Advertisement

Symplectic integrators with adaptive time step applied to runaway electron dynamics

  • Yanyan Shi
  • Yajuan SunEmail author
  • Yang He
  • Hong Qin
  • Jian Liu
Original Paper
  • 25 Downloads

Abstract

Studying the dynamics of runaway electrons has theoretical and practical significance. As the system is highly relativistic, multi-scale and nonlinear, accurate and efficient numerical methods with long-term stability are necessary. In this paper, we develop symplectic methods with adaptive time step and discuss the choice of step-size functions in accordance with the simulation problems for runaway electrons. In the implementation, in order to explore the practical impact of runaway electron dynamics, we use the electromagnetic field and some parameters often used in the study of plasma problems. Numerical results show that the new derived symplectic methods with adaptive time step exhibit good invariant-preserving property and superior stability over longtime integration. Moreover, with appropriate adaptive technique, the numerical efficiency in simulations is improved apparently. They are illustrated in the numerical experiments.

Keywords

Runaway electrons Symplectic methods Adaptive time step 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Funding information

This research was supported by the National Natural Science Foundation of China (11771436, 11261140328, 11305171), by the CAS Program for Interdisciplinary Collaboration Team, the Foundation for Innovative Research Groups of the NNSFC (11321061), and the ITER-China Program (2014GB124005, 2015GB111003), JSPS-NRF-NSFC A3 Foresight Program in the field of Plasma Physics (NSFC-11261140328).

References

  1. 1.
    Dreicer, H.: Electron and ion runaway in a fully ionized gas. Phys. Rev. 115(2), 238–249 (1959)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kulsrud, R.M., Sun, Y.C., Winsor, N.K., Fallon, H.A.: Runaway electrons in a plasma. Phys. Rev. Lett. 31(11), 690 (1973)CrossRefGoogle Scholar
  3. 3.
    Rosenbluth, M., Putvinski, S.: Theory for avalanche of runaway electrons in tokamaks. Nucl. Fusion 37(10), 1355 (1997)CrossRefGoogle Scholar
  4. 4.
    Bartels, H.W.: Impact of runaway electrons. Fusion Eng. Des. 23(4), 323–328 (1994)CrossRefGoogle Scholar
  5. 5.
    Jaspers, R., Cardozo, N.J.L., Donne, A.J.H., Widdershoven, H.L.M., Finken, K.H.: A synchrotron radiation diagnostic to observe relativistic runaway electrons in a tokamak plasma. Rev. Sci. Instrum. 72(1), 466–470 (2001)CrossRefGoogle Scholar
  6. 6.
    Liu, J., Wang, Y., Qin, H.: Collisionless pitch-angle scattering of runaway electrons. Nucl. Fusion 56(6), 064002 (2016)CrossRefGoogle Scholar
  7. 7.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, vol. 31. Springer Science & Business Media, Berlin (2006)zbMATHGoogle Scholar
  8. 8.
    Guan, X., Qin, H., Fisch, N.J.: Phase-space dynamics of runaway electrons in tokamaks. Phys. Plasmas 17(9), 092502 (2010)CrossRefGoogle Scholar
  9. 9.
    Zhou, Z., He, Y., Sun, Y., Liu, J., Qin, H.: Explicit symplectic methods for solving charged particle trajectories. Phys. Plasmas 24(5), 052507 (2017)CrossRefGoogle Scholar
  10. 10.
    Zhang, R., Wang, Y., He, Y., Xiao, J., Liu, J., Qin, H., Tang, Y.: Explicit symplectic algorithms based on generating functions for relativistic charged particle dynamics in time-dependent electromagnetic field. Phys. Plasmas 25(2), 022117 (2018)CrossRefGoogle Scholar
  11. 11.
    Zhang, R., Liu, J., Qin, H., Wang, Y., He, Y., Sun, Y.: Volume-preserving algorithm for secular relativistic dynamics of charged particles. Phys. Plasmas 22(4), 044501 (2015)CrossRefGoogle Scholar
  12. 12.
    He, Y., Sun, Y., Zhang, R., Wang, Y., Liu, J., Qin, H.: High order volume-preserving algorithms for relativistic charged particles in general electromagnetic fields. Phys. Plasmas 23(9), 092109 (2016)CrossRefGoogle Scholar
  13. 13.
    Wesson, J., Campbell, D.J.: Tokamaks, vol. 149. Oxford University Press, London (2011)Google Scholar
  14. 14.
    Wang, Y., Qin, H., Liu, J.: Multi-scale full-orbit analysis on phase-space behavior of runaway electrons in tokamak fields with synchrotron radiation. Phys. Plasmas 23(6), 062505 (2016)CrossRefGoogle Scholar
  15. 15.
    Liu, C., Qin, H., Hirvijoki, E., Wang, Y., Liu, J.: The role of magnetic moment in the collisionless pitch-angle scattering of runaway electrons. arXiv:1804.01971
  16. 16.
    Wang, Y., Liu, J., Qin, H.: Lorentz covariant canonical symplectic algorithms for dynamics of charged particles. Phys. Plasmas 23(12), 122513 (2016)CrossRefGoogle Scholar
  17. 17.
    Jackson, J.D.: Classical Electrodynamics. Wiley, New York (1999)zbMATHGoogle Scholar
  18. 18.
    Eriksson, L., Porcelli, F.: Dynamics of energetic ion orbits in magnetically confined plasmas. Plasma Phys. Controlled Fusion 43(4), R145 (2001)CrossRefGoogle Scholar
  19. 19.
    Porcelli, F., Eriksson, L.-G., Furno, I.: Topological transitions of fast ion orbits in magnetically confined plasmas. Phys. Lett. A 216(6), 289–295 (1996)CrossRefGoogle Scholar
  20. 20.
    Carbajal, L., Delcastillonegrete, D., Spong, D., Seal, S., Baylor, L.: Space dependent, full orbit effects on runaway electron dynamics in tokamak plasmas. Phys. Plasmas 24(4), 39–S202 (2017)CrossRefGoogle Scholar
  21. 21.
    Liu, J., Qin, H., Wang, Y., Yang, G., Zheng, J., Yao, Y., Yifeng, Z., Liu, Z., Liu, X.: Largest particle simulations downgrade the runaway electron risk for iterGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Yanyan Shi
    • 1
    • 2
  • Yajuan Sun
    • 1
    • 2
    Email author
  • Yang He
    • 3
  • Hong Qin
    • 4
    • 5
  • Jian Liu
    • 5
    • 6
  1. 1.LSEC, ICMSEC, Academy of Mathematics and Systems ScienceCASBeijingChina
  2. 2.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingChina
  3. 3.School of Mathematics and PhysicsUniversity of Science and TechnologyBeijingChina
  4. 4.Plasma Physics LaboratoryPrinceton UniversityPrincetonUSA
  5. 5.Department of Engineering and Applied PhysicsUSTCHefeiChina
  6. 6.Key Laboratory of Geospace EnvironmentCASHefeiPeople’s Republic of China

Personalised recommendations