LSFEM Implementation of MHD Numerical Solver

Jan Skála, Miroslav Bárta

Astronomical Institute of Czech Academy of Sciences, Ond?ejov, Czech Republic.
University J. E. Purkinje, ústí nad Labem, Czech Republic.

DOI: 10.4236/am.2012.331250                

Abstract

Many problems in physics are inherently of multi-scale nature. The issues of MHD turbulence or magnetic reconnection, namely in the hot and sparse, almost collision-less astrophysical plasmas, can stand as clear examples. The Finite Element Method (FEM) with adaptive gridding appears to be the appropriate numerical implementation for handling the broad range of scales contained in such high Lundquist-number MHD problems. In spite the FEM is now routinely used in engineering practice in solid-state and fluid dynamics, its usage for MHD simulations has recently only begun and only few implementations exist so far. In this paper we present our MHD solver based on the Least-Square FEM (LSFEM) formulation. We describe the transformation of the MHD equations into form required for finding the LSFEM functional and some practical issues in implementation of the method. The algorithm was tested on selected problems of ideal (non-resistive) and resistive MHD. The tests show the usability of LSFEM for solving MHD equations.

Keywords

Magnetohydrodynamics (MHD); Least-Squares Finite Element Method; Adaptive Mesh Refinement; Magnetic Reconnection; Solar Eruptions; MHD Turbulence

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	E. R. Priest, “Solar Magneto-Hydrodynamics,” Reidel, Dordrecht, 1984.
[2]	K. Shibata and S. Tanuma, “Plasmoid-Induced-Reconnection and Fractal Reconnection,” Earth, Planets, and Space, Vol. 53, 2001, pp. 473-482. http://adsabs.harvard.edu/abs/2001EP%26S...53..473S
[3]	M. Bárta, J. Büchner, M. Karlicky and J. Skála, “Spontaneous Current-Layer Fragmentation and Cascading Reconnection in Solar Flares. I. Model and Analysis,” Astrophysical Journal, Vol. 737, No. 1, 2011. doi:10.1088/0004-637X/737/1/24
[4]	M. Bárta, J. Büchner, M. Karlicky and P. Kotr?, “Spontaneous Current-Layer Fragmentation and Cascading Reconnection in Solar Flares. II. Relation to Observations,” Astrophysical Journal, Vol. 730, No. 1, 2011, 6 pages. doi:10.1088/0004-637X/730/1/47
[5]	O. Mátais, “Course 3: Large-Eddy Simulations of Turbulence,” In: M. Lesieur, A. Yaglom and F. David, Eds., New Trends in Turbulence, Vol. 730, No. 1, 2001, pp. 113-186.
[6]	M. A. Leschziner, “Course 4: Statistical Turbulence Modelling for the Computation of Physically Complex Flows,” In: M. Lesieur, A. Yaglom and F. David, Eds., New Trends in Turbulence, Springer Verlag, Berlin, 2001, p. 187. doi:10.1007/3-540-45674-0_4
[7]	M. J. Berger and J. Oliger, “Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations,” Journal of Computational Physics, Vol. 53, No. 3, 1984, pp. 484-512. doi:10.1016/0021-9991(84)90073-1
[8]	B. Fryxell, K. Olson, P. Ricker, F. X. Timmes, M. Zingale, D. Q. Lamb, P. MacNeice, R. Rosner, J. W. Truran, and H. Tufo, “FLASH: An Adaptive Mesh Hydrodynamics Code for Modeling Astrophysical Thermonuclear Flashes,” Astrophysical Journal Supplements, Vol. 131, No. 1, 2000, pp. 273-334.
[9]	B. van der Holst and R. Keppens, “Hybrid Block-AMR in Cartesian and Curvilinear Coordinates: MHD Applications,” Journal of Computational Physics, Vol. 226, No. 1, 2007, pp. 925-946.
[10]	B. Jiang, “The Least-Squares Finite Element Method,” Springer-Verlag, Berlin, 1998.
[11]	P. P. Bochev and M. D. Gunzburger, “Least-Squares Finite Element Methods,” Springer Science+Business Media, New York, 2009.
[12]	C. R. Sovinec, A. H. Glasser, T. A. Gianakon, D. C. Barnes, R. A. Nebel, S. E. Kruger, D. D. Schnack, S. J. Plimpton, A. Tarditi and M. S. Chu, “Nonlinear Magnetohydrodynamics Simulation Using High-Order Finite Elements,” Journal of Computational Physics, Vol. 195, No. 1, 2004, pp. 355-386. doi:10.1016/j.jcp.2003.10.004
[13]	S. C. Jardin and J. A. Breslau, “Implicit Solution of the Four-Field Extended-Magnetohydrodynamic Equations Using High-Order High-Continuity Finite Elements,” Physics of Plasmas, Vol. 12, No. 5, 2005, Article ID: 056101. doi:10.1063/1.1864992
[14]	V. S. Lukin, “Computational Study of the Internal Kink Mode Evolution and Associated Magnetic Reconnection Phenomena,” Ph.D. Thesis, Princeton University, Princeton, 2008.
[15]	A. H. Glasser and X. Z. Tang, “The SEL Macroscopic Modeling Code,” Computer Physics Communications, Vol. 164, No. 1-3, 2004, pp. 237-243. doi:10.1016/j.cpc.2004.06.034
[16]	J. Büchner and N. Elkina, “Anomalous Resistivity of Current-Driven Isothermal Plasmas Due to Phase Space Structuring,” Physics of Plasmas, Vol. 13, No. 8, 2006, Article ID: 082304.
[17]	B. Kliem, M. Karlicky and A. O. Benz, “Solar Flare Radio Pulsations as a Signature of Dynamic Magnetic Reconnection,” Astronomy & Astrophysics, Vol. 360, 2000, pp. 715-728. http://adsabs.harvard.edu/abs/2000A%26A...360..715K
[18]	T. J. Chung, “Computational Fluid Dynamics,” Cambridge University Press, Cambridge, 2002.
[19]	T. W. H. Sheu and R. K. Lin, “Newton Linearization of the Incompressible Navier-Stokes Equations,” International Journal for Numerical Methods in Fluids, Vol. 44, No. 3, 2004, pp. 297-312. doi:10.1002/fld.639
[20]	H. T. Rathod, K. V. Nagaraja and N. L. Ramesh, “Gauss Legendre Quadrature over a Triangle,” Journal of the Indian Institute of Science, Vol. 84, 2004, pp. 183-188. http://journal.library.iisc.ernet.in/vol200405/paper6/rathod.pdf
[21]	W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes,” Cambridge University Press, Cambridge, 2007.
[22]	D. Ryu and T. W. Jones, “Numerical Magetohydrodynamics in Astrophysics: Algorithm and Tests for One-Dimensional Flow,” Astrophysical Journal, Vol. 442, No. 1, 1995, pp. 228-258. doi:10.1086/175437
[23]	S. A. Orszag and C. M. Tang, “Small-Scale Structure of Two-Dimensional Magnetohydrodynamic Turbulence,” Journal of Fluid Mechanics, Vol. 90, No. 1, 1979, pp. 129-143. doi:10.1017/S002211207900210X