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Efficient Numerical Schemes for a Two-Species Keller-Segel Model and Investigation of Its Blowup Phenomena in 3D

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Abstract

We consider in this paper numerical approximation and simulation of a two-species Keller-Segel model. The model enjoys an energy dissipation law, mass conservation and bound or positivity preserving for the population density of two species. We construct a class of very efficient numerical schemes based on the generalized scalar auxiliary variable with relaxation which preserve unconditionally the essential properties of the model at the discrete level. We conduct a sequence of numerical tests to validate the properties of these schemes, and to study the blow-up phenomena of the model in a three-dimensional domain in parabolic-elliptic form and parabolic-parabolic form.

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Notes

  1. In the particular case with \(\tau =\alpha =0\), the equation (1.3) should be adjusted to \(-\beta \Delta c = \gamma _{1}\rho _{1}+ \gamma _{2}\rho _{2} - \gamma _{1} \langle \rho _{1} \rangle - \gamma _{2} \langle \rho _{2} \rangle \) with \(\langle \rho _{i} \rangle = \frac{1}{| \Omega |} \int _{\Omega} \rho _{i} d\boldsymbol{x}\) (\(i=1,2\)), due to the compatibility with the boundary conditions involved [23, 29].

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Funding

This work is supported in part by NSFC grant 12371409.

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Authors and Affiliations

  1. School of Mathematical Sciences and Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing, Xiamen University, Xiamen, Fujian, 361005, China

    Xueling Huang

  2. Eastern Institute for Advanced Study, Eastern Institute of Technology, Ningbo, Zhejiang, 315200, P.R. China

    Jie Shen

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  1. Xueling Huang
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Correspondence to Jie Shen.

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Dedicated to Professor Shi Jin’s 60th Birthday

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Huang, X., Shen, J. Efficient Numerical Schemes for a Two-Species Keller-Segel Model and Investigation of Its Blowup Phenomena in 3D. Acta Appl Math 190, 10 (2024). https://doi.org/10.1007/s10440-024-00647-0

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