Numerical Solutions of Stiff Chemical Reaction Problems using Hybrid Block Backward Differentiation Formula
DOI:
https://doi.org/10.37934/arnht.25.1.100115Keywords:
Stiff ODEs, Chemical reaction problems, Hybrid method, Block backward differentiation formulaAbstract
This research paper introduces an advanced approach to address the numerical challenges associated with stiff chemical reaction problems. We propose employing a Hybrid Diagonally Implicit Block Backward Differentiation Formula coupled with strategically placed off-step points to improve the accuracy and efficiency of numerical solutions. Stiff chemical reactions, commonly encountered in various industrial processes, require advanced numerical techniques to precisely capture rapid changes in concentrations. Our hybrid formulation enhances stability and computational efficiency by building on the diagonally implicit structure of block backward differentiation formulas, offering improved performance for solving stiff chemical reaction problems. Under a specific selection of a free parameter, the method is found to possess both zero-stability and A−stability properties. Convergence analysis demonstrates its ability to accurately approximate exact solutions. Through rigorous experimentation and comparative analysis, this research will illustrate the effectiveness of the developed method in solving stiff ordinary differential equations. The expected outcomes include the development of the new numerical method, its validation through comprehensive numerical experiments and insights into its performance and applicability in diverse science and engineering domains.
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References
Lambert, J.D. (1973). Computational Methods in Ordinary Differential Equations. Wiley.
Manca, D., G. Buzzi-Ferraris, T. Faravelli, and E. Ranzi. “Numerical problems in the solution of oxidation and combustion models.” Combustion Theory and Modelling 5, no. 2 (2001): 185 – 199. https://doi.org/10.1088/1364-7830/5/2/304 DOI: https://doi.org/10.1088/1364-7830/5/2/304
Liang, Long, Song-Charng Kong, Chulhwa Jung, and Rolf D. Reitz. ”Development of a semi-implicit solver for detailed chemistry in internal combustion engine simulations.” Journal of Engineering for Gas Turbines and Power 129, no. 1 (2007): 271 – 278. https://doi.org/10.1115/1.2204979 DOI: https://doi.org/10.1115/1.2204979
Antonelli, Laura, Paola Belardini, Pasqua D’Ambra, Francesco Gregoretti, and Gennaro Oliva. ”A distributed combustion solver for engine simulations on grids.” Journal of Computational and Applied Mathematics 226, no. 2 (2009): 197 – 204. https://doi.org/10.1016/j.cam.2008.08.002 DOI: https://doi.org/10.1016/j.cam.2008.08.002
Ibrahim, Zarina Bibi, Khairil Iskandar Othman, and Mohamed Suleiman. “Implicit r-point block backward differentiation formula for solving first-order stiff ODEs.” Applied Mathematics and Computation 186, no. 1 (2007): 558 – 565. https://doi.org/10.1016/j.amc.2006.07.116 DOI: https://doi.org/10.1016/j.amc.2006.07.116
Abdulganiy, R.I., O.A. Akinfenwa, and S.A. Okunuga. “Construction of L stable second derivative trigonometrically fitted block backward differentiation formula for the solution of oscillatory initial value problems.” African Journal of Science, Technology, Innovation and Development 10, no. 4 (2018): 411 – 419, https://doi.org/10.1080/20421338.2018.1467859 DOI: https://doi.org/10.1080/20421338.2018.1467859
Aksah, Saufianim Jana, Zarina Bibi Ibrahim, and Iskandar Shah Mohd Zawawi. “Stability analysis of singly diagonally implicit block backward differentiation formulas for stiff ordinary differential equations.” Mathematics 7, no. 2 (2019): 211. https://doi.org/10.3390/MATH7020211 DOI: https://doi.org/10.3390/math7020211
Aksah, Saufianim Jana, and Zarina Bibi Ibrahim. “Singly diagonally implicit block backward differentiation formulas for HIV infection of CD4+T cells.” Symmetry 11, no. 5 (2019): 625. https://doi.org/10.3390/sym11050625 DOI: https://doi.org/10.3390/sym11050625
Ibrahim, Zarina Bibi, Nursyazwani Mohd Noor, and Khairil Iskandar Othman. “Fixed coefficient A(α) stable block backward differentiation formulas for stiff ordinary differential equations.” Symmetry 11, no. 7 (2019): 846. https://doi.org/10.3390/sym11070846 DOI: https://doi.org/10.3390/sym11070846
Ijam, Hazizah Mohd, and Zarina Bibi Ibrahim. “Diagonally Implicit block backward differentiation formula with optimal stability properties for stiff ordinary differential equations.” Symmetry 11, (2019): 1342. https://doi.org/10.3390/sym11111342
Ibrahim, Zarina Bibi, and Amiratul Ashikin Nasarudin. “A class of hybrid multistep block methods with A-stability for the numerical solution of stiff ordinary differential equations.” Mathematics 8, no. 6 (2020): 914. https://doi.org/10.3390/math8060914 DOI: https://doi.org/10.3390/math8060914
Zawawi, Iskandar Shah, and Zarina Bibi Ibrahim. ”BBDF-α for solving stiff ordinary differential equations with oscillating solutions.” Tamkang Journal of Mathematics 51, no. 2 (2020): 123 – 136. https://doi.org/10.5556/j.tkjm.51.2020.2964 DOI: https://doi.org/10.5556/j.tkjm.51.2020.2964
Ijam, Hazizah Mohd, Zarina Bibi Ibrahim, Zanariah Abdul Majid, and Norazak Senu. (2020). ”Stability analysis of a diagonally implicit scheme of block backward differentiation formula for stiff pharmacokinetics models.” Advances in Difference Equations, (2020):400. https://doi.org/10.1186/s13662-020-02846-z DOI: https://doi.org/10.1186/s13662-020-02846-z
Jator, S.N., R.K. Sahi, M.I. Akinyemi, and D. Nyonna. “Exponentially fitted block backward differentiation formulas for pricing options.” Cogent Economics and Finance 9, no. 1 (2021): 1875565. https://doi.org/10.1080/23322039.2021.1875565 DOI: https://doi.org/10.1080/23322039.2021.1875565
Rasid, Norshakila Abd, Zarina Bibi Ibrahim, Zanariah Abdul Majid, Fudziah Ismail, and Azman Ismail. “An Efficient Direct Diagonal Hybrid Block Method for Stiff Second Order Differential Equations.” Advanced Structured Materials, no. 166 (2022): 147 – 156. https://doi.org/10.1007/978-3-030-89992-9_14 DOI: https://doi.org/10.1007/978-3-030-89992-9_14
Zainuddin, Nooraini, Zarina Bibi Ibrahim, and Iskandar Shah Mohd Zawawi. “Diagonal Block Method for Stiff Van der Pol Equation.” IAENG International Journal of Applied Mathematics 53, no. 1 (2023).
Ogunniran, Muideen O., Gabriel C. Olaleye, Omotayo A. Taiwo, Ali Shokri, and Kamsing Nonlaopon. “Generalization of a class of uniformly optimized k-step hybrid block method for solving two-point boundary value problems.” Results in Physics 44, (2023): 106147. https://doi.org/10.1016/j.rinp.2022.106147 DOI: https://doi.org/10.1016/j.rinp.2022.106147
Abasi, Naghmeh, Mohamed Suleiman, Neda Abbasi, and Hamisu Musa. ”2-Point block BDF method with off-step points for solving stiff ODEs.” Journal of Soft Computing and Applications, (2014): 1 – 15. https://doi.org/10.5899/2014/jsca-00039 DOI: https://doi.org/10.5899/2014/jsca-00039
Soomro, Hira, Nooraini Zainuddin, Hanita Daud, Joshua Sunday, Noraini Jamaludin, Abdullah Abdullah, Apriyanto Mulono, and Evizal Abdul Kadir. “3-point block backward differentiation formula with an off-step point for the solutions of stiff chemical reaction problems.” Journal of Mathematics Chemistry 61, (2023): 75 – 97. https://doi.org/10.1007/s10910-022-01402-2 DOI: https://doi.org/10.1007/s10910-022-01402-2
Alhassan, Buhari, and Hamisu Musa. ”Diagonally implicit extended 2-point super class of block backward differentiation formula with two off-step points for solving first order stiff initial value problems.” Applied Mathematics and Computational Intelligence 12, no. 1 (2023): 101 – 124. DOI: https://doi.org/10.56919/usci.1222.004
Ebadi, Moosa, and M.Y. Gokhale. “Solving nonlinear parabolic PDEs via extended hybrid BDF methods.” Indian Journal of Pure and Applied Mathematics 45, no. 3 (2014): 395 – 412. https://doi.org/10.1007/s13226-014-0070-y DOI: https://doi.org/10.1007/s13226-014-0070-y
Ibrahim, Iman H., and Fatma M. Yousry. “Hybrid special class for solving differential-algebraic equations.” Numerical Algorithms 69, no. 2 (2015): 301 – 320. https://doi.org/10.1007/s11075-014-9897-x DOI: https://doi.org/10.1007/s11075-014-9897-x
Ebadi, Moosa (2018). “New class of hybrid BDF methods for the computation of numerical solutions of IVPs.” Numerical Algorithms 79, no. 1 (2018): 179 – 193. https://doi.org/10.1007/s11075-017-0433-7 DOI: https://doi.org/10.1007/s11075-017-0433-7
Isa, Syahirbanun, Zanariah Abdul Majid, Fudziah Ismail, and Faranak Rabiei. “Diagonally Implicit Multistep Block Method of Order Four for Solving Fuzzy Differential Equations Using Seikkala Derivatives.” Symmetry 10, no. 2 (2018): 42. https://doi.org/10.3390/sym10020042 DOI: https://doi.org/10.3390/sym10020042
Crockatt, Michael M., Andrew J. Christlieb, C. Kristopher Garrett, and Cory D. Hauck. “Hybrid methods for radiation transport using diagonally implicit Runge–Kutta and space–time discontinuous Galerkin time integration.” Journal of Computational Physics 376, (2019): 455 – 477. https://doi.org/10.1016/j.jcp.2018.09.041 DOI: https://doi.org/10.1016/j.jcp.2018.09.041
Kulikov, G.Yu., and R. Weiner. ”Variable-stepsize doubly quasi-consistent singly diagonally implicit two-step peer pairs for solving stiff ordinary differential equations.” Applied Numerical Mathematics 154, (2020): 223 – 242. https://doi.org/10.1016/j.apnum.2020.04.003 DOI: https://doi.org/10.1016/j.apnum.2020.04.003
Ijam, Hazizah Mohd, Zarina Bibi Ibrahim, Zanariah Abdul Majid, Norazak Senu, and Khairil Iskandar Othman (2019). “ρ-diagonally implicit block backward differentiation method for solving stiff ordinary differential equations.” Journal of Advanced Research in Dynamical & Control Systems 11, Special Issue-12 (2019): 145 – 154. DOI: https://doi.org/10.3390/sym11111342
Azizan, Farah Liyana, Saratha Sathasivam, Muraly Velavan, Nur Rusyidah Azri, and Nur Iffah Rafhanah Abdul Manaf (2024). “Prediction of drug concentration in human bloodstream using Adams-Bashforth-Moulton method.” Semarak Engineering Journal 4, no. 1 (2024): 29 – 46. https://semarakilmu.com.my/journals/index.php/sem_eng/article/view/7420. DOI: https://doi.org/10.37934/araset.29.2.5371
Hairer, E., and Wanner, G. (1996). Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer. DOI: https://doi.org/10.1007/978-3-642-05221-7
Babangida, B., H. Musa, and L.K. Ibrahim. “A new numerical method for solving stiff initial value problems.” Fluid Mechanics 3, no. 2 (2016). https://doi.org/10.4172/2476-2296.1000136 DOI: https://doi.org/10.4172/2476-2296.1000136
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