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Simulating Magnetohydrodynamic Instabilities with Conservative Perturbed MHD Model Using Discontinuous Galerkin Method

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Astronomy and astrophysics (General)

Published online by Cambridge University Press:  27 March 2017

Jun Ma ,
Wenfeng Guo  and
Zhi Yu
Jun Ma*
Affiliation:
Institute of Plasma Physics, Chinese Academy of sciences, Hefei 230031, China Center for Magnetic Fusion Theory, Chinese Academy of Sciences, Hefei 230031, China
Wenfeng Guo*
Affiliation:
Institute of Plasma Physics, Chinese Academy of sciences, Hefei 230031, China Center for Magnetic Fusion Theory, Chinese Academy of Sciences, Hefei 230031, China
Zhi Yu*
Affiliation:
Institute of Plasma Physics, Chinese Academy of sciences, Hefei 230031, China Center for Magnetic Fusion Theory, Chinese Academy of Sciences, Hefei 230031, China
*
*Corresponding author. Email addresses:[email protected] (J. Ma), [email protected] (W. Guo), [email protected] (Z. Yu)
*Corresponding author. Email addresses:[email protected] (J. Ma), [email protected] (W. Guo), [email protected] (Z. Yu)
*Corresponding author. Email addresses:[email protected] (J. Ma), [email protected] (W. Guo), [email protected] (Z. Yu)
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Abstract

In magnetically confined plasma research, the understandings of small and large perturbations at equilibrium are both critical for plasma controlling and steady state operation. Numerical simulations using original MHD model can hardly give clear picture for small perturbations, while non-conservative perturbed MHD model may break conservation law, and give unphysical results when perturbations grow large after long-time computation. In this paper, we present a nonlinear conservative perturbed MHD model by splitting primary variables in original MHD equations into equilibrium part and perturbed part, and apply an approach in the framework of discontinuous Galerkin (DG) spatial discretization for numerical solutions. This enables high resolution of very small perturbations, and also gives satisfactory non-smooth solutions for large perturbations, which are both broadly concerned in magnetically confined plasma research. Numerical examples demonstrate satisfactory performance of the proposed model clearly. For small perturbations, the results have higher resolution comparing with the original MHD model; for large perturbations, the non-smooth solutions match well with existing references, confirming reliability of the model for instability investigations in magnetically confined plasma numerical research.

Keywords

Discontinuous Galerkin methodconservative perturbed MHD model

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65Z05: Applications to physics 85-08: Computational methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 5 , May 2017 , pp. 1429 - 1448
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Sovinec, C., Glasser, A., Gianakon, T., Barnes, D., Nebel, R., Kruger, S., Schnack, D., Plimpton, S., Tarditi, A., Chu, M. and the NIMROD team, Nonlinear magnetohydrodynamics simulation using high-order finite elements, J. Comput. Phys., 195 (2004), 355–386.CrossRefGoogle Scholar
[2] Wang, S. and Ma, Z., influence of toroidal rotation on resistive tearing modes in tokamaks, Phys. Plasmas., 22(2015), 122504.Google Scholar
[3] Park, W., Belova, E., Fu, G., Tang, X., Strauss, H. and Sugiyama, L., Plasma simulation studies using multilevel physics models, Phys. Plasmas., 6(1999), 1796–1803.CrossRefGoogle Scholar
[4] Biskamp, D., Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, (1993).CrossRefGoogle Scholar
[5] Harten, A., Engquist, B., Osher, S. and Chakravarthy, S.. Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys., 71(1987), 231–303.Google Scholar
[6] Shu, C.-W., Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws, NASA/CR-97-206253, ICASE Report No. 97-65, 1997.Google Scholar
[7] Hu, C. and Shu, C.-W., Weighted Essentially Non-oscillatory Schemes on Triangular Meshes, J. Comput. Phys., 150(1999), 97–127.Google Scholar
[8] Shi, J., Hu, C. and Shu, C.-W., A Technique of Treating Negative Weights in WENO Schemes, J. Comput. Phys., 175(2002), 108–127.Google Scholar
[9] Cockburn, B. and Shu, C.-W., The Runge-Kutta local projection P1-discontinuous-Galerkin finite element method for scalar conservation laws, IMA Preprint Series #388, University of Minnesota, 1988.Google Scholar
[10] Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite elementmethod for conservation laws II: General framework, Math. Comput., 52, 186 (1989), 411–435.Google Scholar
[11] Cockburn, B., Lin, S.-Y. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One dimensional systems, IMA Preprint Series #415, University of Minnesota, 1988.Google Scholar
[12] Cockburn, B., Hou, S. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: The multidimensional case, IMA Preprint Series #513, University of Minnesota, 1989.Google Scholar
[13] Cockburn, B. and Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws V: Multi-dimensional systems, J. Comput. Phys., 141 (1998), 199–224.Google Scholar
[14] Cockburn, B. and Shu, C.-W., Runge-Kutta Discontinuous Galerkin Methods for Convection-dominated Problems, NASA/CR-2000-210624, ICASE Report No. 2000-46.Google Scholar
[15] Shu, C.-W., A brief survey on discontinuous Galerkin methods in computational fluid dynamics, Adv. Mech., 43, 6(2013), 541C554.Google Scholar
[16] Dender, A, Kemm, F., Kröner, D., Munz, C.-D., Schnitzer, T. and Wesenberg, M., Hyperbolic Divergence Cleaning for the MHD Equations, J. Comput. Phys., 175 (2002), 645–673.Google Scholar
[17] Altmann, C., An Explicit Discontinuous Galerkin Scheme with Divergence Cleaning for Magnetohydrodynamics, Lecture notes in Computational Science and Engineering, Vol. 76, Springer, Berlin, 357–364, 2011.Google Scholar
[18] Töth, G., The ∇·B=0 constraint in shock-capturing magnetohydrodynamics codes, J. Comput. Phys., 161 (2000), 605–652.Google Scholar
[19] Li, S., High order central scheme on overlapping cells formagneto-hydrodynamic flows with and without constrained transport method, J. Comput. Phys., 227 (2008), 7368–7393.Google Scholar
[20] Li, F., Xu, L. and Yakovlev, S., Central discontinuous Galerkin methods for ideal MHD equations with exactly divergence-free magnetic field, J. Comput. Phys., 230 (2011), 4828–4847.CrossRefGoogle Scholar
[21] Li, F. and Xu, L., Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations, J. Comput. Phys., 231(2012), 2655–2675.Google Scholar
[22] Li, F. and Shu, C.-W., Locally Divergence-Free Discontinuous Galerkin Methods for MHD Equations, J. Sci. Comput., 22 (2005), 413–442.Google Scholar
[23] Yakovlev, S., Xu, L. and Li, F., Locally divergence-free central discontinuous Galerkin methods for ideal MHD equations, J. Comput. Sci-Neth., 4 (2013), 80–91.Google Scholar
[24] Cockburn, B. and Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141 (1998), 199–224.Google Scholar
[25] Gottlieb, S., Shu, C.-W. and Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001), 89–112.CrossRefGoogle Scholar
[26] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in: Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E., Quarteroni, A. (Eds.), Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, Springer, Berlin, 325–432, 1998.CrossRefGoogle Scholar
[27] Keppens, R., Töth, G., Westermann, R. and Goedbloed, J., Kelvin-Helmholtz instability with parallel and anti-parallel magneti fields, J. Plasma Phys., 61, 1(1999), 1–19.Google Scholar
[28] Longcope, D. and Strauss, H., The coalescence instability and the development of current sheets in two dimensional magnetohydrodynamics, Phys. Fluids B, 5(1993), 2858–2869.Google Scholar