High-accuracy positivity-preserving numerical method for Keller-Segel model

Lin Zhang; Yongbin Ge; Xiaojia Yang; Lin Zhang; Yongbin Ge; Xiaojia Yang

doi:10.3934/mbe.2023378

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 8601-8631. doi: 10.3934/mbe.2023378

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Research article

High-accuracy positivity-preserving numerical method for Keller-Segel model

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2.
Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan 750021, China
3.
School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756000, China

Received: 04 January 2023 Revised: 10 February 2023 Accepted: 23 February 2023 Published: 06 March 2023

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.
- Keller-Segel model,
- finite-difference method,
- high-accuracy,
- positivity-preserving,
- finite-time blow-up
Citation: Lin Zhang, Yongbin Ge, Xiaojia Yang. High-accuracy positivity-preserving numerical method for Keller-Segel model[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 8601-8631. doi: 10.3934/mbe.2023378

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Abstract

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.

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