Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Discrete gauge invariant approximations of a time dependent Ginzburg-Landau model of superconductivity
HTML articles powered by AMS MathViewer

by Qiang Du PDF
Math. Comp. 67 (1998), 965-986 Request permission

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
  • Stephen L. Adler and Tsvi Piran, Relaxation methods for gauge field equilibrium equations, Rev. Modern Phys. 56 (1984), no. 1, 1–40. MR 734670, DOI 10.1103/RevModPhys.56.1
  • 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
  • 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.
  • M. Doria, J. Gubernatis, and D. Rainer, Solving the Ginzburg-Landau equations by simulated annealing, Phys. Rev. B, 41 (1990), pp. 6335–6340.
  • 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, DOI 10.1080/00036819408840240
  • 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, DOI 10.1016/0898-1221(94)90091-4
  • 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, DOI 10.1137/1034003
  • Qiang Du, Max Gunzburger, and Janet Peterson, Finite element approximation of a periodic Ginzburg-Landau model for type-$\textrm {II}$ superconductors, Numer. Math. 64 (1993), no. 1, 85–114. MR 1191324, DOI 10.1007/BF01388682
  • Q. Du, M. Gunzburger, and J. Peterson, Solving the Ginzburg-Landau equations by finite element methods, Phy. Rev. B., 46(1992), pp. 9027-9034.
  • Q. Du, M. Gunzburger, and J. Peterson, Computational simulation of type-II superconductivity including pinning phenomena, Phy. Rev. B., 51(1995), pp. 16194-16203.
  • F. Lin and Q. Du, Ginzburg-Landau vortices, dynamics, pinning and hysteresis, SIAM Math. Anal. 28 (1997), pp. 1265–1293.
  • 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).
  • J. Fleckinger-Pelle, H. Kaper, and P. Takac, Dynamics of the Ginzburg-Landau equations of superconductivity, preprint, MCS P565-0296, Argonne National Lab..
  • H. Frahm, S. Ullah and A. Dorsey, Flux dynamics and the growth of the superconducting phase, Phys. Rev. Letters, 66 (1991), pp. 3067-3072.
  • L. Freitag, M. Jones and P. Plassmann, New techniques for parallel simulation of high temperature superconductors, MCS preprint, Argonne National Lab., (1994).
  • J. Garner, M. Spanbauer, R. Benedek, K. Strandburg, S. Wright and P. Plassman, Critical fields of Josephson-coupled superconducting multilayers, preprint.
  • 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.
  • H. Kaper, and M. Kwong, Vortex configurations in type-II superconducting films, J. Com. Phys. 119(1995), pp. 120–131.
  • M. K. Wong, Sweeping algorithms for inverting the discrete Ginzburg-Landau operators, Applied Math. Comp., 53(1993), pp.12d90-150.
  • San Yih Lin and Yi Song Yang, Computation of superconductivity in thin films, J. Comput. Phys. 89 (1990), no. 2, 257–275. MR 1067047, DOI 10.1016/0021-9991(90)90144-P
  • F. Liu, M. Mondello and N. Goldenfeld, Kinetics of the superconducting transition, Phys. Rev. Lett., 66 (1991), pp. 3071–3074.
  • R. A. Nicolaides, Direct discretization of planar div-curl problems, SIAM J. Numer. Anal. 29 (1992), no. 1, 32–56. MR 1149083, DOI 10.1137/0729003
  • 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, DOI 10.1090/S0025-5718-96-00665-5
  • Qi Tang and S. Wang, Time dependent Ginzburg-Landau equations of superconductivity, Phys. D 88 (1995), no. 3-4, 139–166. MR 1360881, DOI 10.1016/0167-2789(95)00195-A
  • 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
  • MR Author ID: 191080
  • Email: madu@uxmail.ust.hk
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 965-986
  • MSC (1991): Primary 65M12, 65M15; Secondary 82D55
  • DOI: https://doi.org/10.1090/S0025-5718-98-00954-5
  • MathSciNet review: 1464143