Positivity-preserving high-order compact difference method for the Keller-Segel chemotaxis model

Lin Zhang; Yongbin Ge; Zhi Wang; Lin Zhang; Yongbin Ge; Zhi Wang

doi:10.3934/mbe.2022319

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 6764-6794. doi: 10.3934/mbe.2022319

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Research article

Positivity-preserving high-order compact difference method for the Keller-Segel chemotaxis model

Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan 750021, China

Received: 27 January 2022 Revised: 29 March 2022 Accepted: 18 April 2022 Published: 05 May 2022

The paper is concerned with development of an accurate and effective positivity-preserving high-order compact difference method for solving the Keller-Segel chemotaxis model, which is a kind of nonlinear parabolic-parabolic system in mathematical biology. Firstly, a stiffly-stable five-step fourth-order fully implicit compact difference scheme is proposed. The new scheme not only has fourth-order accuracy in the spatial direction, but also has fourth-order accuracy in the temporal direction, and the computational strategy for the nonlinear chemotaxis term is provided. Then, a positivity-preserving numerical algorithm is presented, which ensures the non-negativity of cell density at all time without accuracy loss. And a time advancement algorithm is established. Finally, the proposed method is applied to the numerical simulation for chemotaxis phenomena, and the accuracy, stability and positivity-preserving of the new scheme are validated with several numerical examples.
Citation: Lin Zhang, Yongbin Ge, Zhi Wang. Positivity-preserving high-order compact difference method for the Keller-Segel chemotaxis model[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 6764-6794. doi: 10.3934/mbe.2022319

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Abstract

The paper is concerned with development of an accurate and effective positivity-preserving high-order compact difference method for solving the Keller-Segel chemotaxis model, which is a kind of nonlinear parabolic-parabolic system in mathematical biology. Firstly, a stiffly-stable five-step fourth-order fully implicit compact difference scheme is proposed. The new scheme not only has fourth-order accuracy in the spatial direction, but also has fourth-order accuracy in the temporal direction, and the computational strategy for the nonlinear chemotaxis term is provided. Then, a positivity-preserving numerical algorithm is presented, which ensures the non-negativity of cell density at all time without accuracy loss. And a time advancement algorithm is established. Finally, the proposed method is applied to the numerical simulation for chemotaxis phenomena, and the accuracy, stability and positivity-preserving of the new scheme are validated with several numerical examples.

References

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