Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications
In this study, a class of nonlinear heterogeneous reaction–diffusion system (RDS) has been considered that arises in modeling epidemiological interactions, environmental sciences, and chemical and ecological systems. Numerical and analytic solutions for this kind of variable medium nonlinear RDS are...
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MDPI AG
2025-01-01
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| Series: | Mathematics |
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| author | Samima Akhter Md. Ariful Islam Arif Rubayyi T. Alqahtani Samir Kumar Bhowmik |
| author_facet | Samima Akhter Md. Ariful Islam Arif Rubayyi T. Alqahtani Samir Kumar Bhowmik |
| author_sort | Samima Akhter |
| collection | DOAJ |
| description | In this study, a class of nonlinear heterogeneous reaction–diffusion system (RDS) has been considered that arises in modeling epidemiological interactions, environmental sciences, and chemical and ecological systems. Numerical and analytic solutions for this kind of variable medium nonlinear RDS are challenging. This article developed a few highly accurate numerical schemes for such problems. For the spatial integration of the heterogeneous RDS, a few finite difference schemes, a Bernstein collocation scheme, and a Fourier transform scheme were employed. The stability and accuracy analysis of the spectral schemes were studied to confirm the order of convergence of the approximation. A few methods of lines were then used for the temporal integration of the resulting semidiscrete model. It was confirmed theoretically that the spectral/pseudo-spectral method is very efficient and highly accurate for such a model. A fast and efficient solver for the resulting full discrete system is highly desired. A Newton–GMRES–Multigrid solver was applied for the resulting full discrete system. It is demonstrated in tabular form that a multigrid accelerated Newton–GMRES solver outperforms most linear solvers for such a model. |
| format | Article |
| id | doaj-art-4dc76e93dd2448cdbe273fe2f2c07f12 |
| institution | DOAJ |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-4dc76e93dd2448cdbe273fe2f2c07f122025-08-20T02:48:09ZengMDPI AGMathematics2227-73902025-01-0113335510.3390/math13030355Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with ApplicationsSamima Akhter0Md. Ariful Islam Arif1Rubayyi T. Alqahtani2Samir Kumar Bhowmik3Department of Mathematics, University of Dhaka, Dhaka 1000, BangladeshDepartment of Mathematics, University of Dhaka, Dhaka 1000, BangladeshDepartment of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi ArabiaDepartment of Mathematics, University of Dhaka, Dhaka 1000, BangladeshIn this study, a class of nonlinear heterogeneous reaction–diffusion system (RDS) has been considered that arises in modeling epidemiological interactions, environmental sciences, and chemical and ecological systems. Numerical and analytic solutions for this kind of variable medium nonlinear RDS are challenging. This article developed a few highly accurate numerical schemes for such problems. For the spatial integration of the heterogeneous RDS, a few finite difference schemes, a Bernstein collocation scheme, and a Fourier transform scheme were employed. The stability and accuracy analysis of the spectral schemes were studied to confirm the order of convergence of the approximation. A few methods of lines were then used for the temporal integration of the resulting semidiscrete model. It was confirmed theoretically that the spectral/pseudo-spectral method is very efficient and highly accurate for such a model. A fast and efficient solver for the resulting full discrete system is highly desired. A Newton–GMRES–Multigrid solver was applied for the resulting full discrete system. It is demonstrated in tabular form that a multigrid accelerated Newton–GMRES solver outperforms most linear solvers for such a model.https://www.mdpi.com/2227-7390/13/3/355heterogeneous reaction diffusion systemepidemicsfinite differencesBernstein polynomialFourier transformsmultigrid preconditioning |
| spellingShingle | Samima Akhter Md. Ariful Islam Arif Rubayyi T. Alqahtani Samir Kumar Bhowmik Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications Mathematics heterogeneous reaction diffusion system epidemics finite differences Bernstein polynomial Fourier transforms multigrid preconditioning |
| title | Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications |
| title_full | Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications |
| title_fullStr | Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications |
| title_full_unstemmed | Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications |
| title_short | Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications |
| title_sort | efficient numerical schemes for a heterogeneous reaction diffusion system with applications |
| topic | heterogeneous reaction diffusion system epidemics finite differences Bernstein polynomial Fourier transforms multigrid preconditioning |
| url | https://www.mdpi.com/2227-7390/13/3/355 |
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