Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T02:32:29.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Moving Finite Element Simulations for Reaction-Diffusion Systems

Published online by Cambridge University Press:  03 June 2015

Guanghui Hu ,
Zhonghua Qiao  and
Tao Tang
Guanghui Hu*
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA
Zhonghua Qiao*
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong
Tao Tang*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong
*
URL:myweb.polyu.edu.hk/∼zqiao/, Email: ghhu@math.msu.edu
Corresponding author. Email: zqiao@inet.polyu.edu.hk
Email: ttang@math.hkbu.edu.hk
Get access

Abstract

This work is concerned with the numerical simulations for two reaction-diffusion systems, i.e., the Brusselator model and the Gray-Scott model. The numerical algorithm is based upon a moving finite element method which helps to resolve large solution gradients. High quality meshes are obtained for both the spot replication and the moving wave along boundaries by using proper monitor functions. Unlike [33], this work finds out the importance of the boundary grid redistribution which is particularly important for a class of problems for the Brusselator model. Several ways for verifying the quality of the numerical solutions are also proposed, which may be of important use for comparisons.

Keywords

65M5065M60Reaction-diffusion systemsBrusselator modelGray-Scott modelmoving finite element method
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 3 , June 2012 , pp. 365 - 381
Copyright
Copyright © Global-Science Press 2012

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]Van Dam, A. and Zegeling, P. A., Balanced monitoring of flow phenomena in moving mesh methods, Commun. Comput. Phys., 7 (2010), pp. 138–170.Google Scholar
[2]Doelman, A., Kaper, T. J. and Zegeling, P. A., Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity., 10 (1997), pp. 523–563.Google Scholar
[3]Gierer, A. and Meinhardt, H., A theory of biological pattern formation, Kybernetik., 12 (1972), pp. 30–39.CrossRefGoogle ScholarPubMed
[4]Granero, M. I., Porati, A. and Zanacca, D., A bifurcation analysis of pattern formation in a diffusion governed morphogenetic field, J. Math. Biol., 4 (1977), pp. 21–27.CrossRefGoogle Scholar
[5]Gray, P. and Scott, S. K., Sustained oscillations and other exotic patterns of behavior in isothermal reaction, J. Phys. Chem., 59 (1985), pp. 22–32.Google Scholar
[6]Hairer, E., Norsett, S. P. and Wanner, G., Solving Ordinary Differential Equations, I. Nonstiff Problems, Springer, Berlin, 1987.Google Scholar
[7]Harrison, L. and Holloway, D., Order and localization in reaction-diffusion patter, Phys. A., 222 (1995), pp. 210–233.Google Scholar
[8]Holloway, D., Reaction-Diffusion Theory of Localized Structures with Applications to Vertebrate Organogenesis, Ph. D. thesis, Department of Chemistry, University of British Columbia, Vancouver, Canada, 1995.Google Scholar
[9]Hunding, A., Morphogen prepatterns during mitosis and cytokinesis in flattened cells: three-dimensional Turing structures of reaction-diffusion systems in cylindrical coordinates, J. Theoret. Biol., 114 (1985), pp. 571–588.Google Scholar
[10]Hundsdorfer, W. H. and Verwer, J. G., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer, 2003.Google Scholar
[11]Iron, D. and Ward, M. J., The dynamics of multispike solutions to the one-dimensional Gierer-Meinhardt model, SIAM J. Appl. Math., 62 (2002), pp. 1924–1951.CrossRefGoogle Scholar
[12]Iron, D., Ward, M. J. and Wei, J., The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D., 150 (2001), pp. 25–62.CrossRefGoogle Scholar
[13]Li, R. and Liu, W., Google Scholar
[14]Li, R., Tang, T. and Zhang, P. W., Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys., 170 (2001), pp. 562–588.Google Scholar
[15]Li, R., Tang, T. and Zhang, P. W., A moving mesh finite element algorithm for singular problems in two and three space dimensions, J. Comput. Phys., 177 (2002), pp. 365–393.Google Scholar
[16]Madzvamuse, A., Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J. Comput. Phys., 214 (2006), pp. 239–263.CrossRefGoogle Scholar
[17]Madzvamuse, A. and Maini, P. K., Velocity-induced numerical solutions of reaction-diffusion systems on fixed and growing domains, J. Comput. Phys., 225 (2007), pp. 100–119.CrossRefGoogle Scholar
[18]Madzvamuse, A., Maini, P. K. and Wathen, A. J., A moving grid finite element method applied to a model biological pattern generator, J. Comput. Phys., 190 (2003), pp. 478–500.CrossRefGoogle Scholar
[19]Madzvamuse, A., Maini, P. K. and Wathen, A. J., A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains, J. Sci. Comput., 24 (2005), pp. 247–262.CrossRefGoogle Scholar
[20]Meinhardt, H., Models of Biological Pattern Formation, Academic Press, London, 1982.Google Scholar
[21]Meinhardt, H., The Algorithmic Beauty of Sea Shells, Springer-Verlag, Berlin, 1995.Google Scholar
[22]Nicolis, G. and Prigogine, I., Self-Organization in Non-Equilibrium System: From Dissipative Structures to Order Through Fluctuations, Wiley, New York, 1977.Google Scholar
[23]Qiao, Z., Numerical investigations of the dynamical behaviors and instabilities for the Gierer-Meinhardt system, Commun. Comput. Phys., 3 (2008), pp. 406–426.Google Scholar
[24]Qiao, Z., Zhang, Z. and Tang, T., An adaptive time-stepping strategy for the molecular beam epitaxy models, SIAM J. Sci. Comput., 33 (2011), pp. 1395–1414.Google Scholar
[25]Schnakenberg, J., Simple chemical reaction systems with limit cycle behavior, J. Theoret. Biol., 81 (1979), pp. 389–400.CrossRefGoogle Scholar
[26]Sun, W., Tang, T., Ward, M. J. and Wei, J., Numerical challenges for resolving spike dynamics for two reaction-diffusion systems, Studies. Appl. Math., 111 (2003), pp. 41–84.CrossRefGoogle Scholar
[27]Sun, W., Ward, M. J. and Russell, R., The slow dynamics of two-spike solutions for the Gray-Scott and Gierer-Meinhardt systems: competition and oscillatory instabilities, SIAM J. Appl. Dyn. Syst., 4 (2005), pp. 904–953.Google Scholar
[28]Turing, A., The chemical basis of morphogenesis, Phil. Trans. R. Soc. B., 237 (1952), pp. 32– 72.Google Scholar
[29]Ward, M. J., Asymptotic methods for reaction-diffusion systems: past and present, Bull. Math. Biol., 68 (2006), pp. 1151–1167.CrossRefGoogle ScholarPubMed
[30]Ward, M. J., Mcinereny, D. and Houston, P., The dynamics and pinning o f a spike for a reaction-diffusion system, SIAM J. Appl. Math., 62 (2002), pp. 1297–1328.CrossRefGoogle Scholar
[31]Wei, J. and Winter, M., On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal., 30 (1999), pp. 1241–1263.Google Scholar
[32]Wei, J. and Winter, M., Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case, J. Nonlinear. Sci., 11 (2001), pp. 415–458.Google Scholar
[33]Zegeling, P. A. and Kok, H. P., Adaptive moving mesh computations for reaction-diffusion systems, J. Comput. Appl. Math., 168 (2004), pp. 519–528.Google Scholar
[34]Zhang, Z. and Qiao, Z., An adaptive time-stepping strategy for the Cahn-Hilliard equation, Commun. Comput. Phys., 11 (2011), pp. 1261–1278.Google Scholar
[35]Baines, M. J., Hubbard, M. E. and Jimack, P. K., Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations, Commun. Comput. Phys., vol., 10 (2011), pp. 509–576.CrossRefGoogle Scholar
[36]Zhang, Y., Wang, H. and Tang, T., Simulating two-phase viscoelastic flows using moving finite element methods, Commun. Comput. Phys., 7 (2010), pp. 333–349.Google Scholar