Institute of Applied Mathematics and Mechanics, Ningxia University, Yinchuan 750021, China.
Math Biosci Eng. 2022 May 5;19(7):6764-6794. doi: 10.3934/mbe.2022319.
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.
本文致力于开发一种精确有效的保正算符的高阶紧致差分方法，以求解 Keller-Segel 趋化模型，这是一种数学生物学中的非线性抛物-抛物系统。首先，提出了一个刚性稳定的五阶四步全隐紧致差分格式。新格式不仅在空间方向上具有四阶精度，而且在时间方向上也具有四阶精度，并给出了非线性趋化项的计算策略。然后，提出了一个保正数值算法，该算法在不损失精度的情况下保证了细胞密度在所有时间的非负性。并建立了一个时间推进算法。最后，将所提出的方法应用于趋化现象的数值模拟，通过几个数值例子验证了新格式的精度、稳定性和保正性。
