Numerical Study of Nonlinear Reaction-Diffusion Equation in CatalyticPellet Model Using Finite Element Method
DOI:
https://doi.org/10.23918/eajse.v11i2p15Keywords:
Catalytic Pellet Modeling, Numerical Analysis, Finite Element Method, Stability and ConvergenceAbstract
This study explores numerical solutions to nonlinear reaction-diffusion equations, with a focus on modeling concentration profiles in catalytic pellets, crucial for many chemical engineering applications. The problem is discretised and investigated in both the temporal and spatial domains using finite element method. A weak formulation is developed, and the existence and uniqueness of the finite element method solution are established. Stability and convergence of the numerical schemes are rigorously analyzed, and Crank–Nicolson-based discretization is implemented for enhanced accuracy. Numerical results illustrate the effectiveness of finite element method and finite difference method, with close agreement between the methods. This comparison highlights finite element method’s potential advantages in catalytic process modeling, showcasing its effectiveness in solving nonlinear reaction-diffusion PDEs
References
[1] El Danaf, T. S. Efficient and accurate numerical treatment of Huxley equation. International Journal of Numerical Methods for Heat & Fluid Flow. 2011, 21(3), 282–292. https://doi.org/10.1108/09615531111108468.
[2] Sadeeq, M. I. & Sabawi, Y. A. Cubic B-spline finite element method for solving Whitham BroerKaup shallow water model. International Journal of Computational Methods. 2025. https://doi.org/10.1142/S021987622550032X.
[3] Thomée, V. Galerkin Finite Element Methods for Parabolic Problems. 2nd ed. Berlin: SpringerVerlag, 2006. https://link.springer.com/book/10.1007/3-540-33122-0.
[4] Atouani, N. & Omrani, K. Galerkin finite element method for the Rosenau-RLW equation. Computers & Mathematics with Applications. 2013, 66(3), 289–303. https://doi.org/10.1016/j.camwa.2013.04.029.
[5] Ciarlet, P. G. The Finite Element Method for Elliptic Problems. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2002.
[6] Chung, S. K. & Ha, S. N. Finite element Galerkin solutions for the Rosenau equation. Applicable Analysis. 1994, 54(1-2), 39–56. https://doi.org/10.1080/00036819408840267.
[7] Hartman, P. Ordinary Differential Equations. SIAM, 2002.
[8] Sabawi, Y. A. A posteriori error analysis in finite element approximation for fully discrete semilinear parabolic problems. In Finite Element Methods and Their Applications. IntechOpen, 2020.https://doi.org/10.5772/intechopen.94369.
[9] Sabawi, Y. A. A Posteriori L∞(L2) + L2(H1)–Error bounds in discontinuous galerkin methods for semidiscrete semilinear parabolic interface problems. Baghdad Science Journal. 2021, 18(3),0522. https://doi.org/10.21123/bsj.2021.18.3.0522.
[10] Pirdawood, M. A. & Sabawi, Y. A. High-order solution of Generalized Burgers–Fisher Equationusing compact finite difference and DIRK methods. Journal of Physics: Conference Series. 2021,1999, 012088. https://doi.org/10.1088/1742-6596/1999/1/012088.
[11] Sabawi, Y. A., Pirdawood, M. A. & Sadeeq, M. I. A compact fourth-order implicit-explicitRunge-Kutta type method for solving diffusive Lotka–Volterra system. Journal of Physics: Conference Series. 2021, 1999, 012103. https://doi.org/10.1088/1742-6596/1999/1/012103.
[12] Cangiani, A., Georgoulis, E. H. & Sabawi, Y. A. Convergence of an adaptive discontinuousGalerkin method for elliptic interface problems. Journal of Computational and Applied Mathematics. 2020, 367, 112397.
[13] Cangiani, A., Georgoulis, E. & Sabawi, Y. Adaptive discontinuous Galerkin methods for ellipticinterface problems. Mathematics of Computation. 2018, 87(314), 2675–2707.
[14] Evans, L. C. Partial Differential Equations. 2nd ed. American Mathematical Society, 2022.
[15] Brezis, H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer Science & Business Media, 2011.
[16] Al-Bairaqdar, O.T., Sabawi, Y.A. and Hasso, M.S., 2025. Efficient numerical solution of theAllen–Cahn equation via a non-lumping Galerkin finite element method. AIP Advances, 15(7).https://doi.org/10.1063/5.0262503.
[17] Quarteroni, A. Numerical Models for Differential Problems. Vol. 2. Springer, 2009.
[18] Kesavan, S. Topics in Functional Analysis and Applications. 1989.
[19] Cho, Y. Modeling of Cone-Shaped Pellets for Catalytic Reactors. Journal of Chemical Engineering of Japan. 2024, 57(1), 2317457. https://doi.org/10.1080/00219592.2024.2317457.
[20] Cho, Y. Modeling of porous catalytic pellets with various morphologies suspended in batch reactor and CSTR from analytical solutions of reaction-diffusion equations. Journal of Chemical Engineering of Japan. 2023, 56(1), 2247783. https://doi.org/10.1080/00219592.2023.2247783.
[21] Kreyszig, E. Introductory Functional Analysis with Applications. John Wiley & Sons, 1991.
[22] Rufai, M. A., Carpentieri, B. & Ramos, H. A New Hybrid Block Method for Solving First-Order
Differential System Models in Applied Sciences and Engineering. Fractal and Fractional. 2023,
7(10), 703. https://doi.org/10.3390/fractalfract7100703.
[23] Tannhäuser, K., Serrao, P. H. & Kozinov, S. A three-dimensional collocation finite element
method for higher-order electromechanical coupling. Computers & Structures. 2024, 291,107219. https://doi.org/10.1016/j.compstruc.2023.107219.
[24] Mohammad, N. A., Sabawi, Y. A. & Hasso, M. S. Numerical solution based on the Haar wavelet collocation method for partial integro-differential equations of Volterra type. Arab Journal of
Basic and Applied Sciences. 2024, 31(1), 614–628. https://doi.org/10.1080/25765299.2024.2419145.
[25] Mohammad, N. A., Sabawi, Y. A. & Hasso, M. S. A compact finite-difference and Haar wavelets
collocation technique for parabolic volterra integro-differential equations. Physica Scripta. 2024,
99(12), 125251. https://doi.org/10.1088/1402-4896/ad8d3d.
[26] Hellman, F., Målqvist, A. & Mosquera, M. Well-posedness and finite element approximation of
mixed dimensional partial differential equations. BIT Numerical Mathematics. 2024, 64(1), 2.
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