Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T09:16:07.314Z Has data issue: false hasContentIssue false
  • English
  • Français

Solving Vlasov-Poisson-Fokker-Planck Equations using NRxx method

Part of: Numerical analysis: Ordinary differential equations Equations of mathematical physics and other areas of application Optics, electromagnetic theory - Basic methods

Published online by Cambridge University Press:  07 February 2017

Yanli Wang  and
Shudao Zhang
Yanli Wang*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, P.R. China
Shudao Zhang*
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100094, P.R. China
*
*Corresponding author. Email addresses:wang_yanli@iapcm.ac.cn (Y. Wang), zhang_shudao@iapcm.ac.cn (S. Zhang)
*Corresponding author. Email addresses:wang_yanli@iapcm.ac.cn (Y. Wang), zhang_shudao@iapcm.ac.cn (S. Zhang)
Get access

Abstract

We present a numerical method to solve the Vlasov-Poisson-Fokker-Planck (VPFP) system using the NRxx method proposed in [4, 7, 9]. A globally hyperbolic moment system similar to that in [23] is derived. In this system, the Fokker-Planck (FP) operator term is reduced into the linear combination of the moment coefficients, which can be solved analytically under proper truncation. The non-splitting method, which can keep mass conservation and the balance law of the total momentum, is used to solve the whole system. A numerical problem for the VPFP system with an analytic solution is presented to indicate the spectral convergence with the moment number and the linear convergence with the grid size. Two more numerical experiments are tested to demonstrate the stability and accuracy of the NRxx method when applied to the VPFP system.

Keywords

Moment closureFokker-Planck operatorVPFP systemspectral convergence

MSC classification

Secondary: 35Q83: Vlasov-like equations 65L20: Stability and convergence of numerical methods 78M05: Method of moments
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 3 , March 2017 , pp. 782 - 807
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] Allen, E.J. and Victory, H.D. Jr. A computational investigation of the random particle method for numerical solution of the kinetic Vlasov-Poisson-Fokker -Planck equations. Physica A, 209(3):318–346, 1994.CrossRefGoogle Scholar
[2] Bouchut, F.. Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions. J. Funct. Anal, 111:239–258, 1993.CrossRefGoogle Scholar
[3] Bouchut, F.. Smoothing effect for the non-linear Vlasov-Poisson-Fokker-Planck system. J. Differ. Eq, 122:225–238, 1995.Google Scholar
[4] Cai, Z., Fan, Y., and Li, R.. Globally hyperbolic regularization of Grad's moment system in one dimensional space. Comm. Math Sci., 11(2):547–571, 2013.Google Scholar
[5] Cai, Z., Fan, Y., and Li, R.. Globally hyperbolic regularization of Grad's moment system. Comm. Pure Appl. Math., 67(3):464–518, 2014.Google Scholar
[6] Cai, Z., Fan, Y., Li, R., Lu, T., and Wang, Y.. Quantum hydrodynamics models by moment closure of wigner equation. J. Math. Phys., 53:103503, 2012.CrossRefGoogle Scholar
[7] Cai, Z. and Li, R.. Numerical regularized moment method of arbitrary order for Boltzmann-BGK equation. SIAM J. Sci. Comput., 32(5):2875–2907, 2010.Google Scholar
[8] Cai, Z., Li, R., and Qiao, Z.. Globally hyperbolic regularized moment method with applications to microflow simulation. Comput. Fluids, 81:95–109, 2013.CrossRefGoogle Scholar
[9] Cai, Z., Li, R., and Wang, Y.. An efficient NRxx method for Boltzmann-BGK equation. J. Sci. Comput., 50(1):103–119, 2012.CrossRefGoogle Scholar
[10] Cai, Z., Li, R., and Wang, Y.. Numerical regularized moment method for high Mach number flow. Commun. Comput. Phys., 11(5):1415–1438, 2012.Google Scholar
[11] Cai, Z., Li, R., and Wang, Y.. Solving Vlasov equations using NRxx method. SIAM J. Sci. Comput., 35(6):A2807–A2831, 2013.Google Scholar
[12] Cercignani, C., Gamba, I.M., JeromeW, J.W., and Shu, C.-W.. Device benchmark comparisons via kinetic, hydrodynamic, and high-hield models. Comput. Method. Appl. M, 181(4):381–392, 2000.Google Scholar
[13] Cheng, C. Z. and Knorr, G.. The integration of the Vlasov equation in configuration space. J. Comput. Phys, 22:330–351, 1976.Google Scholar
[14] Degond, P.. Global existence of smooth solutions of Vlasov-Fokker-Planck equation in 1 and 2 space dimensions. Ann. Scient. Éc. Norm. Sup. 4c Série, 19:519–542, 1986.Google Scholar
[15] Filbet, F. and Pareschi, L.. Nummerical solution of the Fokker-Planck-Landau equation by spectral methods. Comm. Math. Sciences, 1(1):208–209, 2003.Google Scholar
[16] Fok, J., Guo, B., and Tang, T.. Combined Hermite spectral-finite difference method for the Fokker-Planck equation. Math. Comput, 71(240):1497–1528, 2002.Google Scholar
[17] Goudon, T., Nieto, J., Poupaud, F., and Soler, J.. Multidimensional high-field limit of the electrosatic Vlasov-Poisson-Fokker-Planck system. J. Diff. Eq., 213:418–442, 2005.Google Scholar
[18] Havlak, K.J. and Victory, H.D. Jr. The numerical analysis of random particle methods applied to Vlasov-Poisson Fokker-Planck kinetic equations. SIAM J. Number. Anal, 1:291–317, 1996.Google Scholar
[19] Havlak, K.J. and Victory, H.D. Jr. On deterministic particle methods for solving Vlasov–Poisson–Fokker–Planck systems. SIAM J. Number. Anal, 35(4):1473–1519, 1998.CrossRefGoogle Scholar
[20] Hu, J., Jin, S., and Yan, B.. A numerical scheme for the quantum Fokker-Planck-Landau equation efficient in the fluid regime. Commn. Comp. Phys., 12:1541–1561, 2012.CrossRefGoogle Scholar
[21] Hu, Z., Li, R., Lu, T., Wang, Y., and Yao, W.. Simulation of an n +-n-n + diode by using globally-hyperbolically-closed high-order moment models. J. Sci. Comput, 59(3):761–774, 2014.Google Scholar
[22] Jin, S. and Wang, L.. An asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck system in the high field regime. Acta. Math. Sci, 31(6):2219–2232, 2011.CrossRefGoogle Scholar
[23] Li, R., Lu, T., Wang, Y., and Yao, W.. Numerical method for high order hyperbolic moment system of Wigner equation. Commun. Comput. Phys, 15(3):569–595, 2013.CrossRefGoogle Scholar
[24] Neunzert, H., Pulvirenti, M., and Triolo, L.. On the Vlasov-Fokker-Planck equation. Math. Meth. Appl. Sci., 6:527538, 1984.Google Scholar
[25] Poupaud, F. and Soler, J.. Parabolic limit and stability of the Vlasov–Fokker–Planck system. Math. Mol. Meth. Appl. S, 10:1027–1045, 2000.Google Scholar
[26] Rautmann, R.. On the uniqueness and stability of weak solutions of a Fokker-Planck-Vlasov equation. In Numerical Treatment of Differential Equations in Applications, pages 141–150. Springer, 1978.Google Scholar
[27] Schaeffer, J.. A difference scheme for the Vlasov-Poisson-Fokker-Planck system. Research Report No. 97-NA-004, Department of Mathematical Sciences, Carnegie Mellon University, 1997.Google Scholar
[28] Schaeffer, J.. Convergence of a difference scheme for the Vlasov–Poisson–Fokker–Planck system in one dimension. SIAM J. Number. Anal, 35(3):1149–1175, 1998.Google Scholar
[29] Schumer, J. W. and Holloway, J. P.. Vlasov simulation using velocity-scaled Hermite representations. J. Comput. Phys., 144(2):626–661, 1998.Google Scholar
[30] Triolo, L.. Global existence for the Vlasov-Poisson/Fokker-Planck equation in many dimensions, for small data. Math. Method. Appl. Sci, 10(4):487–497, 1988.Google Scholar
[31] Wollman, S. and Ozizmir, E.. Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in one dimension. J. Comput. Phys, 202:602–644, 2005.Google Scholar
[32] Wollman, S. and Ozizmir, E.. Numerical approximation of the Vlasov–Poisson–Fokker–Planck system in two dimensions. J. Comput. Phys, 228:6629–6669, 2009.Google Scholar
[33] Yan, B. and Jin, S.. A successive penalty based asymptotic-preserving scheme for kinetic equations. SIAM J. Sci. Comput., 35:A150–A172, 2013.Google Scholar
[34] Dal Maso, G., LeFloch, P. G., and Murat, F.. Definition and weak stability of nonconservative products. J. Math. Pures Appl., 74(6):483–548, 1995.Google Scholar
[35] Rhebergen, S., Bokhove, O., and van der Vegt, J. J. W.. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J. Comput. Phys., 227(3):1887–1922, 2008.Google Scholar
[36] Cai, Z., Li, R., and Qiao, Z.. NRxx simulation of microflows with Shakhov model. SIAM J. Sci. Comput., 34(1):A339–A369, 2012.Google Scholar
[37] Wollman, S. and Ozizmir, E.. A deterministic particle method for the Vlasov–Poisson–Fokker–Planck equation in one dimension. J. Comput. Phys, 213:316–365, 2008.Google Scholar