×

Automatic exploration techniques of numerical bifurcation diagrams illustrated by the Ginzburg-Landau equation. (English) Zbl 1431.37066

Summary: This paper considers the extreme type-II Ginzburg-Landau equations, a nonlinear PDE model that describes the states of a wide range of superconductors. For two-dimensional domains, a robust method is developed that performs a numerical continuation of the equations, automatically exploring the whole solution landscape. The strength of the applied magnetic field is used as the bifurcation parameter. Our branch switching algorithm is based on Lyapunov-Schmidt reduction, but we will show that for an important class of domains an alternative method based on the equivariant branching lemma can be applied as well. The complete algorithm has been implemented in Python and tested for multiple examples. For each example a complete solution landscape was constructed, showing the robustness of the algorithm.

MSC:

37M20 Computational methods for bifurcation problems in dynamical systems
35Q56 Ginzburg-Landau equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys. JETP, 5 (1957), pp. 1174-1182.
[2] B. Baelus and F. Peeters, Dependence of the vortex configuration on the geometry of mesoscopic flat samples, Phys. Rev. B, 65 (2002), pp. 1-12.
[3] W. Beyn, A. Champneys, E. Doedel, W. Govaerts, Y. Kuznetsov, and B. Sandstede, Numerical Continuation, and Computation of Normal Forms, in Handbook of Dynamical Systems, Vol. 2, Elsevier, New York, 2002, pp. 149-219. · Zbl 1034.37048
[4] E. H. Boutyour, H. Zahrouni, M. Potier-Ferry, and M. Boudi, Bifurcation points and bifurcated branches by an asymptotic numerical method and Padé approximants, Internat. J. Numer. Methods Engrg., 60 (2004), pp. 1987-2012. · Zbl 1069.74020
[5] T. Chang and D. C. Resasco, Generalized deflated block-elimination, SIAM J. Numer. Anal., 23 (1986), pp. 913-924. · Zbl 0624.65024
[6] E. Charalampidis, P. Kevrekidis, and P. Farrell, Computing stationary solutions of the two-dimensional Gross-Pitaevskii equation with deflated continuation, Commun. Nonlinear Sci. Numer. Simul., 54 (2018), pp. 482-499. · Zbl 1510.35291
[7] R. Córdoba, T. I. Baturina, J. Sesé, et al., Magnetic field-induced dissipation-free state in superconducting nanostructures, Nature Commun., 4 (2013).
[8] G. Cramer, Introduction à l’analyse des lignes courbes algébriques, chez les frères Cramer et C. Philibert, 1750.
[9] J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997. · Zbl 0879.65017
[10] A. Dhooge, W. Govaerts, and Y. Kuznetsov, MatCont: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), pp. 141-164. · Zbl 1070.65574
[11] E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, Congr. Numer., 30 (1981), pp. 265-284. · Zbl 0511.65064
[12] M. Dorojevets and Z. Chen, Fast pipelined storage for high-performance energy-efficient computing with superconductor technology, in Proceedings of the 12th International Conference Expo on Emerging Technologies for a Smarter World, 2015, pp. 1-6, https://doi.org/10.1109/CEWIT.2015.7338159.
[13] D. Draelants, P. Klosiewicz, J. Broeckhove, and W. Vanroose, Solving general auxin transport models with a numerical continuation toolbox in Python: PyNCT, in Hybrid Systems Biology, Lecture Notes in Comput. Sci., 9271, Springer, New York, 2015, pp. 211-225. · Zbl 1412.92061
[14] Q. Du, M. Gunzburger, and J. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev., 34 (1992), pp. 54-81. · Zbl 0787.65091
[15] L. Embon, Y. Anahory, and Ž. L. Jelić, Imaging of super-fast dynamics and flow instabilities of superconducting vortices, Nature Commun., 8 (2017), 85.
[16] T. Filippov, A. Sahu, A. F. Kirichenko, I. V. Vernik, M. Dorojevets, C. L. Ayala, and O. A. Mukhanov, 20 GHz operation of an asynchronous wave-pipelined RSFQ arithmetic-logic unit, Physics Procedia, 36 (2012), pp. 59-65.
[17] B. Fornberg, Generation of finite difference formulas on arbitrarily spaced grids, Math. Comp., 51 (1988), pp. 699-706. · Zbl 0701.65014
[18] A. Gaul, M. Gutknecht, J. Liesen, and R. Nabben, Deflated and Augmented Krylov Subspace Methods: Basic Facts and a Breakdown-free Deflated MINRES, in MINRES, MATHEON preprint 759, Technical University Berlin, 2011. · Zbl 1273.65049
[19] A. Gaul, M. Gutknecht, J. Liesen, and R. Nabben, A framework for deflated and augmented Krylov subspace methods, SIAM J. Matrix Anal. Appl., 34 (2013), pp. 495-518. · Zbl 1273.65049
[20] A. Gaul and N. Schlömer, Preconditioned Recycling Krylov Subspace Methods for Self-Adjoint Problems, https://arxiv.org/abs/1208.0264, 2012. · Zbl 1327.65059
[21] A. Gaul and N. Schlömer, PyNosh: Python Framework for Nonlinear Schrödinger Equations, https://github.com/nschloe/pynosh, 2013.
[22] M. Golubitsky and I. Stewart, The Symmetry Perspective, Birkhäuser, Basel, 2002. · Zbl 1031.37001
[23] B. Goodman, Type II superconductors, Rep. Progr. Phys., 29 (1966), pp. 445-483.
[24] W. Govaerts, Stable solvers and block elimination for bordered systems, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 469-483. · Zbl 0736.65015
[25] R. Groh, D. Avitabile, and A. Pirrera, Generalised path-following for well-behaved nonlinear structures, Comput. Methods Appl. Mech. Engrg, 331 (2018), pp. 394-426. · Zbl 1439.74144
[26] D. S. Holmes, A. L. Ripple, and M. A. Manheimer, Energy-efficient superconducting computing-power budgets and requirements, IEEE Trans. Appl. Superconductivity, 23 (2013), https://doi.org/10.1109/TASC.2013.2244634.
[27] R. Hoyle, Pattern Formation, Cambridge University Press, Cambridge, UK, 2006. · Zbl 1087.00001
[28] J. Huitfeldt, Nonlinear Eigenvalue Problems–Prediction of Bifurcation Points and Branch Switching, Tech. report, Chalmers University of Technology and University of Göteborg, 1991.
[29] H. Keller, Lectures on Numerical Methods in Bifurcation Problems, Springer-Verlag, Berlin, 1986.
[30] R. Kleiner, D. Koelle, F. Ludwig, and J. Clarke, Superconducting quantum interference devices: State of the art and applications, Proc. IEEE, 92 (2004), pp. 1534-1548.
[31] R. Kouhia and M. Mikkola, Tracing the equilibrium path beyond compound critical points, Internat. J. Numer. Methods Engrg., 46 (1999), pp. 1049-1074. · Zbl 0967.74065
[32] Z. Mei, Numerical Bifurcation Analysis for Reaction-Diffusion Equations, Springer, New York, 2000. · Zbl 0952.65105
[33] Z. Mei and A. Schwarzer, Scaling solution branches of one-parameter bifurcation problems, J. Math. Anal. Appl., 204 (1996), pp. 102-123. · Zbl 0873.35008
[34] A. Murphy, D. V. Averin, and A. Bezryadin, Nanoscale superconducting memory based on the kinetic inductance of asymmetric nanowire loops, New J. Phys., 19 (2017).
[35] A. Salinger et al., Loca 1.1 Library of Continuation Algorithms: Theory and Implementation Manual, Tech. report SAND2002-0396, Sandia National Laboratories, Albuquerque, NM, 2002.
[36] A. G. Salinger, E. A. Burroughs, R. P. Pawlowski, E. T. Phipps, and L. A. Romero, Bifurcation tracking algorithms and software for large scale applications, Internat. J. Bifur. Chaos, 15 (2005), pp. 1015-1032. · Zbl 1076.65118
[37] N. Schlömer, D. Avitabile, and W. Vanroose, Numerical bifurcation study of superconducting patterns on a square, SIAM J. Appl. Dyn. Syst., 11 (2012), pp. 447-477. · Zbl 1253.82125
[38] N. Schlömer, M. V. Milošev\ić, B. Partoens, and W. Vanroose, Exploration of Stable and Unstable Vortex Patterns in a Superconductor Under a Magnetic Disc, https://arxiv.org/abs/1304.8081, 2013.
[39] N. Schlömer and W. Vanroose, An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg-Landau problem, J. Comput. Phys., 234 (2013), pp. 560-572. · Zbl 1284.35412
[40] R. Seydel, From Equilibrium to Chaos: Practical Bifurcation and Stability Analysis, Springer, New York, 1988. · Zbl 0652.34059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.