Abstract
In this study, an efficient numerical method is proposed for the numerical solution of a class of two-dimensional initial-boundary value problems governed by a non-linear system of partial differential equations known as a Brusselator system. The method proposed is based on a combination of higher-order Compact Finite Difference (CFD) scheme and stable time integration scheme which is known as adaptive step-size Runge-Kutta method. The performance of adaptive step-size Runge-Kutta (RK) formula of third-order accurate in time and Compact Finite Difference scheme of sixth-order in space are investigated. The proposed method has been compared with the studies in the literature. Several test problems are considered to check the accuracy and efficiency of the method and reveal that the method is an efficient and reliable alternative to approximate the Brusselator system.
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References
Mittal, R.C., Kumar, S., Jiwari, R.: A cubic B-spline quasi-interpolation algorithm to capture the pattern formation of coupled reaction-diffusion models. Eng. Comput. 38, 1375–1391 (2021). https://doi.org/10.1007/s00366-020-01278-3
Prigogine, I., Lefever, R.S.: Symmetry breaking instabilities in dissipative systems. II. J. Chem. Phys. 48(4), 1695–1700 (1968)
Prigogine, I., Nicolis, G.: Self-organisation in nonequilibrium systems. In: Dissipative Structures to Order through Fluctuations, pp. 339–426 (1977)
Lefever, R.S., Nicolis, G.: Chemical instabilities and sustained oscillations. J. Theor. Biol. 30(4), 267–284 (1971)
Al-Islam, S., Ali, A., Al-Haq, S.: A computational modeling of the behavior of the two-dimensional reaction-diffusion Brusselator system. Appl. Math. Model. 34(12), 3896–3909 (2010)
Kumar, S., Jiwari, R., Mittal, R.C.: Numerical simulation for computational modelling of reaction-diffusion Brusselator model arising in chemical processes. J. Math. Chem. 57, 149–179 (2019)
Alqahtani, A.M.: Numerical simulation to study the pattern formation of reaction-diffusion Brusselator model arising in triple collision and enzymatic. J. Math. Chem. 56(6), 1543–1566 (2011)
Jiwari, R., Yuan, J.: A computational modeling of two dimensional reaction-diffusion Brusselator system arising in chemical processes. J. Math. Chem. 52(6), 1535–1551 (2011)
Mittal, R.C., Rohila, R.: Numerical simulation of reaction-diffusion systems by modified cubic B-spline differential quadrature method. Chaos, Solitons Fractals 92, 9–19 (2016)
Onarcan, A.T., Adar, N., Dag, I.: Trigonometric cubic B-spline collocation algorithm for numerical solutions of reaction-diffusion equation systems. J. Math. Chem. 37(5), 6848–6869 (2018)
Haq, S., Ali, I., Nisar, K.S.: A computational study of two-dimensional reaction-diffusion Brusselator system with applications in chemical processes. Alex. Eng. J. 60(5), 4381–4392 (2021)
Mittal, R.C., Jiwari, R.: Numerical solution of two-dimensional reaction-diffusion Brusselator system. Appl. Math. Comput. 217, 5404–5415 (2011)
Saad, K.M.: Fractal-fractional Brusselator chemical reaction. Chaos, Solitons Fractals 150(12), 111087 (2011)
Jena, R.M., Chakraverty, S., Rezazadeh, H., Domiri, G.D.: On the solution of time-fractional dynamical model of Brusselator reaction-diffusion system arising in chemical reactions. Math. Methods Appl. Sci. 43(7), 3903–3913 (2010)
Hoffmann, K.A., Chiang, S.T.: Computational Fluid Dynamics, vol. 3, 4th edn, pp. 117–137. A Publication of Engineering System™ (2000)
Press, W.H., Teukolsky, S.A.: Adaptive Stepsize Runge-Kutta integration. Comput. Phys. 6, 188 (1992). https://doi.org/10.1063/1.4823060
Lu, Y.Y.: Numerical Methods for Differential Equations, Department of Mathematics City University of Hong Kong Kowloon, Hong Kong, pp. 13–16
Lele, S.K.: Compact finite difference schemes with spectral-like solution. J. Comput. Phys. 103, 16–42 (1992)
Cicek, Y., Gucuyenen, K.N., Bahar, E., Gurarslan, G., Tanoğlu, G.: A new numerical algorithm based on Quintic B-Spline and adaptive time integrator for coupled Burger’s equation. Comput. Methods Differ. Equ. 11, 130–142 (2022)
İmamoğlu, K.N., Korkut, S.Ö., Gurarslan, G., Tanoğlu, G.: A reliable and fast mesh-free solver for the telegraph equation. Comput. Appl. Math. 41(5), 1–24 (2022)
Bahar, E., Gurarslan, G.: B-spline method of lines for simulation of contaminant transport in groundwater. Water 12(6), 1607 (2020)
Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6(1) (1980)
Dormand, J.R., Prince, P.J.: High order embedded Runge-Kutta formulae, Department of Mathematics and Statistics, Teesside Polytechnic, Middlesbrough, Cleveland, TS1 3BA, U.K
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Ismoilov, S., Gurarslan, G., Tanoğlu, G. (2023). Higher-Order and Stable Numerical Scheme for Nonlinear Diffusion System via Compact Finite Difference and Adaptive Step-Size Runge-Kutta Methods. In: Hemanth, D.J., Yigit, T., Kose, U., Guvenc, U. (eds) 4th International Conference on Artificial Intelligence and Applied Mathematics in Engineering. ICAIAME 2022. Engineering Cyber-Physical Systems and Critical Infrastructures, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-031-31956-3_3
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