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Year 2021, Volume: 42 Issue: 1, 75 - 87, 29.03.2021

Sıla Övgü Korkut Uysal , Yeşim Çiçek

https://doi.org/10.17776/csj.695738

Abstract

Thanks

Yazarlar, Editör ve hakemlere, ayırdıkları zaman, anlayışlı yaklaşımları ve çok değerli katkıları için şimdiden teşekkür ediyorlar.

References

  • [1] Lefever R., Nicolis G., Chemical instabilities and sustained oscillations, Journal of Theoretical Biology, 30 (1971) 267-284.
  • [2] Nicolis G., Prigogine I., Self-Organization in Non-Equilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley/Interscience, New York, London, Sydney, Toronto (1977).
  • [3] Prigogine I., Lefever R., Symmetries breaking instabilities in dissipative systems II. The Journal of Chemical Physics, 48 (1968) 1695-1700.
  • [4] Tyson J. J., Some further studies of nonlinear oscillations in chemical systems, The Journal of Chemical Physics, 58 (1973) 3919-3930.
  • [5] Turing A., The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London Series B, Biological Sciences, 237 (1952) 37-72.
  • [6] Tyson J. J., The Belousov-Zhabotinskii Reaction (Lecture Notes in Biomathematics), 10, Springer-Verlag, Heidelberg, 1976.
  • [7] Ang W-T., The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution, Engineering Analysis with Boundary Elements, 27 (2003) 897-903.
  • [8] Adomian G., The diffusion-Brusselator equation, Computers & Mathematics with Applications, 29 (5) (1995) 1-3.
  • [9] Twizell E. H., Gumel A.B., Cao Q., A second-order scheme for the “Brusselator” reaction-diffusion system, Journal of Mathematical Chemistry, 26 (1999) 297-316.
  • [10] Wazwaz A.-M., The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model, Applied Mathematics and Computation, 110 (2000) 251-264.
  • [11] Jiwari R., Yuan J., A computational modeling of two dimensional reaction–diffusion Brusselator system arising in chemical processes, Journal of Mathematical Chemistry, 52 (2014) 1535-1551.
  • [12] Mittal R.C., Jiwari R., Numerical solution of two-dimensional reaction-diffusion Brusselator system, Applied Mathematics and Computation, 217 (2011) 5404-5415.
  • [13] Shirzadi A., Sladek V., Sladek J., A Meshless Simulations for 2D Nonlinear Reaction-diffusion Brusselator System, Computer Modeling in Engineering and Sciences, 95 (2013) 259-282.
  • [14] Siraj-ul-Islam Ali A., Haq S., A computational modeling of the behavior of the two-dimensional reaction-diffusion Brusselator system, Applied Mathematical Modelling, 34 (2010) 3896-3909.
  • [15] Yadav O. P., Jiwari R., A finite element approach for analysis and computational modelling of coupled reaction diffusion models, Numerical Methods for Partial Differential Equations, 35 (2019) 830-850.
  • [16] Yadav O. P., Jiwari R., A finite element approach to capture Turing patterns of autocatalytic Brusselator model, Journal of Mathematical Chemistry, 57 (2019) 769-789.
  • [17] Kumar S., Jiwari, R., Mittal R.C., Numerical simulation for computational modelling of reaction-diffusion Brusselator model arising in chemical processes, Journal of Mathematical Chemistry, 57 (2019) 149-179.
  • [18] Bhatt H., Chowdhury A., Comparative analysis of numerical methods for the multidimensional Brusselator system, Open Journal of Mathematical Sciences, 3 (2019) 262-272.
  • [19] Asante-Asamani E. O., Wade B. A., A Dimensional Splitting of ETD Schemes for Reaction-Diffusion Systems, Communications in Computational Physics, 19 (2016) 1343-1356.
  • [20] McLachlan R., Quispel G., Splitting methods, Acta Numerica, 11 (2002) 341-434.
  • [21] Hosseini R., Tatari M., Some splitting methods for hyperbolic PDEs, Applied Numerical Mathematics, 146 (2019) 361-378.
  • [22] Zurnacı F., Gücüyenen-Kaymak N., Seydaoğlu M., Tanoğlu G., Convergence analysis and numerical solution of the Benjamin-Bona-Mahony equation by Lie-Trotter splitting, Turkish Journal of Mathematics, 42 (2018) 1471-1483.
  • [23] Seydaoğlu M., An accurate approximation algorithm for Burgers’ equation in the presence of small viscosity, Journal of Computational and Applied Mathematics, 344 (2018) 473-481.
  • [24] Hundsdorfer W., Verwer J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. 1st ed. Springer, Berlin, Heidelberg, (2003) 116-121.
  • [25] Mickens R.E., Nonstandard finite difference schemes for reaction-diffusion equation, Numerical Methods for Partial Differential Equations, 15 (1999) 201-214.
  • [26] Singh R., Brusselator as a Reaction-Diffusion, Master Dissertation, Homi Bhabha National Institute, The Institute of Mathematical Sciences, (2008).
  • [27] Manaa S. A., Saeed R. K., Easif F. H., Numerical Solution of Brusselator Model by Finite Difference Method, Journal of Applied Sciences Research, 6 (11) (2010) 1632-1646.

Numerical solution of the brusselator model by time splitting method

Year 2021, Volume: 42 Issue: 1, 75 - 87, 29.03.2021

Sıla Övgü Korkut Uysal , Yeşim Çiçek

https://doi.org/10.17776/csj.695738

Abstract

One of the significant models in chemical reactions with oscillations is the Brusselator model. This model essentially describes a nonlinear reaction-diffusion equation. Brusselator system arises in applications of many physical and chemical models. In this study, the Brusselator model is solved numerically with the help of a time-splitting method. Consistency and stability of the method are proved with the help of auxiliary lemmas. Additionally, the positivity preservation of the method is analyzed. The accuracy of the presented method is also tested on numerical examples and all theoretical results are supported by the tables and figures.

Keywords

Local error analysis Brusselator model Splitting method Non-linear partial differential equation Reaction-Diffusion equation

References

  • [1] Lefever R., Nicolis G., Chemical instabilities and sustained oscillations, Journal of Theoretical Biology, 30 (1971) 267-284.
  • [2] Nicolis G., Prigogine I., Self-Organization in Non-Equilibrium Systems: From Dissipative Structures to Order through Fluctuations, Wiley/Interscience, New York, London, Sydney, Toronto (1977).
  • [3] Prigogine I., Lefever R., Symmetries breaking instabilities in dissipative systems II. The Journal of Chemical Physics, 48 (1968) 1695-1700.
  • [4] Tyson J. J., Some further studies of nonlinear oscillations in chemical systems, The Journal of Chemical Physics, 58 (1973) 3919-3930.
  • [5] Turing A., The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London Series B, Biological Sciences, 237 (1952) 37-72.
  • [6] Tyson J. J., The Belousov-Zhabotinskii Reaction (Lecture Notes in Biomathematics), 10, Springer-Verlag, Heidelberg, 1976.
  • [7] Ang W-T., The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution, Engineering Analysis with Boundary Elements, 27 (2003) 897-903.
  • [8] Adomian G., The diffusion-Brusselator equation, Computers & Mathematics with Applications, 29 (5) (1995) 1-3.
  • [9] Twizell E. H., Gumel A.B., Cao Q., A second-order scheme for the “Brusselator” reaction-diffusion system, Journal of Mathematical Chemistry, 26 (1999) 297-316.
  • [10] Wazwaz A.-M., The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model, Applied Mathematics and Computation, 110 (2000) 251-264.
  • [11] Jiwari R., Yuan J., A computational modeling of two dimensional reaction–diffusion Brusselator system arising in chemical processes, Journal of Mathematical Chemistry, 52 (2014) 1535-1551.
  • [12] Mittal R.C., Jiwari R., Numerical solution of two-dimensional reaction-diffusion Brusselator system, Applied Mathematics and Computation, 217 (2011) 5404-5415.
  • [13] Shirzadi A., Sladek V., Sladek J., A Meshless Simulations for 2D Nonlinear Reaction-diffusion Brusselator System, Computer Modeling in Engineering and Sciences, 95 (2013) 259-282.
  • [14] Siraj-ul-Islam Ali A., Haq S., A computational modeling of the behavior of the two-dimensional reaction-diffusion Brusselator system, Applied Mathematical Modelling, 34 (2010) 3896-3909.
  • [15] Yadav O. P., Jiwari R., A finite element approach for analysis and computational modelling of coupled reaction diffusion models, Numerical Methods for Partial Differential Equations, 35 (2019) 830-850.
  • [16] Yadav O. P., Jiwari R., A finite element approach to capture Turing patterns of autocatalytic Brusselator model, Journal of Mathematical Chemistry, 57 (2019) 769-789.
  • [17] Kumar S., Jiwari, R., Mittal R.C., Numerical simulation for computational modelling of reaction-diffusion Brusselator model arising in chemical processes, Journal of Mathematical Chemistry, 57 (2019) 149-179.
  • [18] Bhatt H., Chowdhury A., Comparative analysis of numerical methods for the multidimensional Brusselator system, Open Journal of Mathematical Sciences, 3 (2019) 262-272.
  • [19] Asante-Asamani E. O., Wade B. A., A Dimensional Splitting of ETD Schemes for Reaction-Diffusion Systems, Communications in Computational Physics, 19 (2016) 1343-1356.
  • [20] McLachlan R., Quispel G., Splitting methods, Acta Numerica, 11 (2002) 341-434.
  • [21] Hosseini R., Tatari M., Some splitting methods for hyperbolic PDEs, Applied Numerical Mathematics, 146 (2019) 361-378.
  • [22] Zurnacı F., Gücüyenen-Kaymak N., Seydaoğlu M., Tanoğlu G., Convergence analysis and numerical solution of the Benjamin-Bona-Mahony equation by Lie-Trotter splitting, Turkish Journal of Mathematics, 42 (2018) 1471-1483.
  • [23] Seydaoğlu M., An accurate approximation algorithm for Burgers’ equation in the presence of small viscosity, Journal of Computational and Applied Mathematics, 344 (2018) 473-481.
  • [24] Hundsdorfer W., Verwer J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. 1st ed. Springer, Berlin, Heidelberg, (2003) 116-121.
  • [25] Mickens R.E., Nonstandard finite difference schemes for reaction-diffusion equation, Numerical Methods for Partial Differential Equations, 15 (1999) 201-214.
  • [26] Singh R., Brusselator as a Reaction-Diffusion, Master Dissertation, Homi Bhabha National Institute, The Institute of Mathematical Sciences, (2008).
  • [27] Manaa S. A., Saeed R. K., Easif F. H., Numerical Solution of Brusselator Model by Finite Difference Method, Journal of Applied Sciences Research, 6 (11) (2010) 1632-1646.
There are 27 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Sıla Övgü Korkut Uysal 0000-0003-4784-2013

Yeşim Çiçek 0000-0001-5438-4685

Publication Date March 29, 2021
Submission Date February 27, 2020
Acceptance Date December 16, 2020
Published in Issue Year 2021Volume: 42 Issue: 1

Cite

APA Korkut Uysal, S. Ö., & Çiçek, Y. (2021). Numerical solution of the brusselator model by time splitting method. Cumhuriyet Science Journal, 42(1), 75-87. https://doi.org/10.17776/csj.695738