Advertisement

A Highly Accurate Time–Space Pseudospectral Approximation and Stability Analysis of Two Dimensional Brusselator Model for Chemical Systems

  • A. K. MittalEmail author
  • L. K. Balyan
Original Paper
  • 40 Downloads

Abstract

In this paper, the authors investigate the numerical solutions of two-dimensional reaction–diffusion equations with Neumann boundary conditions, known as Brusselator model, using Chebyshev pseudospectral method. The proposed methods are established in both time and space to approximate the solutions and prove the stability analysis of the equations. Higher order Chebyshev differential matrix is used in discretizing Brusselator model. The methods convert the Brusselator model into a system ofS algebraic equations, which are solved using Newton–Raphson method and obtained different types of patterns. Error of the proposed method is presented in terms of \( L_{\infty }\) and \( L_{2}\) error norms. In support of theoretical results, the methods are implemented on two problems and found highly accurate and stable approximations. The detailed comparison of the proposed method with various other methods are also given.

Keywords

Brusselator model Reaction–diffusion equations Chebyshev–Gauss–Lobbato(CGL) points Pseudospectral method Newton–Raphson method 

Notes

Acknowledgements

The first author thankfully acknowledges to the Ministry of Human Resource Development, India, for providing financial support for this research.

References

  1. 1.
    Adomian, G.: The diffusion-Brusselator equation. Comput. Math. Appl. 29(5), 1–3 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alkhalaf, S.: Third-order approximate solution of chemical reaction–diffusion Brusselator system using optimal homotopy asymptotic method. Adv. Math. Phys. 2017, 1–8 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andreianov, B., Bendahmane, M., Ruiz-Baier, R.: Analysis of a finite volume method for a cross-diffusion model in population dynamics. Math. Models Methods Appl. Sci. 21(02), 307–344 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ang, W.-T.: The two-dimensional reaction–diffusion brusselator system: a dual-reciprocity boundary element solution. Eng. Anal. Bound. Elem. 27(9), 897–903 (2003)CrossRefGoogle Scholar
  5. 5.
    Auchmuty, J., Nicolis, G.: Bifurcation analysis of nonlinear reaction–diffusion equations-I. Evolution equations and the steady state solutions. Bull. Math. Biol. 37(4), 323–365 (1975)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Biazar, J., Ayati, Z.: A numerical solution of reaction–diffusion Brusselator system by ADM. J. Nat. Sci. Sustain. Technol. 1(2), 263–270 (2008)Google Scholar
  7. 7.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Corporation, North Chelmsford (2001)zbMATHGoogle Scholar
  8. 8.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Thomas Jr., A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (2012)zbMATHGoogle Scholar
  9. 9.
    Chowdhury, M., Hassan, T., Mawa, S.: A new application of homotopy perturbation method to the reaction–diffusion Brusselator model. Proc. Soc. Behav. Sci. 8, 648–653 (2010)CrossRefGoogle Scholar
  10. 10.
    Dehghan, M., Abbaszadeh, M.: Variational multiscale element free galerkin (VMEFG) and local discontinuous galerkin (LDG) methods for solving two-dimensional Brusselator reaction–diffusion system with and without cross-diffusion. Comput. Method Appl. Mech. Eng. 300, 770–797 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Elyukhina, I.: Nonlinear stability analysis of the full Brusselator reaction–diffusion model. Theor. Found. Chem. Eng. 48(6), 806–812 (2014)CrossRefGoogle Scholar
  12. 12.
    Ersoy, O., Dag, I.: Numerical solutions of the reaction–diffusion system by using exponential cubic b-spline collocation algorithms. Open Phys. 13(1), 414–427 (2015)CrossRefGoogle Scholar
  13. 13.
    Ghergu, M., Rădulescu, V.: Turing patterns in general reaction–diffusion systems of Brusselator type. Commun. Contemp. Math. 12(04), 661–679 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Herschkowitz-Kaufman, M., Nicolis, G.: Localized spatial structures and nonlinear chemical waves in dissipative systems. J. Chem. Phys. 56(5), 1890–1895 (1972)CrossRefGoogle Scholar
  15. 15.
    Hu, G., Qiao, Z., Tang, T.: Moving finite element simulations for reaction–diffusion systems. Adv. Appl. Math. Mech. 4(3), 365–381 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Islam, S., Ali, A., Haq, S.: A computational modeling of the behavior of the two-dimensional reaction–diffusion brusselator system. Appl. Math. Model. 34(12), 3896–3909 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jia, Y., Li, Y., Wu, J.: Coexistence of activator and inhibitor for Brusselator diffusion system in chemical or biochemical reactions. Appl. Math. Lett. 53, 33–38 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jiwari, R., Tomasiello, S., Tornabene, F.: A numerical algorithm for computational modelling of coupled advection–diffusion–reaction systems. Eng. Comput. 35(3), 1383–1401 (2018)CrossRefGoogle Scholar
  19. 19.
    Khani, F., Samadi, F., Hamedi-Nezhad, S.: New exact solutions of the Brusselator reaction diffusion model using the exp-function method. Math. Probl. Eng. 2009, 1–9 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kumar, S., Khan, Y., Yildirim, A.: A mathematical modeling arising in the chemical systems and its approximate numerical solution. Asia Pac. J. Chem. Eng. 7(6), 835–840 (2012)CrossRefGoogle Scholar
  21. 21.
    Lefever, R.: Dissipative structures in chemical systems. J. Chem. Phys. 49(11), 4977–4978 (1968)CrossRefGoogle Scholar
  22. 22.
    Lefever, R., Nicolis, G.: Chemical instabilities and sustained oscillations. J. Theor. Biol. 30(2), 267–284 (1971)CrossRefGoogle Scholar
  23. 23.
    Lin, Z., Ruiz-Baier, R., Tian, C.: Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion. J. Comput. Phys. 256, 806–823 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mittal, R., Jiwari, R.: Numerical solution of two-dimensional reaction–diffusion Brusselator system. Appl. Math. Comput. 217(12), 5404–5415 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Mohammadi, M., Mokhtari, R., Schaback, R.: A meshless method for solving the 2D Brusselator reaction–diffusion system. Comput. Model. Eng. Sci. 101, 113–138 (2014)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Pena, B., Perez-Garcia, C.: Stability of turing patterns in the Brusselator model. Phys. Rev. E 64(5), 056213 (2001)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Peng, R., Wang, M.: Pattern formation in the Brusselator system. J. Math. Anal. Appl. 309(1), 151–166 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Prigogine, I., Lefever, R.: Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48(4), 1695–1700 (1968)CrossRefGoogle Scholar
  29. 29.
    Ruuth, S.J.: Implicit-explicit methods for reaction–diffusion problems in pattern formation. J. Math. Biol. 34(2), 148–176 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shirzadi, A., Sladek, V., Sladek, J.: A meshless simulations for 2D nonlinear reaction–diffusion Brusselator system. CMES Comput. Model. Eng. Sci. 95(4), 259–282 (2013)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Stoltz, S. M.: Pattern formation in the Brusselator model of chemical reactions (Doctoral dissertation, University of Pretoria) (2016)Google Scholar
  32. 32.
    Twizell, E.H., Gumel, A.B., Cao, Q.: A second-order scheme for the “Brusselator” reaction–diffusion system. J. Math. Chem. 26(4), 297–316 (1999)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Vandewalle, S., Piessens, R.: Numerical experiments with nonlinear multigrid waveform relaxation on a parallel processor. Appl. Numer. Math. 8(2), 149–161 (1991)CrossRefGoogle Scholar
  34. 34.
    Wazwaz, A.-M.: The decomposition method applied to systems of partial differential equations and to the reaction–diffusion Brusselator model. Appl. Math. Comput. 110(2–3), 251–264 (2000)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Discipline of MathematicsIIITDMJabalpurIndia

Personalised recommendations