High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source

Panpan Xu; Yongbin Ge; Lin Zhang; Panpan Xu; Yongbin Ge; Lin Zhang

doi:10.3934/nhm.2023065

Networks and Heterogeneous Media

2023, Volume 18, Issue 4: 1471-1492. doi: 10.3934/nhm.2023065

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Research article

High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source

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

Received: 24 March 2023 Revised: 18 May 2023 Accepted: 22 May 2023 Published: 03 July 2023

In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated.
- Keller-Segel chemotaxis model,
- logistic source,
- self-diffusion terms,
- cross-diffusion terms,
- finite difference method,
- high-order accuracy
Citation: Panpan Xu, Yongbin Ge, Lin Zhang. High-order finite difference approximation of the Keller-Segel model with additional self- and cross-diffusion terms and a logistic source[J]. Networks and Heterogeneous Media, 2023, 18(4): 1471-1492. doi: 10.3934/nhm.2023065

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Abstract

In this paper, we consider the Keller-Segel chemotaxis model with self- and cross-diffusion terms and a logistic source. This system consists of a fully nonlinear reaction-diffusion equation with additional cross-diffusion. We establish some high-order finite difference schemes for solving one- and two-dimensional problems. The truncation error remainder correction method and fourth-order Padé compact schemes are employed to approximate the spatial and temporal derivatives, respectively. It is shown that the numerical schemes yield second-order accuracy in time and fourth-order accuracy in space. Some numerical experiments are demonstrated to verify the accuracy and reliability of the proposed schemes. Furthermore, the blow-up phenomenon and bacterial pattern formation are numerically simulated.

References

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