MODIFIED RUNGE-KUTTA ALGORITHM FOR OSCILLATORY CHEMICAL KINETICS

JULIUS OSATO  EHIGIE

JULIUS OSATO EHIGIE DEPARTMENT OF MATHEMATICS, UNIVERSITY OF LAGOS, AKOKA, LAGOS STATE, NIGERIA.

Keywords: Exponential fitting, Reaction rate equations, Runge-Kutta methods, Chemical Kinetics, Oscillatory dynamics

Abstract

Oscillatory dynamics frequently arise in the kinetics of chemical reactions, posing significant challenges to traditional numerical methods in capturing the dynamics. In this study, we introduce a class of modified Runge- Kutta methods specifically designed to solve such oscillatory chemical kinetics problems. Based on exponential fitting conditions for Runge-Kutta methods, we adapt several classical Runge-Kutta methods to better accommodate oscillatory behaviour. Three distinct classes of these modified methods based on some choosen parameters are proposed and presented. To evaluate their effectiveness, we conduct numerical experiments on two practical problems characterised by oscillatory chemical reactions. The results clearly demonstrate that the modified Runge-Kutta methods outperform their classical counter- parts when applied to oscillatory systems.

References

[1] Albrecht, P. (1987). The extension of the theory of A-methods to RK methods. In Numerical Treatment of Differential Equations, Proceedings 4th Seminar NUMDIFF (Vol. 4, pp. 9–18).
[2] Albrecht, P. (1987). A new theoretical approach to Runge–Kutta methods. SIAM Journal on Numerical Analysis, 24(2), 391–406.
[3] Al-fayyadh, K. A., Fawzi, F. A., & Hussain, K. A. (2025). Exponentially Fitted-Diagonally Implicit Runge-Kutta Method for Direct Solution of Fifth-Order Ordinary Differential Equations. Journal of Al-Qadisiyah for Computer Science and Mathematics, 17(1), 142- 158.
[4] Amiri, S. (2019). Some drift exponentially fitted stochastic Runge-Kutta methods for solving Itˆo SDE systems. Bulletin of the Belgian Mathematical Society, 26(3), 431-451.
[5] Belousov, B. P. (1951). A periodic reaction and its mechanism. Oscillation and Travelling Waves in Chemical Systems.
[6] Bibik, Y. V. (2007). The second Hamiltonian structure for a special case of the Lotka- Volterra equations. Computational Mathematics and Mathematical Physics, 47, 1285–1294.
[7] Bray, W. C., & Liebhafsky, H. A. (1931). Reactions involving hydrogen peroxide, iodine and iodate ion. I. Introduction. Journal of the american chemical society, 53(1), 38–44.
[8] Briggs, T. S., & Rauscher, W. C. (1973). An oscillating iodine clock. Journal of chemical Education, 50(7), 496.
[9] J.C. Butcher (2016). Numerical Methods for Ordinary Differential Equations, third ed., John Wiley & Sons Ltd.
[10] Calvo, M., Franco, J. M., Montijano, J. I., & R´andez, L. (1996). Explicit Runge-Kutta methods for initial value problems with oscillating solutions. Journal of Computational and Applied mathematics, 76(1-2), 195–212.
[11] Cardelli, L. (2008). From processes to ODEs by chemistry. In Fifth Ifip International Con- ference On Theoretical Computer Science–Tcs 2008 (pp. 261-281). Boston, MA: Springer US.
[12] Chen, Z., Li, J., Zhang, R., & You, X. (2015). Exponentially Fitted Two-Derivative Runge- Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems. Computational and mathematical methods in medicine, 2015(1), 689137.
[13] Conte, D., & Frasca-Caccia, G. (2022). Exponentially fitted methods that preserve conservation laws. Communications in Nonlinear Science and Numerical Simulation, 109, 106334.
[14] D’Ambrosio, R., Moccaldi, M., Paternoster, B., & Rossi, F. (2018). Adapted numeri- cal modelling of the Belousov–Zhabotinsky reaction. Journal of Mathematical Chemistry, 56(10), 2876-2897.
[15] Ehigie, J. O., Diao, D., Zhang, R., Fang, Y., Hou, X., & You, X. (2018). Exponentially fitted symmetric and symplectic DIRK methods for oscillatory Hamiltonian systems. Journal of Mathematical Chemistry, 56, 1130–1152.
[16] Ehigie, J. O., Luan, V. T., Okunuga, S. A., & You, X. (2022). Exponentially fitted two- derivative DIRK methods for oscillatory differential equations. Applied Mathematics and Computation, 418, 126770.
[17] Ehigie, J. O., & Okunuga, S. A. (2018). A new collocation formulation for the block Falkner-type methods with trigonometric coefficients for oscillatory second order ordinary differential equations. Afrika Matematika, 29, 531–555.
[18] Ghawadri, N., Senu, N., Ismail, F., & Ibrahim, Z. B. (2018, November). Exponentially- fitted forth-order explicit modified Runge-Kutta type method for solving third-order ODEs. In Journal of Physics: Conference Series (Vol. 1132, No. 1, p. 012016). IOP Publishing.
[19] Goodwin, B. C. (1965). Oscillatory behavior in enzymatic control processes. Advances in enzyme regulation, 3, 425–437.
[20] Hairer, E., Lubich, C., & Wanner, G. Geometric numerical integration: structure- preserving algorithms for ordinary differential equations (Springer, Berlin Heidelberg, 2006).
[21] Hering, R. H. (1990). Oscillations in Lotka-Volterra systems of chemical reactions. Journal of mathematical chemistry, 5(2), 197–202.
[22] Horn, F., & Jackson, R. (1972). General mass action kinetics. Archive for rational mechanics and analysis, 47, 81–116.
[23] Ixaru, L. G., Berghe, G. V., & De Meyer, H. (2002). Frequency evaluation in exponential fitting multistep algorithms for ODEs. Journal of Computational and Applied Mathematics, 140(1-2), 423-434.
[24] Lee, K. C., Senu, N., Ahmadian, A., & Ibrahim, S. N. I. (2022). High-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge–Kutta-type methods for solving third-order oscillatory problems. Mathematical Sciences, 16(3), 281-297.
[25] Lotka, A. J. (1920). Undamped oscillations derived from the law of mass action. Journal of the American Chemical Society, 42(8), 1595–1599.
[26] Naseri, A., Khalilzadeh, H., & Sheykhizadeh, S. (2016). Tutorial review: Simulation of oscillating chemical reactions using Microsoft excel macros. Analytical and Bioanalytical Chemistry Research, 3(2), 169–185.
[27] Nicholis, G. Prigogine, I. (1977) Self-Organization in Nonequilibrium Systems.
[28] Petrusevski, V. M., Stojanovska, M. I., & Sˇoptrajanov, B. T. (2007). Oscillating Reactions: Two Analogies. Science Education Review, 6(2), 68–73.
[29] Senu, N., Lee, K. C., Wan Ismail, W. F., Ahmadian, A., Ibrahim, S. N., & Laham, M. (2021). Improved Runge-Kutta method with trigonometrically-fitting technique for solving oscillatory problem. Malaysian Journal of Mathematical Sciences, 15(2), 253-266.fourth- order
[30] Simos, T. E. (1998). An exponentially-fitted Runge-Kutta method for the numerical inte- gration of initial-value problems with periodic or oscillating solutions. Computer Physics Communications, 115(1), 1–8.
[31] Tiwari, S., & Pandey, R. K. (2021). Revised version of exponentially fitted pseudo-Runge- Kutta method. International Journal of Computing Science and Mathematics, 13(2), 116- 125.

[32] Van de Vyver, H. (2005). Modified explicit Runge–Kutta methods for the numerical so- lution of the Schr¨odinger equation. Applied mathematics and computation, 171(2), 1025– 1036.
[33] Berghe, G. V., De Meyer, H., Van Daele, M., & Van Hecke, T. (1999). Exponentially-fitted explicit Runge–Kutta methods. Computer Physics Communications, 123(1-3), 7–15.
[34] Vigo-Aguiar, J., & Ramos, H. (2015). On the choice of the frequency in trigonometrically- fitted methods for periodic problems. Journal of computational and Applied Mathematics, 277, 94–105.
[35] Zhabotinsky, A. M. (1964). Periodic liquid phase reactions, Proc. Ac. Sci. USSR 157, 392–95.
[36] Zhai, H. Y., Zhai, W. J., & Chen, B. Z. (2018). A class of implicit symmetric symplectic and exponentially fitted Runge–Kutta–Nystr¨om methods for solving oscillatory problems. Advances in Difference Equations, 2018, 1-16.
[37] Zhang, R., Jiang, W., Ehigie, J. O., Fang, Y., & You, X. (2017). Novel phase-fitted sym- metric splitting methods for chemical oscillators. Journal of Mathematical Chemistry, 55, 238–258.