenum Press, New York and London, 2nd edition (1974).Google Scholar
[3] Vahedi, V. and Surendra, M., “A Monte Carlo collision model for the particle-in-cell method: applications to argon and oxygen discharges,” Comput. Phys. Commun., vol. 87 (1995), no. 1, pp. 179–198.Google Scholar
[4] Filbet, F. and Sonnendrücker, E., “Comparison of Eulerian Vlasov solvers,” Comput. Phys. Commun., vol. 150 (2003), no. 3, pp. 247–266.Google Scholar
[5] Filbet, F., Sonnendrücker, E., and Bertrand, P., “Conservative numerical schemes for the Vlasov equation,” J. Comput. Phys., vol. 172 (2001), no. 1, pp. 166–187.Google Scholar
[6] Degon, P., Deluzet, F., Navoret, F., and Sun, A.B., “Asymptotic-preserving particle-in-cell method for the VlasovCPoisson system near quasineutrality,” J. Comput. Phys., vol. 229 (2010), no. 1, pp. 5630–5652.Google Scholar
[7] Degon, P., Deluzet, F., Navoret, F., and Sun, A.B., “Asymptotic-preserving particle-in-cell method for the VlasovCMaxwell systemin the quasi-neutral limit,” J. Comput. Phys., vol. 330 (2017), no. 1, pp. 467–492.Google Scholar
[8] Degon, P., Deluzet, F., “Asymptotic-Preserving methods and multiscale models for plasma physics,” arXiv preprint, (2016).Google Scholar
[9] Qiu, J.-M. and Shu, C.-W., “Conservative semi-Lagrangian finite difference WENO formulations with applications to the Vlasov equation,” Commun. Comput. Phys., vol. 10 (2011), no. 4, p. 979.Google Scholar
[10] Guo, W. and Qiu, J.-M., “Hybrid semi-Lagrangian finite element-finite differencemethods for the Vlasov equation,” J. Comput. Phys., vol. 234 (2013), pp. 108–132.Google Scholar
[11] Xiong, T., Qiu, J.-M., Xu, Z., and Christlieb, A., “High order maximum principle preserving semi-Lagrangian finite differenceWENO schemes for the Vlasov equation,” J. Comput. Phys., vol. 273 (2014), pp. 618–639.Google Scholar
[12] Powell, K. G., Roe, P. L., Linde, T. J., Gombosi, T. I., and De Zeeuw, D. L., “A solution-adaptive upwind scheme for ideal magnetohydrodynamics,” J. Comput. Phys., vol. 154 (1999), no. 2, pp. 284–309.CrossRefGoogle Scholar
[13] Brio, M. and Wu, C. C., “An upwind differencing scheme for the equations of ideal magnetohydrodynamics,” J. Comput. Phys., vol. 75 (1988), no. 2, pp. 400–422.Google Scholar
[14] Xu, K., “Gas-kinetic theory-based flux splitting method for ideal magnetohydrodynamics,” J. Comput. Phys., vol. 153 (1999), no. 2, pp. 334–352.Google Scholar
[15] Araya, D. B., Ebersohn, F. H., Anderson, S. E., and Girimaji, S. S., “Magneto-gas kineticmethod for nonideal magnetohydrodynamics flows: verification protocol and plasma jet simulations,” J. Fluids Eng., vol. 137 (2015), no. 8, p. 081302.Google Scholar
[16] Shumlak, U. and Loverich, J., “Approximate Riemann solver for the two-fluid plasma model,” J. Comput. Phys., vol. 187 (2003), no. 2, pp. 620–638.Google Scholar
[17] Hakim, A., Loverich, J., and Shumlak, U., “A high resolution wave propagation scheme for ideal two-fluid plasma equations,” J. Comput. Phys., vol. 219 (2006), no. 1, pp. 418–442.Google Scholar
[18] Loverich, J. and Shumlak, U., “Adiscontinuous Galerkinmethod for the full two-fluid plasma model,” J. Comput. Phys., vol. 169 (2005), no. 1, pp. 251–255.Google Scholar
[19] Loverich, J., Hakim, A., and Shumlak, U., “A discontinuous Galerkin method for ideal twofluid plasma equations,” J. Comput. Phys., vol. 9 (2011), no. 02, pp. 240–268.Google Scholar
[20] Srinivasan, B. and Shumlak, U., “Analytical and computational study of the ideal full twofluid plasma model and asymptotic approximations for Hall-magnetohydrodynamics,” Phys. Plasmas, vol. 18 (2011), no. 9, p. 092113.Google Scholar
[21] Crestetto, A., Crouseilles, N., and Lemou, M., “Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles,” Kin. Rel. Mod., vol. 5 (2012), no. 4, pp. 787–816.Google Scholar
[22] Dimarco, G., Mieussens, L., and Rispoli, V., “An asymptotic preserving automatic domain decomposition method for the Vlasov-Poisson-BGK system with applications to plasmas,” J. Comput. Phys., vol. 274 (2014), pp. 122–139.Google Scholar
[23] Jin, S. and Yan, B., “A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation,” J. Comput. Phys., vol. 230 (2011), no. 17, pp. 6420–6437.Google Scholar
[24] Dimarco, G., Li, Q., Pareschi, L., and Yan, B., “Numerical methods for plasma physics in collisional regimes,” J. Plasma Phys., vol. 81 (2015), no. 1, 305810106Google Scholar
[25] Degond, P., Deluzet, F., Navoret, L., Sun, A.-B., and Vignal, M.-H., “Asymptotic-preserving particle-in-cell method for the Vlasov–Poisson system near quasineutrality,” J. Comput. Phys., vol. 229 (2010), no. 16, pp. 5630–5652.Google Scholar
[26] Xu, K. and Huang, J., “A unified gas-kinetic scheme for continuum and rarefied flows,” J. Comput. Phys., vol. 229 (2010), no. 20, pp. 7747–7764.Google Scholar
[27] Huang, J., Xu, K., and Yu, P., “A unified gas-kinetic scheme for continuum and rarefied flows II: Multi-dimensional cases,” Commun. Comput. Phys., vol. 12 (2012), no. 3, pp. 662–690.Google Scholar
[28] Huang, J., Xu, K., and Yu, P., “A unified gas-kinetic scheme for continuum and rarefied flows III: Microflow simulations,” Commun. Comput. Phys., vol. 14 (2013), no. 5, pp. 1147–1173.Google Scholar
[29] Sun, W., Jiang, S., and Xu, K., “An asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations,” J. Comput. Phys., vol. 285 (2015), pp. 265–279.Google Scholar
[30] Sun, W., Jiang, S., Xu, K., and Li, S., “An asymptotic preserving unified gas kinetic scheme for frequency-dependent radiative transfer equations,” J. Comput. Phys., vol. 302 (2015), pp. 222–238.CrossRefGoogle Scholar
[31] Guo, Z. and Xu, K., “Discrete unified gas kinetic scheme for multiscale heat transfer based on the phonon Boltzmann transport equation,” Int. J. Heat Mass, vol. 102 (2016), pp. 944–958.Google Scholar
[32] Xu, K., “Direct modeling for computational fluid dynamics: construction and application of unified gas-kinetic schemes,” World Scientific, Singapore (2015).Google Scholar
[33] Andries, P., Aoki, K., and Perthame, B., “A consistent BGK-type model for gas mixtures,” J. Stat. Phys., vol. 106 (2002), no. 5-6, pp. 993–1018.Google Scholar
[34] Morse, T.F., “Energy and momentum exchange between nonequipartition gases,” Phys. Fluids, vol. 6 (1963), no. 10, pp. 1420–1427.Google Scholar
[35] Liu, C., Xu, K., Sun, Q., and Cai, Q., “A unified gas-kinetic scheme for continuum and rarefied flows IV: full Boltzmann andmodel equations,” J. Comput. Phys., vol. 314 (2016), pp. 305–340.Google Scholar
[36] Munz, C.-D., Omnes, P., Schneider, R., Sonnendrücker, E., and Voss, U., “Divergence correction techniques for Maxwell solvers based on a hyperbolic model,” J. Comput. Phys., vol. 161 (2000), no. 2, pp. 484–511.Google Scholar
[37] LeVeque, R.J., “ Finite volume methods for hyperbolic problems,” Cambridge university press (2002).Google Scholar
[38] Orszag, S. A. and Tang, C.-M., “Small-scale structure of two-dimensional magnetohydrodynamic turbulence,” J. Fluid Mech., vol. 90 (1979), no. 01, pp. 129–143.Google Scholar
[39] Tang, H.-Z. and Xu, K., “A high-order gas-kinetic method for multidimensional ideal magnetohydrodynamics,” J. Comput. Phys., vol. 165 (2000), no. 1, pp. 69–88.Google Scholar
[40] Parker, E.N., “Sweet's mechanism for merging magnetic fields in conducting fluids,” J. Geophys. Res., vol. 62 (1957), no. 4, pp. 509–520.CrossRefGoogle Scholar
[41] Birn, J., Drake, J., Shay, M., Rogers, B., Denton, R., Hesse, M., Kuznetsova, M., Ma, Z., Bhattacharjee, A., Otto, A., et al., “Geospace environmental modeling (GEM) magnetic reconnection challenge,” J. Geophys. Res.-Space, vol. 106 (2001), no. A3, pp. 3715–3719.Google Scholar
[42] Hesse, M., Birn, J., and Kuznetsova, M., “Collisionless magnetic reconnection: Electron processes and transportmodeling,” J. Geophys. Res.-Space, vol. 106 (2001), no. A3, pp. 3721–3735.Google Scholar
[43] Birn, J. and Hesse, M., “Geospace environment modeling (GEM) magnetic reconnection challenge: Resistive tearing, anisotropic pressure and hall effects,” J. Geophys. Res.-Space, vol. 106 (2001), no. A3, pp. 3737–3750.Google Scholar
[44] Ma, Z. and Bhattacharjee, A., “Hallmagnetohydrodynamic reconnection: The geospace environment modeling challenge,” J. Geophys. Res.-Space, vol. 106 (2001), no. A3, pp. 3773–3782.Google Scholar
[45] Pritchett, P., “Geospace environment modeling magnetic reconnection challenge: Simulations with a full particle electromagnetic code,” J. Geophys. Res.-Space, vol. 106 (2001), no. A3, pp. 3783–3798.Google Scholar
[46] Kuznetsova, M. M., Hesse, M., and Winske, D., “Collisionless reconnection supported by nongyrotropic pressure effects in hybrid and particle simulations,” J. Geophys. Res.-Space, vol. 106 (2001), no. A3, pp. 3799–3810.Google Scholar