Skip to main content
    • Aa
    • Aa

Numerical Methods for Two-Fluid Dispersive Fast MHD Phenomena

  • Bhuvana Srinivasan (a1), Ammar Hakim (a2) and Uri Shumlak (a1)

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.

Corresponding author
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] R. J. LeVeque , Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.

[3] A. Hakim , Extended MHD modelling with the ten-moment equations, J. Fusion. Energy., 27 (2008), 36–43.

[4] A. Hakim , J. Loverich , and U. Shumlak , A high resolution wave propagation scheme for ideal two-fluid plasma equations, J. Comput. Phys., 219 (2006), 418–442.

[6] B. Cockburn , and C.-W. Shu , Runge-Kutta discontinous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), 173–261.

[7] B. Cockburn , G. E. Karniadakis , and C.-W. Shu , 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.

[9] B. Cockburn , and C.-W. Shu , TVB runge-Kutta local projection discontinous Galerkin finite element for conservation laws II-general framework, Math. Comput., 52(186) (1989), 411–435.

[10] C.-W. Shu , Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9(6) (1988), 1073–1084.

[11] F. Bassi , and S. Rebay , 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.

[12] J. T. Oden , I. Babuŝka , and C. E. Baumann , A discontinuous hp finite element method for diffusion problems, J. Comput. Phys., 146 (1998), 491–519.

[13] B. Cockburn , F. Li , and C.-W. Shu , Locally divergence-free discontinuous Galerkin methods for the Maxwell equations, J. Comput. Phys., 194(2) (2004), 588–610.

[16] D. Levy , C.-W. Shu , and J. Yan , Local discontinuous Galerkin methods for nonlinear dispersive equations, J. Comput. Phys., 196 (2004), 751–772.

[17] M. Zhang , and C.-W. Shu , An analysis of and a comparison between the discontinuous galerkin and the spectral finite volume methods, Comput. Fluids., 34 (2005), 581–592.

[18] J. Loverich , and U. Shumlak , A discontinuous Galerkin method for the full two-fluid plasma model, Comput. Phys. Commun., 169 (2005), 251–255.

[19] U. Shumlak , and J. Loverich , Approximate Riemann solver for the two-fluid plasma model, J. Comput. Phys., 187 (2003), 620–638.

[20] R. J. Spiteri , and S. J. Ruuth , A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40(2) (2002), 469–491.

[22] B. Cockburn , S. Hou , and C.-W. Shu , The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws iv: the multidimensional case, Math. Comput., 54 (1990), 545–581.

[23] Lilia Krivodonova , Limiters for high-order discontinuous Galerkin methods, J. Comput. Phys., 226 (2007), 879–896.

[25] C. D. Munz et al., Divergence correction techniques for Maxwell solvers based on a hyperbolic model, J. Comput. Phys., 161 (2000), 484–511.

[26] S. Baboolal , Finite-difference modeling of solitons induced by a density hump in a plasma multi-fluid, Math. Comput. Sim., 55 (2001), 309–316.

[27] J. Loverich , and U. Shumlak , Nonlinear full two-fluid study of m=0 sausage instabilities in an axisymmetric Z-pinch, Phys. Plasmas., 13 (2006), 082310.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 13 *
Loading metrics...

Abstract views

Total abstract views: 77 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th August 2017. This data will be updated every 24 hours.