Hostname: page-component-7479d7b7d-767nl Total loading time: 0 Render date: 2024-07-09T15:04:26.588Z Has data issue: false hasContentIssue false
  • English
  • Français

pde2path - A Matlab Package for Continuation and Bifurcation in 2D Elliptic Systems

Published online by Cambridge University Press:  28 May 2015

Hannes Uecker ,
Daniel Wetzel  and
Jens D. M. Rademacher
Hannes Uecker*
Affiliation:
Institut für Mathematik, Universität Oldenburg, 26111 Oldenburg, Germany
Daniel Wetzel*
Affiliation:
Institut für Mathematik, Universität Oldenburg, 26111 Oldenburg, Germany
Jens D. M. Rademacher*
Affiliation:
Universität Bremen, Fachbereich Mathematik, Postfach 33 04 40, 28359 Bremen, Germany
*
Corresponding author.Email address:hannes.uecker@uni-oldenburg.de
Corresponding author.Email address:daniel.wetzel@uni-oldenburg.de
Corresponding author.Email address:rademach@math.uni-bremen.de
Get access

Abstract

pde2path is a free and easy to use Matlab continuation/bifurcation package for elliptic systems of PDEs with arbitrary many components, on general two dimensional domains, and with rather general boundary conditions. The package is based on the FEM of the Matlab pdetoolbox, and is explained by a number of examples, including Bratu’s problem, the Schnakenberg model, Rayleigh-Bénard convection, and von Karman plate equations. These serve as templates to study new problems, for which the user has to provide, via Matlab function files, a description of the geometry, the boundary conditions, the coefficients of the PDE, and a rough initial guess of a solution. The basic algorithm is a one parameter arclength-continuation with optional bifurcation detection and branch-switching. Stability calculations, error control and mesh-handling, and some elementary time-integration for the associated parabolic problem are also supported. The continuation, branch-switching, plotting etc are performed via Matlab command-line function calls guided by the AUTO style. The software can be downloaded from www.staff.uni-oldenburg.de/hannes.uecker/pde2path, where also an online documentation of the software is provided such that in this paper we focus more on the mathematics and the example systems.

Keywords

35J4735J6035B2265N30Elliptic systemscontinuation and bifurcationfinite element method
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 7 , Issue 1 , February 2014 , pp. 58 - 106
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Allgower, E. and Georg, K., Numerical Continuation Methods, Springer, 1990.Google Scholar
[2] R. E. Bank, PLTMG, http://ccom.ucsd.edu/~reb/software.html.Google Scholar
[3] Beyn, W.-J., Kless, W., and hÜmmler, V, Continuation of low-dimensional invariant subspaces in dynamical systems of large dimension, Fiedler, , Bernold, (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer Berlin, 47–72, 2001.Google Scholar
[4] Bindel, D., Demmel, J., and Friedman, M., Continuation of invariant subspaces in large bifurcation problems, SIAM J. Sci. Comput., 30(2) (2008), pp. 637–656.CrossRefGoogle Scholar
[5] Bratu, G., Sur les equations integrales non lineares, Bull. Soc. Math. France, 42 (1914), pp. 113–142.Google Scholar
[6] Chien, C.-S., Gong, S.-Y., and Mei, Z., Mode jumping in the von Kármán equations, SIAM J. Sci. Comput., 22(4) (2000), pp. 1354–1385.Google Scholar
[7] Doedel, E. J., Lecture notes on numerical analysis of nonlinear equations, Krauskopf, , Bernd, (ed.) et al., Numerical Continuation Methods for Dynamical Systems, Path Following and Boundary Value Problems, 1–49, Springer, 2007.Google Scholar
[8] Dohnal, T. and Uecker, H., Coupled mode equations and gap solitonsfor the 2D Gross-Pitaevsky equation with a non-separable periodic potentia, Phys. D, 238 (2009), pp. 860–879.Google Scholar
[9] Engeln-MÜllges, G. and Uhlig, F., Numerical Algorithms with C, Springer Berlin, 1996.Google Scholar
[10] Ermentrout, B., XPP-Aut, www.math.pitt.edu/~bard/xpp/xpp.htm.Google Scholar
[11] Doedel, E. et al., AUTO: continuation and bifurcation software for ordinary differential equations, http://cmvl.cs.concordia.ca/auto/.Google Scholar
[12] Georg, K., Matrix-free numerical continuation and bifurcation, Numer. Funct. Anal. Optim., 22 (2001), pp. 303–320.Google Scholar
[13] Gervais, J., Oukit, A., and Pierre, R., Finite element analysis of the buckling and mode jumping of a rectangular plate, Dyn. Stab. Syst., 12(3) (1997), pp. 161–185.Google Scholar
[14] Govaerts, W., MatCont, http://sourceforge.net/projects/matcont/. Google Scholar
[15] Govaertsm, W., Numerical methods for bifurcations of dynamical equilibria, SIAM, 2000.Google Scholar
[16] Heil, M. and Hazel, A.L., oomph, http://oomph-lib.maths.man.ac.uk/doc/html/.Google Scholar
[17] Hirschberg, P. and Knobloch, E., Mode interactions in large aspect ratio convection, J. Nonlinear Sci., 7 (1997), pp. 537–556.Google Scholar
[18] Iron, D. and Ward, M. J., A metastable spike solution for a nonlocal reaction-diffusion model, SIAP, 60(3) (2000), pp. 778–802.CrossRefGoogle Scholar
[19] Keller, H. B., Numerical solution of bifurcation and nonlinear eigenvalue problems, App. Bifur. Theory Proc. Adv. Semin., Madison/Wis, 1976, 359–384, 1977.Google Scholar
[20] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, 3rd ed, Springer, 2004.Google Scholar
[21] Lashkin, V.M., Two-dimensional multisolitons and azimuthons in Bose-Einstein condensates, Phys. Rev. A, 77 (2008), 025602.Google Scholar
[22] Lashkin, V.M., Ostrovskaya, E. A., Desyatnikov, A. S., and Kivshar, Yu.S., Vector azimuthons in two-component Bose-Einstein condensates, Phys. Rev. A, 80 (2009), 013615–6.Google Scholar
[23] Maini, P. K., Myerscough, M. R., Murray, J.D., and Winters, K. H., Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern formation, Bull. Math. Biol., 53 (1991), pp. 701–719.Google Scholar
[24] Mittelmann, H. D., Multilevel continuation techniques for nonlinear boundary value problems with parameter dependence, Appl. Math. Comput., 19 (1986), pp. 265–282.Google Scholar
[25] Murray, J. D., Mathematical Biology, Springer-Verlag, Berlin, 1989.CrossRefGoogle Scholar
[26] pde2path homepage, www.staff.uni-oldenburg.de/hannes.uecker/pde2path.Google Scholar
[27] PrÜfert, U., PDE Toolbox, tu-freiberg.de/fakult4/iec/mitarbeiter/profil/Uwe_Pruefert_ntfd.Google Scholar
[28] Rose, K. C., Battogtokh, D., Mikhailov, A., Imbihl, R., Engel, W, and Bradshaw, A. M., Cellular structures in catalytic reactions with global coupling, Phys. Rev. Lett., 76 (1996), pp. 3582–3585.CrossRefGoogle ScholarPubMed
[29] Salinger, A., LOCA, http://www.es.sandia.gov/LOCA/.Google Scholar
[30] Schaeffer, D. and Golubitsky, M., Boundary conditions and mode jumping in the buckling of a rectangular plate, Commun. Math. Phys., 69 (1979), pp. 209–236.Google Scholar
[31] Schilder, F. and Dankowicz, H., coco, http://sourceforge.net/projects/cocotooIs/.Google Scholar
[32] Schnakenberg, J., Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol., 81(3) (1979), pp. 389–400.Google Scholar
[33] Seydel, R., Practical Bifurcation and Stability Analysis, 3rd ed., Springer, 2010.Google Scholar
[34] Stollenwerk, L., Gurevich, S. V., Laven, J. G., and Purwins, H.-G., Transition from bright to dark dissipative solitons in dielectric barrier gas-discharge, Euro. Phys. J. D, 42 (2007), pp. 273–278.Google Scholar
[35] Matlab PDE Toolbox, online documentation.Google Scholar
[36] Uecker, H. and Wetzel, D., Numerical results for snaking of patterns over patterns in some 2D Selkov-Schnakenberg Reaction-Diffusion systems, SIAM J. Appl. Dyna. Syst., to appear, 2014.Google Scholar
[37] Woesler, R., SchÜtz, P., Bode, M., Or-Guil, M., and Purwins, H.-G., Oscillations of fronts and front pairs in two- and three-component reaction-diffusion systems, Phys. D, 91 (1996), pp. 376–405.Google Scholar
[38] www.sercoassurance.com/entwife/introduction.html, 2001.Google Scholar