Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T12:47:53.298Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Methods for Two-Fluid Dispersive Fast MHD Phenomena

Published online by Cambridge University Press:  20 August 2015

Bhuvana Srinivasan ,
Ammar Hakim  and
Uri Shumlak
Bhuvana Srinivasan*
Affiliation:
Aerospace and Energetics Research Program, University of Washington, Seattle, WA 98195, USA
Ammar Hakim*
Affiliation:
Tech-X Corporation, 5621 Arapahoe Avenue Suite A, Boulder, CO 80303, USA
Uri Shumlak*
Affiliation:
Aerospace and Energetics Research Program, University of Washington, Seattle, WA 98195, USA
*
Corresponding author.Email:srinbhu@u.washington.edu
Email:ammar@txcorp.com
Email:shumlak@u.washington.edu
Get access

Abstract

The finite volume wave propagation method and the finite element Runge-Kutta discontinuous Galerkin (RKDG) method are studied for applications to balance laws describing plasma fluids. The plasma fluid equations explored are dispersive and not dissipative. The physical dispersion introduced through the source terms leads to the wide variety of plasma waves. The dispersive nature of the plasma fluid equations explored separates the work in this paper from previous publications. The linearized Euler equations with dispersive source terms are used as a model equation system to compare the wave propagation and RKDG methods. The numerical methods are then studied for applications of the full two-fluid plasma equations. The two-fluid equations describe the self-consistent evolution of electron and ion fluids in the presence of electromagnetic fields. It is found that the wave propagation method, when run at a CFL number of 1, is more accurate for equation systems that do not have disparate characteristic speeds. However, if the oscillation frequency is large compared to the frequency of information propagation, source splitting in the wave propagation method may cause phase errors. The Runge-Kutta discontinuous Galerkin method providesmore accurate results for problems near steady-state aswell as problems with disparate characteristic speeds when using higher spatial orders.

Keywords

35Q3535Q6165L6065M0882D10Two-fluid plasma model5 momentdiscontinuous Galerkinhigh-resolution wave propagationdispersive source termshyperbolic conservation lawsdispersive EulersolitonZ-pinch
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 1 , July 2011 , pp. 183 - 215
Copyright
Copyright © Global Science Press Limited 2011

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]Kulikoviskii, A. G., Pogorelov, N. V., and Semenov, A. Y., Mathematical Aspects of Numerical Solutions of Hyperbolic Systems, Chapman and Hall/CRC, 2001.Google Scholar
[2]LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.Google Scholar
[3]Hakim, A., Extended MHD modelling with the ten-moment equations, J. Fusion. Energy., 27 (2008), 36–43.Google Scholar
[4]Hakim, A., Loverich, J., and Shumlak, U., A high resolution wave propagation scheme for ideal two-fluid plasma equations, J. Comput. Phys., 219 (2006), 418–442.CrossRefGoogle Scholar
[5]Bale, D., LeVeque, R. J., Mitran, S., and Rossmanith, J. A., A wave-propagation method for conservation laws and balance laws with spatially varying flux functions, SIAM J. Sci. Comput., 24 (2002), 955–978.Google Scholar
[6]Cockburn, B., and Shu, C.-W., Runge-Kutta discontinous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), 173–261.Google Scholar
[7]Cockburn, B., Karniadakis, G. E., and Shu, C.-W., The Development of Discontinuous Galerkin Methods, in: Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture notes in Computational Science and Engineering, Volume 11, Springer, 2000.Google Scholar
[8]Reed, W. H., and Hill, T. R., Triangular mesh methods for the neutron transport equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973.Google Scholar
[9]Cockburn, B., and Shu, C.-W., TVB runge-Kutta local projection discontinous Galerkin finite element for conservation laws II-general framework, Math. Comput., 52(186) (1989), 411–435.Google Scholar
[10]Shu, C.-W., Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9(6) (1988), 1073–1084.Google Scholar
[11]Bassi, F., and Rebay, S., A high order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131 (1997), 267–279.Google Scholar
[12]Oden, J. T., Babuŝka, I., and Baumann, C. E., A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., 146 (1998), 491–519.Google Scholar
[13]Cockburn, B., Li, F., and Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys., 194(2) (2004), 588–610.Google Scholar
[14]Hesthaven, J. S., and Warburton, T., High-order nodal discontinuous Galerkin methods for the Maxwell eigenvalue problem, Royal. Soc. London., 362 (2004), 493–524.Google Scholar
[15]Li, F., and Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for MHD equations, J. Sci. Comput., 22-23 (2003), 413–442.Google Scholar
[16]Levy, D., Shu, C.-W., and Yan, J., Local discontinuous Galerkin methods for nonlinear dispersive equations, J. Comput. Phys., 196 (2004), 751–772.Google Scholar
[17]Zhang, M., and Shu, C.-W., An analysis of and a comparison between the discontinuous galerkin and the spectral finite volume methods, Comput. Fluids., 34 (2005), 581–592.Google Scholar
[18]Loverich, J., and Shumlak, U., A discontinuous Galerkin method for the full two-fluid plasma model, Comput. Phys. Commun., 169 (2005), 251–255.Google Scholar
[19]Shumlak, U., and Loverich, J., Approximate Riemann solver for the two-fluid plasma model, J. Comput. Phys., 187 (2003), 620–638.Google Scholar
[20]Spiteri, R. J., and Ruuth, S. J., A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40(2) (2002), 469–491.Google Scholar
[21]Gottlieb, S., Ketcheson, D. I., and Shu, C-W., High order strong stability preserving time discretizations, J. Sci. Comput., 38(3) (2008), 251–289.Google Scholar
[22]Cockburn, B., Hou, S., and Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws iv: the multidimensional case, Math. Comput., 54 (1990), 545–581.Google Scholar
[23]Krivodonova, Lilia, Limiters for high-order discontinuous Galerkin methods, J. Comput. Phys., 226 (2007), 879–896.Google Scholar
[24]Qiu, J., and Shu, C.-W., Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM J. Sci>. Comput., 26 (2005), 907–929.Google Scholar
[25]Munz, C. D.et al., Divergence correction techniques for Maxwell solvers based on a hyperbolic model, J. Comput. Phys., 161 (2000), 484–511.Google Scholar
[26]Baboolal, S., Finite-difference modeling of solitons induced by a density hump in a plasma multi-fluid, Math. Comput. Sim., 55 (2001), 309–316.Google Scholar
[27]Loverich, J., and Shumlak, U., Nonlinear full two-fluid study of m=0 sausage instabilities in an axisymmetric Z-pinch, Phys. Plasmas., 13 (2006), 082310.Google Scholar