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International Journal of Nonlinear Sciences and Numerical Simulation

Editor-in-Chief: Birnir, Björn

Editorial Board: Angheluta-Bauer, Luiza / Chen, Xi / Chou, Tom / Grauer, Rainer / Marzocchella, Antonio / Rangarajan, Govindan / Trivisa, Konstantina / Weikard, Rudi / Yang, Xu


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2191-0294
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Volume 16, Issue 6

Issues

Robust IMEX Schemes for Solving Two-Dimensional Reaction–Diffusion Models

Kolade M. Owolabi
  • Corresponding author
  • Department of Mathematical Sciences, Federal University of Technology, Akure PMB 704, Akure, Ondo State, Nigeria
  • Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa
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Published Online: 2015-09-05 | DOI: https://doi.org/10.1515/ijnsns-2015-0004

Abstract

In this paper, numerical simulations of two-dimensional reaction–diffusion (for single and multi-species) models are considered for pattern formation processes. The nature of our problems permits the use of two classical approaches. These semi-linear partial differential equations are split into a linear equation which contains the highly stiff part of the problem, and a nonlinear part that is expected to be varying slowly than the linear part. For the spatial discretization, we introduce higher-order symmetric finite difference scheme, and the resulting ordinary differential equations are then solved with the use of the family of implicit–explicit (IMEX) schemes. Stability properties of these schemes as well as the linear stability analysis of the problems are well presented. Numerical examples and results are also given to illustrate the accuracy and implementation of the methods.

Keywords: autocatalysis; Ginzburg–Landau model; Gray-Scott model; systems; IMEX methods; pattern formation; predator–prey; reaction–diffusion; stability

MSC® 2010: 92B05; 92C15; 35K55; 65M20; 76R50

References

  • [1] P. K. Maini, K. J. Painter, and H. N. P. Chau, Spatial pattern formation in chemical and biological systems, J. Chem Soc., Faraday Trans. 93 (1997), 3601–3610.Google Scholar

  • [2] T. T. Marquez-Lago and P. Padilla, A selection criterion for patterns in reaction-diffusion systems, Theor. Biol. Med. Model, 11 (2014), doi:10.1186/1742-4682-11-7.Crossref

  • [3] M. Garvie, Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bull. Math. Biol. 69 (2007), 931–956.Google Scholar

  • [4] Q. Nie, F. Y. M. Wan, Y. Zhang, and X. Liu, Compact integration factor methods in high spatial dimensions, J. Comput. Phys. 227 (2008), 5238–5255.Google Scholar

  • [5] K. M. Owolabi, Efficient numerical methods to solve some reaction-diffusion problems arising in biology, PhD thesis, University of the Western Cape, 2013.Google Scholar

  • [6] W. Bao, Q. Du, and Y. Zhang, The dynamics and interaction of quantized vortices in Ginzburg-Landau-Schroedinger equations, SIAM J. Appl. Math. 67 (2007), 1740–1775.Google Scholar

  • [7] A. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput. 26 (2005), 1214–1233.Google Scholar

  • [8] K. M. Owolabi and K. C. Patidar, Higher-order time-stepping methods for time-dependent reaction-diffusion equations arising in biology, Appl. Math. Comput. 240 (2014), 30–50.Google Scholar

  • [9] S. M. Cox and P. C. Matthews, Exponential time differencing for stiff systems, J. Comput. Phys. 176 (2002), 430–455.Google Scholar

  • [10] S. Krogstad, Generalized integrating factor methods for stiff PDEs, J. Comput. Phys. 203 (2005), 72–88.Google Scholar

  • [11] M. Crouzeix, Une méthod multipas implicite-explicite pour l‘approximation des équations d‘évolution paraboliques, Numer. Math. 35 (1980), 257–276.Google Scholar

  • [12] J. M. Varah, Stability restrictions on second order, three level finite difference schemes for parabolic equations, SIAM J. Numer. Anal. 17 (1980), 300–309.Google Scholar

  • [13] U. M. Ascher, S. J. Ruth, and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math. 25 (1997), 151–167.Google Scholar

  • [14] W. Hundsdorfer and S. J. Ruuth, Imex extensions of linear multistep monotonicity and boundedness properties, J. Comput. Phys. 225 (2007), 2016–2042.Google Scholar

  • [15] I. Grooms and K. Julien, Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation, J. Comput. Phys. 230 (2011), 3630–3560.Google Scholar

  • [16] D. Li, C. Zhang, W. Wang, and Y. Zhang, Implicit-explicit predictor-corrector schemes for nonlinear parabolic differential equations, Appl. Math. Model. 35 (2011), 2711–2722.Google Scholar

  • [17] U. M. Ascher, S. J. Ruth, and B. T. R. Wetton, Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal. 32 (1995), 797–823.Google Scholar

  • [18] S. O. Fatunla, Numerical methods for IVP’s in ordinary differential equation, Academic Press Inc., New York, 1988.Google Scholar

  • [19] J. D. Lambert and A. Watson, Symmetric multistep method for periodic initial value problems, J. Inst. Math. Appl. 18 (1976), 189–202.Google Scholar

  • [20] G. Beylkin, J. M. Keiser, and L. Vozovoi, A new class of time discretization schemes for the solution of nonlinear PDEs, J. Comput. Phys. 147 (1998), 362–387.Google Scholar

  • [21] J. C. Butcher, The numerical analysis of ordinary differential equations, Runge-Kutta and general linear methods, John Wiley and Sons, Chichester, 1987.

  • [22] E. Hairer and G. Wanner, Solving ordinary differential equations II: stiff and differential-algebraic problems, Springer-Verlag, New York, 1991.Google Scholar

  • [23] K. M. Owolabi, An efficient implicit optimal order formula for direct integration of second order orbital problems, Int. J. Nonlinear Sci. 16 (2013), 175–184.Google Scholar

  • [24] M. Robinson, IMEX method convergence for a parabolic equation, J Diff. Eqn. 241 (2007), 225–236.Google Scholar

  • [25] A. C. Newell and J. A. Whitehead, Review of the finite bandwidth concept, in: H. Leipholz (ed.), Proceedings of the International Union of Theoretical and Applied Mechanics, Symposium on Instability of Continuous Systems, 1969, pp. 279–303, Springer-Verlag, Berlin, 1971.Google Scholar

  • [26] A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38 (1969), 279–303.Google Scholar

  • [27] N. Akhmediev and A. Ankiewicz, Dissipative solitons in the CGLE and Swift-Hohenberg equations, Springer, Berlin, 2005.Google Scholar

  • [28] F. Bérad and S. C. Mancas, Spatiotemporal two-dimensional solitons in the complex Ginzburg-Landau equations, Advances and Applications in Fluid Mech. 8 (2011), 141–156.

  • [29] C. Cartes, J. Cisternas, O. Descalzi, and H. R. Brand, Model of a two-dimensional extended chaotic system: Evidence of diffusing dissipative solitons, Phys. Rev. Lett. 109 (2012), 178303.Google Scholar

  • [30] H. Bernard, A. Islas, and C. M. Schober, Conservation of phase properties using exponential integrators on the cubic Schrödinger equation, J. Comput. Phys. 225 (2007), 284–299.Google Scholar

  • [31] F. de la Hoz and F. Vadilo, An exponential time differencing method for the nonlinear Schrödinger equation, Comput. Phys. Commun. 179 (2008), 449–456.Google Scholar

  • [32] I. S. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys. 74 (2002).

  • [33] W. van Saarloos, Spatiotemporal patterns in nonequilibrium complex systems, Santa Fe Institute Studies in the Sciences of Complexity, Proceedings XXI, Addison-Wesley, Reading, 1994.

  • [34] A. Doelman, R. A. Gardner, and T. J. Kaper, Stability analysis of singular patterns in the 1d GS model: a matched asymptotic approach, Phys. D Nonlinear Phenomena. 122 (1998), 1–36.Google Scholar

  • [35] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Lsolas and other forms of multistability, Chem. Eng. Sci. 38 (1983), 29–43.Google Scholar

  • [36] P. Gray and S. K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system Chem. Eng. Sci. 39 (1984), 1087–1097.Google Scholar

  • [37] P. Gray and S. K. Scott, Sustained oscillations and other exotic patterns of behaviour in isothermal reactions, J. Phys. Chem. 89 (1985), 22–32.Google Scholar

  • [38] A. Doelman, T. J. Kaper, and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, J. Nonlinear Sci. 10 (1997), 523–563.Google Scholar

  • [39] J. D. Murray, Mathematical biology I: an introduction, Springer-Verlag, New York, 2002.Google Scholar

  • [40] J. E. Pearson, Complex patterns in a simple system, Science. 261 (1993), 189–192.Google Scholar

  • [41] J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in R2, Phys. D. 176 (2003), 147–180.Google Scholar

  • [42] A. Doelman and H. van der Ploeg, Homoclinic stripe patterns, SIAM J. Appl. Dyn. Syst. 1 (2002), 65–104.Google Scholar

  • [43] Y. Nishiura and D. Ueyama, Spatio-temporal chaos for the Gray-Scott model, Phys. D. 150 (2001), 137–162.Google Scholar

  • [44] J. D. Murray, Mathematical biology II: spatial models and biomedical applications, Springer-Verlag, Berlin, 2003.Google Scholar

  • [45] H. Shoji, Y. Iwasa, and S. Kondo, Stripes, spots, or reversed spots in two-dimensional Turing systems, J. Theor. Biol. 224 (2003), 339–350.Google Scholar

  • [46] A. Doelman, A. Gardner, and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc. 155 (2002), 737, xii+64.Google Scholar

  • [47] C. Liu, X. Fu, L. Liu, X. Ren, C. K. Chau, S. Li, L. Xiang, H. Zeng, G. Chen, L. H. Tang, et al., Sequential establishment of stripe patterns in an expanding cell population, Science. 334 (2011), 238–241.Google Scholar

  • [45] K. M. Owolabi and K. C. Patidar, Numerical solution of singular patterns in one-dimensional Gray-Scott-like models, Int. J. Nonlinear Sci. Numer. Simul. (2014), doi:10.1515/ijnsns–2013–0124.Crossref

  • [49] A. Munteanu and R. V. Sole, Pattern formation in noisy self-replicating spots, Int. J. Bifurcat. Chaos. 16 (2006), 3679.Google Scholar

  • [50] A. J. Lotka, The elements of physical biology, Williams and Wilkins, Baltimore, 1925.Google Scholar

  • [51] V. Volterra, Fluctuation in abundance of the species considered mathematically, Nature. 118 (1926), 558–560.Google Scholar

  • [52] V. Volterra, Variations and fluctuations of the numbers of individuals in animal and species living together, Reprinted in 1931 in R.N. Chapman, Animal Ecology, McGraw-Hill, New York, 1926.

  • [46] K. M. Owolabi and K. C. Patidar, Robust numerical simulation of reaction-diffusion models arising in mathematical ecology, in: G. Akrivis, V. Dougalis, S. Gallopoulos, A. Hadjidimos, I. Kotsireas, C. Makridakis and Y. Saridakis (eds.), Proceedings of NumAn2014 Conference on Numerical Analysis. Recent Approaches to Numerical Analysis: Theory, Methods & Applications, Chania, Greece; 2–5 September 2014, 222–227.,ISBN: 978–960–8475–21–1.

  • [54] G. Sun, G. Zhang, Z. Jin, and L. Li, Predator cannibalism can give rise to regular spatial pattern in a predator-prey system, Nonlinear Dyn. 58 (2009), 75–84.Google Scholar

  • [55] W. Wang, L. Zhang, H. Wang, and Z. Li, Pattern formation of a predator-prey system with Ivlev-type function response, Ecol. Model. 221 (2010), 131–140.Google Scholar

About the article

Received: 2015-01-08

Accepted: 2015-08-07

Published Online: 2015-09-05

Published in Print: 2015-10-01


Funding: This research was supported by the South African National Research Foundation.


Citation Information: International Journal of Nonlinear Sciences and Numerical Simulation, Volume 16, Issue 6, Pages 271–284, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: https://doi.org/10.1515/ijnsns-2015-0004.

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