Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-16T09:21:42.988Z Has data issue: false hasContentIssue false
  • English
  • Français

A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations

Published online by Cambridge University Press:  20 August 2015

John Loverich ,
Ammar Hakim  and
Uri Shumlak
John Loverich*
Affiliation:
Tech-X Corporation, 5621 Arapahoe Avenue Suite A, Boulder CO, 80303, USA
Ammar Hakim*
Affiliation:
Tech-X Corporation, 5621 Arapahoe Avenue Suite A, Boulder CO, 80303, USA
Uri Shumlak*
Affiliation:
University of Washington, Aerospace and Energetics Research Program, Seattle, WA 98195-2250, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

A discontinuous Galerkin method for the ideal 5 moment two-fluid plasma system is presented. The method uses a second or third order discontinuous Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping scheme. The method is benchmarked against an analytic solution of a dispersive electron acoustic square pulse as well as the two-fluid electromagnetic shock [1] and existing numerical solutions to the GEM challenge magnetic reconnection problem [2]. The algorithm can be generalized to arbitrary geometries and three dimensions. An approach to maintaining small gauge errors based on error propagation is suggested.

Keywords

35Q3535Q6165L60Plasmatwo-fluid5 momentdiscontinuous Galerkinelectrostatic shockelectromagnetic shockmagnetic reconnection
Type
Research Article
Information
Communications in Computational Physics , Volume 9 , Issue 2 , February 2011 , pp. 240 - 268
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]Shumlak, U., Loverich, J., Approximate Riemann solver for the two-fluid plasma model, Journal of Computational Physics 187 (2003) 620–638.Google Scholar
[2]Birn, J., et al., Geospace environmental modeling (gem) magnetic reconnection challenge, Journal of Geophysical Research 106 (A3) (2001) 3715–3719.Google Scholar
[3]Freidberg, J. P., Ideal Magnetohydrodynamics, Plenum Press, 1987.CrossRefGoogle Scholar
[4]Bellan, P. M., Spheromaks, Imperial College Press, 2000.Google Scholar
[5]Hakim, A., Shumlak, U., Two-fluid physics and field-reversed configurations, Physics of Plasmas 14 (5) (2007) 055911.Google Scholar
[6]Connor, J. W., Pressure gradient turbulent transport and collisionless reconnection, Plasma Physics and Controlled Fusions 35 (1993) 757–763.Google Scholar
[7]Krall, N. A., Liewer, P. C., Low-frequency instabilities in magnetic pulses, Physical Review A 4 (5) (1971) 2094–2103.Google Scholar
[8]Davidson, R. C., Gladd, N. T., Anomalous transport properties associated with the lower-hybrid-drift instability, The Physics of Fluids 18 (10) (1975) 1327–1335.Google Scholar
[9]Loverich, J., Shumlak, U., Nonlinear full two-fluid study of m = 0 sausage instabilities in an axisymmetric z pinch, Physics of Plasmas 13 (8) (2006) 082310.Google Scholar
[10]Solovev, L., Dynamics of a cylindrical z pinch, Soviet Journal of Plasma Physics 10 (5) (1984) 602–605.Google Scholar
[11]Haines, M., The physics of the dense z-pinch in theory and in experiment with application to fusion reactor, Physica Scripta T2/2 (1982) 380–390.Google Scholar
[12]Jones, O. S., Shumlak, U., Eberhardt, D. S., An implicit scheme for nonideal magnetohydro-dynamics, Journal of Computational Physics 130 (1997) 231–242.Google Scholar
[13]Sovinec, C., et al., Nonlinear magnetohydrodynamics simulation using high-order finite elements, Journal of Computational Physics 195 (2004) 355–386.CrossRefGoogle Scholar
[14]Bhattacharjee, A., Center for magnetic reconnection studies: Present status, future plans, PSACI PAC Presentation, Princeton (June 2003).Google Scholar
[15]Breslau, J., Jardin, S., A parallel algorithm for global magnetic reconnection studies, Computer Physics Communications 151 (2003) 8–24.CrossRefGoogle Scholar
[16]Huba, J. D., Hall magnetohydrodynamics - a tutorial, in: Buchner, M. S. J., Dunn, C.T. (Ed.), Space Plasma Simulation, Springer, 2003, pp. 166–192.Google Scholar
[17]Park, W., et al., Plasma simulation studies using multilevel physics models, Physics of Plasmas 6 (5) (1999) 1796–1803.Google Scholar
[18]Baboolal, S., Finite-difference modeling of solitons induced by a density hump in a plasma multi-fluid, Mathematics and Computers in Simulation 55 (2001) 309–316.Google Scholar
[19]Schneider, R., Munz, C. D., The approximation of two-fluid plasma flow with explicit upwind schemes, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 8 (1995) 399–416.Google Scholar
[20]Rambo, P., Denavit, J., Time-implicit fluid simulation of collisional plasmas, Journal of Computational Physics 98 (1991) 317–331.Google Scholar
[21]Mason, R., An electromagnetic field algorithm for 2d implicit plasma simulation, Journal of Computational Physics 71 (1987) 429–473.Google Scholar
[22]Mason, R., Cranfill, C., Hybrid two-dimensional electron transport in self-consistent electromagnetic fields, IEEE Transactions on Plasma Science 14 (1) (1986) 45–52.Google Scholar
[23]Hakim, A., Loverich, J., Shumlak, U., A high resolution wave propagation scheme for ideal two-fluid plasma equations, J. Comput. Phys. 219 (1) (2006) 418–442.Google Scholar
[24]Cockburn, B., Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Mathematics of Computation 52 (1989) 411–435.Google Scholar
[25]Cockburn, B., Lin, S.-Y., Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, Journal of Computational Physics 84 (1989) 90–113.CrossRefGoogle Scholar
[26]Cockburn, B., Hou, S., Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: Multidimensional case, Mathematics of Computation 54 (1990) 545–581.Google Scholar
[27]Cockburn, B., Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, Journal of Computational Physics 141 (1998) 199–224.Google Scholar
[28]Hesthaven, J. S., Warburton, T., Nodal high-order methods on unstructured grids, Journal of Computational Physics 181 (2002) 186–221.Google Scholar
[29]Cockburn, B., Li, F., Shu, C.-W., Locally divergence-free discontinuous Galerkin methods for Maxwell’s equations, Journal of Computational Physics 194 (2004) 588–610.Google Scholar
[30]Warburton, T. C., Karniadakis, G. E., A discontinuous Galerkin method for the viscous MHD equations, Journal of Computational Physics 152 (1999) 608–641.CrossRefGoogle Scholar
[31]Lin, G., Karniadakis, G., A discontinuous Galerkin method for two-temperature plasmas, Comput. Methods Appl. Mech. Engrg. 195 (2006) 3504–3527.Google Scholar
[32]Mangeney, A., Califano, F., Cavazzoni, C., Travnicek, P., A numerical scheme for the integration of the Vlasov-Maxwell system of equations, Journal of Computational Physics 179 (2002) 495–538.Google Scholar
[33]Cockburn, B., Karniadakis, G. E., Shu, C.-W. (Eds.), Discontinuous Galerkin Methods, Springer, 2000.Google Scholar
[34]Munz, C. D., Schneider, R., Vos, U., A finite-volume method for the Maxwell equations in the time domain, Siam Journal of Scientific Computing 22 (2000) 449–475.Google Scholar
[35]Umeda, T., Omura, Y., Tominaga, T., Matsumoto, H., A new charge conservation method in electromagnetic particle-in-cell simulations, Computer Physics Communications 156 (2003) 73–85.Google Scholar
[36]Villasenor, J., Buneman, O., Rigorous charge conservation for local electromagnetic field solvers, Computer Physics Communications 69 (2-3) (1992) 306–316.Google Scholar
[37]Mardahl, P. J., Verboncoeur, J. P., Charge conservation in electromagnetic PIC codes; spectral comparison of Boris/DADI and Langdon-Marder methods, Computer Physics Communications 106 (1997) 219–229.Google Scholar
[38]Li, S., High order central scheme on overlapping cells for magneto-hydrodynamic flows with and without constrained transport method, J. Comput. Phys. 227 (15) (2008) 7368–7393.Google Scholar
[39]Balsara, D. S., Divergence-free reconstruction of magnetic fields and WENO schemes for magnetohydrodynamics, J. Comput. Phys. 228 (14) (2009) 5040–5056.CrossRefGoogle Scholar
[40]Gardiner, T. A., Stone, J. M., An unsplit Godunov method for ideal MHD via constrained transport in three dimensions, J. Comput. Phys. 227 (8) (2008) 4123–4141.CrossRefGoogle Scholar
[41]Londrillo, P., Zanna, L. D., On the divergence-free condition in Godunov-type schemes for ideal magnetohydrodynamics: the upwind constrained transport method, J. Comput. Phys. 195 (1) (2004) 17–48.Google Scholar
[42]Durran, D., Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Springer, 1998.Google Scholar
[43]LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002.Google Scholar
[44]Biswas, R., Devine, K. D., Flaherty, J. E., Parallel, adaptive finite element methods for conservation laws, Applied Numerical Mathematics 14 (1994) 255–283.Google Scholar
[45]Brio, M., Wu, C. C., An upwind differencing scheme for the equations of ideal magnetohy-drodynamics, Journal of Computational Physics 75 (1988) 400–422.Google Scholar
[46]Loverich, J., A Finite Volume Algorithm for the Two-Fluid Plasma System in One Dimension, MSc. Thesis, University of Washington (2003).Google Scholar
[47]Loverich, J., Shumlak, U., A discontinuous Galerkin method for the full two-fluid plasma model, Computer Physics Communications 169 (2005) 251–255.Google Scholar
[48]Hesse, M., Birn, J., Kusnetsova, M., Collisionless magnetic reconnection: Electron processes and transport modeling, Journal of Geophysical Research 106 (2001) 3721–3735.CrossRefGoogle Scholar
[49]Shay, M. A., Drake, J. F., Rogers, B. N., Denton, R. E., Alfvenic collisionless magnetic reconnection and the Hall term, Journal of Geophysical Research 106 (2001) 3759–3772.CrossRefGoogle Scholar