High-accuracy positivity-preserving numerical method for Keller-Segel model
The Keller-Segel model is a time-dependent nonlinear partial differential system, which couples a reaction-diffusion-chemotaxis equation with a reaction-diffusion equation; the former describes cell density, and the latter depicts the concentration of chemoattractants. This model plays a vital role...
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AIMS Press
2023-03-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2023378?viewType=HTML |
| _version_ | 1827977420773261312 |
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| author | Lin Zhang Yongbin Ge Xiaojia Yang |
| author_facet | Lin Zhang Yongbin Ge Xiaojia Yang |
| author_sort | Lin Zhang |
| collection | DOAJ |
| description | The Keller-Segel model is a time-dependent nonlinear partial differential system, which couples a reaction-diffusion-chemotaxis equation with a reaction-diffusion equation; the former describes cell density, and the latter depicts the concentration of chemoattractants. This model plays a vital role in the simulation of the biological processes. In view of the fact that most of the proposed numerical methods for solving the model are low-accuracy in the temporal direction, we aim to derive a high-precision and stable compact difference scheme by using a finite difference method to solve this model. First, a fourth-order backward difference formula and compact difference operators are respectively employed to discretize the temporal and spatial derivative terms in this model, and a compact difference scheme with the space-time fourth-order accuracy is proposed. To keep the accuracy of its boundary with the same order as the main scheme, a Taylor series expansion formula with the Peano remainder is used to discretize the boundary conditions. Then, based on the new scheme, a multigrid algorithm and a positivity-preserving algorithm which can guarantee the fourth-order accuracy are established. Finally, the accuracy and reliability of the proposed method are verified by diverse numerical experiments. Particularly, the finite-time blow-up, non-negativity, mass conservation and energy dissipation are numerically simulated and analyzed. |
| first_indexed | 2024-04-09T21:05:46Z |
| format | Article |
| id | doaj.art-fc8ae4a2a91f44bbb4ee1ae5bc68b2a2 |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-04-09T21:05:46Z |
| publishDate | 2023-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-fc8ae4a2a91f44bbb4ee1ae5bc68b2a22023-03-29T01:40:49ZengAIMS PressMathematical Biosciences and Engineering1551-00182023-03-012058601863110.3934/mbe.2023378High-accuracy positivity-preserving numerical method for Keller-Segel modelLin Zhang0Yongbin Ge1Xiaojia Yang 21. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China 2. Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan 750021, China2. Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan 750021, China3. School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756000, ChinaThe Keller-Segel model is a time-dependent nonlinear partial differential system, which couples a reaction-diffusion-chemotaxis equation with a reaction-diffusion equation; the former describes cell density, and the latter depicts the concentration of chemoattractants. This model plays a vital role in the simulation of the biological processes. In view of the fact that most of the proposed numerical methods for solving the model are low-accuracy in the temporal direction, we aim to derive a high-precision and stable compact difference scheme by using a finite difference method to solve this model. First, a fourth-order backward difference formula and compact difference operators are respectively employed to discretize the temporal and spatial derivative terms in this model, and a compact difference scheme with the space-time fourth-order accuracy is proposed. To keep the accuracy of its boundary with the same order as the main scheme, a Taylor series expansion formula with the Peano remainder is used to discretize the boundary conditions. Then, based on the new scheme, a multigrid algorithm and a positivity-preserving algorithm which can guarantee the fourth-order accuracy are established. Finally, the accuracy and reliability of the proposed method are verified by diverse numerical experiments. Particularly, the finite-time blow-up, non-negativity, mass conservation and energy dissipation are numerically simulated and analyzed.https://www.aimspress.com/article/doi/10.3934/mbe.2023378?viewType=HTMLkeller-segel modelfinite-difference methodhigh-accuracypositivity-preservingfinite-time blow-up |
| spellingShingle | Lin Zhang Yongbin Ge Xiaojia Yang High-accuracy positivity-preserving numerical method for Keller-Segel model Mathematical Biosciences and Engineering keller-segel model finite-difference method high-accuracy positivity-preserving finite-time blow-up |
| title | High-accuracy positivity-preserving numerical method for Keller-Segel model |
| title_full | High-accuracy positivity-preserving numerical method for Keller-Segel model |
| title_fullStr | High-accuracy positivity-preserving numerical method for Keller-Segel model |
| title_full_unstemmed | High-accuracy positivity-preserving numerical method for Keller-Segel model |
| title_short | High-accuracy positivity-preserving numerical method for Keller-Segel model |
| title_sort | high accuracy positivity preserving numerical method for keller segel model |
| topic | keller-segel model finite-difference method high-accuracy positivity-preserving finite-time blow-up |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2023378?viewType=HTML |
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