Skip to main content
SpringerLink
Log in

A Fully Implicit Parallel Solver for MHD Instabilities in a Tokamak

  • Research
  • Published:
Journal of Fusion Energy Aims and scope Submit manuscript

Abstract

Aiming at the long-term and high-precision simulation of the magnetohydrodynamic (MHD) instabilities in the tokamak model, we developed a parallelized solver based on a fully implicit difference scheme. A 4th-order precision difference scheme and the Newton–Krylov method are employed in the proposed solver for both the flow and the electromagnetic field. To achieve high parallel efficiency, we adopt a strategy based on the spatial domain decomposition to partition the large Jacobian matrices in the iteration, and a buffer area based on the grid density is utilized to minimize the memory and time consumption. The accuracy of the methodology is verified, and the numerical results are validated by comparison with recognized results. The numerical results of the tearing mode instability in the tokamak model have demonstrated the precision and reliability of the algorithm, and the high parallel efficiency has been proven by the scalability test on the platform with up to 1280 threads, showing significant potential in the large-scale simulation of MHD problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data Availability

The datasets used or analyzed during the current study are available from the corresponding author upon reasonable request.

References

  1. T.C. Hender, J.C. Wesley, J. Bialek, A. Bondeson, A.H. Boozer, R.J. Buttery, A. Garofalo, T.P. Goodman, R.S. Granetz, Y. Gribov, Chapter 3: MHD stability, operational limits and disruptions. Nucl. Fusion 39(12), 2251–2389 (1999)

    Article  Google Scholar 

  2. R.J.L. Haye, D.A. Humphreys, J.R. Ferron, T.C. Luce, F.W. Perkins, C.C. Petty, R. Prater, E.J. Strait, A.S. Welander, Higher stable beta by use of pre-emptive electron cyclotron current drive on DIII-D. Nucl. Fusion 45(11), L37–L41 (2005)

    Article  Google Scholar 

  3. K. Nagasaki, A. Isayama, S. Ide, Team, J. T., Stabilization effect of early ECCD on a neoclassical tearing mode in the JT-60U tokamak. Nucl. Fusion 43(10), L7–L10 (2003)

    Article  Google Scholar 

  4. Y. Yoshioka, S. Kinoshha, T. Kobayashi, Numerical study of magnetic island suppression by RF waves in large tokamaks. Nucl. Fusion 24(5), 565–572 (1984)

    Article  Google Scholar 

  5. R.J. La Haye, Neoclassical tearing modes and their control. Phys. Plasmas 13(5), 055501 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  6. M. Chu, H. Ikezi, T. Jensen, Suppression of tearing mode growth by externally imposed resonant magnetic islands. Phys. Fluids 27(2), 472–474 (1984)

    Article  ADS  Google Scholar 

  7. E. Lazzaro, M. Nave, Feedback control of rotating resistive modes. Phys. Fluids 31(6), 1623–1629 (1988)

    Article  ADS  Google Scholar 

  8. La Haye, R. J., "Stabilization of neoclassical tearing modes in tokamaks by radio frequency current drive," Proceedings AIP Conference Proceedings, AIP, pp. 361–368.

  9. R. Buttery, R. la Haye, P. Gohil, G. Jackson, H. Reimerdes, E. Strait, Team D.-D, The influence of rotation on the β N threshold for the 2/1 neoclassical tearing mode in DIII-D. Phys. Plasmas 15(5), 056115 (2008)

    Article  ADS  Google Scholar 

  10. A. Aydemir, D. Barnes, Three-dimensional nonlinear incompressible MHD calculations. J. Comput. Phys. 53(1), 100–123 (1984)

    Article  ADS  Google Scholar 

  11. Z. Ma, A. Bhattacharjee, Hall magnetohydrodynamic reconnection: the geospace environment modeling challenge. J. Geophys. Res. Space Phys. 106(A3), 3773–3782 (2001)

    Article  ADS  Google Scholar 

  12. J.W. Haverkort, H.J. de Blank, G.T.A. Huysmans, J. Pratt, B. Koren, Implementation of the full viscoresistive magnetohydrodynamic equations in a nonlinear finite element code. J. Comput. Phys. 316, 281–302 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  13. W. Zhang, Z.W. Ma, S. Wang, Hall effect on tearing mode instabilities in tokamak. Phys. Plasmas 24(10), 102510 (2017)

    Article  ADS  Google Scholar 

  14. H.R. Strauss, Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19(1), 134–140 (1976)

    Article  ADS  Google Scholar 

  15. W. Park, D. Monticello, R. White, S. Jardin, Nonlinear saturation of the internal kink mode. Nucl. Fusion 20(9), 1181 (1980)

    Article  ADS  Google Scholar 

  16. C. Cheng, M. Chance, NOVA: a nonvariational code for solving the MHD stability of axisymmetric toroidal plasmas. J. Comput. Phys. 71(1), 124–146 (1987)

    Article  ADS  Google Scholar 

  17. H. Lütjens, J.-F. Luciani, The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas. J. Comput. Phys. 227(14), 6944–6966 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  18. B.D. Dudson, M.V. Umansky, X.Q. Xu, P.B. Snyder, H.R. Wilson, BOUT++: a framework for parallel plasma fluid simulations. Comput. Phys. Commun. 180(9), 1467–1480 (2009)

    Article  ADS  Google Scholar 

  19. H. Lütjens, J.-F. Luciani, XTOR-2F: a fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks. J. Comput. Phys. 229(21), 8130–8143 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  20. S. Wang, Z. Ma, Influence of toroidal rotation on resistive tearing modes in tokamaks. Phys. Plasmas 22(12), 122504 (2015)

    Article  ADS  Google Scholar 

  21. de Moura, C. A., and Kubrusly, C. S., 2012, The Courant–Friedrichs–Lewy (CFL) Condition: 80 Years After Its Discovery, Birkhäuser Boston.

  22. S. Jardin, Computational methods in plasma physics (CRC Press, 2010)

    Book  Google Scholar 

  23. B. Coppi, J.M. Greene, J.L. Johnson, Resistive instabilities in a diffuse linear pinch. Nucl. Fusion 6(2), 101 (1966)

    Article  Google Scholar 

  24. A. Glasser, J. Greene, J. Johnson, Resistive instabilities in general toroidal plasma configurations. Phys. Fluids 18(7), 875–888 (1975)

    Article  ADS  Google Scholar 

  25. S.C. Jardin, Review of implicit methods for the magnetohydrodynamic description of magnetically confined plasmas. J. Comput. Phys. 231(3), 822–838 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  26. M.M. Lopes, R. Deiterding, A.K. Fontes Gomes, O. Mendes, M.O. Domingues, An ideal compressible magnetohydrodynamic solver with parallel block-structured adaptive mesh refinement. Comput. Fluids 173, 293–298 (2018)

    Article  MathSciNet  Google Scholar 

  27. D.S. Harned, W. Kerner, Semi-implicit method for three-dimensional compressible magnetohydrodynamic simulation. J. Comput. Phys. 60(1), 62–75 (1985)

    Article  ADS  Google Scholar 

  28. D.D. Schnack, D.C. Barnes, Z. Mikic, D.S. Harned, E.J. Caramana, Semi-implicit magnetohydrodynamic calculations. J. Comput. Phys. 70(2), 330–354 (1987)

    Article  ADS  Google Scholar 

  29. D.S. Harned, D.D. Schnack, Semi-implicit method for long time scale magnetohydrodynamic computations in three dimensions. J. Comput. Phys. 65(1), 57–70 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  30. K. Lerbinger, J.F. Luciani, A new semi-implicit method for MHD computations. J. Comput. Phys. 97(2), 444–459 (1991)

    Article  ADS  Google Scholar 

  31. 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, M.S. Chu, Nonlinear magnetohydrodynamics simulation using high-order finite elements. J. Comput. Phys. 195(1), 355–386 (2004)

    Article  ADS  Google Scholar 

  32. H. Lütjens, J.-F. Luciani, "XTOR-2F: a fully implicit Newton-Krylov solver applied to nonlinear 3D extended MHD in tokamaks. J Comput Phys 229, 8130–8143 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  33. A. Marx, H. Lütjens, Hybrid parallelization of the XTOR-2F code for the simulation of two-fluid MHD instabilities in tokamaks. Comput. Phys. Commun. 212, 90–99 (2017)

    Article  ADS  Google Scholar 

  34. H.W. Zhang, J. Zhu, Z.W. Ma, G.Y. Kan, X. Wang, W. Zhang, Acceleration of three-dimensional Tokamak magnetohydrodynamical code with graphics processing unit and OpenACC heterogeneous parallel programming. Int. J. Comput. Fluid Dyn. 33(10), 393–406 (2019)

    MathSciNet  Google Scholar 

  35. Arge, E., Bruaset, A. M., and Langtangen, H. P., 1997, Modern software tools for scientific computing.

Download references

Acknowledgements

Haowei Zhang provided technical suggestions to this work.

Funding

This research was funded by the national key R&D program for international collaboration, under grant 2020YFA0712502. The Natural Science Foundation of China (NSFC), grant 11972384, and Guangdong Science and Technology Fund, grant 2021B1515310001, also supported this work.

Author information

Authors and Affiliations

  1. School of Aeronautics and Astronautics, Sun Yet-Sen University, Guangzhou, 510275, China

    Qinghe Yao, Zichao Jiang, Zhuolin Wang & Junyang Jiang

  2. Department of Physics, Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou, 310027, China

    Zhiwei Ma

Authors
  1. Qinghe Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zichao Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhuolin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Junyang Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zhiwei Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Prof. QY led the research of this project and was responsible for the writing of the central part of the paper. Dr. ZJ and Mr. JJ finished the data collation and analysis. Prof. ZM provided theoretical support for this work.

Corresponding author

Correspondence to Qinghe Yao.

Ethics declarations

Competing interests

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yao, Q., Jiang, Z., Wang, Z. et al. A Fully Implicit Parallel Solver for MHD Instabilities in a Tokamak. J Fusion Energ 42, 31 (2023). https://doi.org/10.1007/s10894-023-00369-5

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10894-023-00369-5

Keywords

Search

Navigation