Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



Discrete gauge invariant approximations of
a time dependent Ginzburg-Landau model of superconductivity

Author: Qiang Du
Journal: Math. Comp. 67 (1998), 965-986
MSC (1991): Primary 65M12, 65M15; Secondary 82D55
MathSciNet review: 1464143
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present here a mathematical analysis of a nonstandard difference method for the numerical solution of the time dependent Ginzburg-Landau models of superconductivity. This type of method has been widely used in numerical simulations of the behavior of superconducting materials. We also illustrate some of their nice properties such as the gauge invariance being retained in discrete approximations and the discrete order parameter having physically consistent pointwise bound.

References [Enhancements On Off] (What's this?)

  • 1. Stephen L. Adler and Tsvi Piran, Relaxation methods for gauge field equilibrium equations, Rev. Modern Phys. 56 (1984), no. 1, 1–40. MR 734670,
  • 2. Zhiming Chen and K.-H. Hoffmann, Numerical studies of a non-stationary Ginzburg-Landau model for superconductivity, Adv. Math. Sci. Appl. 5 (1995), no. 2, 363–389. MR 1360996
  • 3. G. Crabtree, G. Leaf, H. Kaper, V. Vinokur, A. Koshelev, D. Braun, D. Levine, W. Kkwok, and J. Fendrich, Time-dependent Ginzburg-Landau simulations of vortex guidance by twin boundaries, Physica C., 263(1996), pp. 401-408.
  • 4. M. Doria, J. Gubernatis, and D. Rainer, Solving the Ginzburg-Landau equations by simulated annealing, Phys. Rev. B, 41 (1990), pp. 6335-6340.
  • 5. Qiang Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal. 53 (1994), no. 1-2, 1–17. MR 1379180,
  • 6. Q. Du, Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity, Comput. Math. Appl. 27 (1994), no. 12, 119–133. MR 1284135,
  • 7. Qiang Du, Max D. Gunzburger, and Janet S. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev. 34 (1992), no. 1, 54–81. MR 1156289,
  • 8. Qiang Du, Max Gunzburger, and Janet Peterson, Finite element approximation of a periodic Ginzburg-Landau model for type-𝐼𝐼 superconductors, Numer. Math. 64 (1993), no. 1, 85–114. MR 1191324,
  • 9. Q. Du, M. Gunzburger, and J. Peterson, Solving the Ginzburg-Landau equations by finite element methods, Phy. Rev. B., 46(1992), pp. 9027-9034.
  • 10. Q. Du, M. Gunzburger, and J. Peterson, Computational simulation of type-II superconductivity including pinning phenomena, Phy. Rev. B., 51(1995), pp. 16194-16203.
  • 11. F. Lin and Q. Du, Ginzburg-Landau vortices, dynamics, pinning and hysteresis, SIAM Math. Anal. 28 (1997), pp. 1265-1293. CMP 98:02
  • 12. Q. Du, R. Nicolaides, and X. Wu, Analysis and convergence of a covolume approximation of the Ginzburg-Landau model of superconductivity, to appear in SIAM Numer. Anal., (1997).
  • 13. J. Fleckinger-Pelle, H. Kaper, and P. Takac, Dynamics of the Ginzburg-Landau equations of superconductivity, preprint, MCS P565-0296, Argonne National Lab..
  • 14. H. Frahm, S. Ullah and A. Dorsey, Flux dynamics and the growth of the superconducting phase, Phys. Rev. Letters, 66 (1991), pp. 3067-3072.
  • 15. L. Freitag, M. Jones and P. Plassmann, New techniques for parallel simulation of high temperature superconductors, MCS preprint, Argonne National Lab., (1994).
  • 16. J. Garner, M. Spanbauer, R. Benedek, K. Strandburg, S. Wright and P. Plassman, Critical fields of Josephson-coupled superconducting multilayers, preprint.
  • 17. W. Gropp, H. Kaper, G. Leaf, D. Levine, M. Plumbo, and V. Vinokur, Numerical simulation of vortex dynamics in type-II superconductors, J. Comp. Phys. 123(1996), pp. 254-266. CMP 96:07
  • 18. H. Kaper, and M. Kwong, Vortex configurations in type-II superconducting films, J. Com. Phys. 119(1995), pp. 120-131.
  • 19. M. K. Wong, Sweeping algorithms for inverting the discrete Ginzburg-Landau operators, Applied Math. Comp., 53(1993), pp.12d90-150.
  • 20. San Yih Lin and Yi Song Yang, Computation of superconductivity in thin films, J. Comput. Phys. 89 (1990), no. 2, 257–275. MR 1067047,
  • 21. F. Liu, M. Mondello and N. Goldenfeld, Kinetics of the superconducting transition, Phys. Rev. Lett., 66 (1991), pp. 3071-3074.
  • 22. R. A. Nicolaides, Direct discretization of planar div-curl problems, SIAM J. Numer. Anal. 29 (1992), no. 1, 32–56. MR 1149083,
  • 23. R. A. Nicolaides and X. Wu, Analysis and convergence of the MAC scheme. II. Navier-Stokes equations, Math. Comp. 65 (1996), no. 213, 29–44. MR 1320897,
  • 24. Qi Tang and S. Wang, Time dependent Ginzburg-Landau equations of superconductivity, Phys. D 88 (1995), no. 3-4, 139–166. MR 1360881,
  • 25. M. Tinkham, Introduction to Superconductivity, 2nd edition, McGraw-Hill, New York, 1994.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 65M12, 65M15, 82D55

Retrieve articles in all journals with MSC (1991): 65M12, 65M15, 82D55

Additional Information

Qiang Du
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong and Department of Mathematics, Iowa State University, Ames, Iowa 50011

Keywords: Ginzburg-Landau model of superconductivity, time dependent equations, nonstandard difference approximations, gauge invariance, convergence, error estimates
Received by editor(s): June 13, 1996
Received by editor(s) in revised form: February 19, 1997
Additional Notes: Research is supported in part by the U. S. National Science Foundation grant MS-9500718 and in part by the HKUST grant DAG 95/96.SC18.
Article copyright: © Copyright 1998 American Mathematical Society