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Positivity-preserving and asymptotic preserving method for 2D Keller-Segal equations


Authors: Jian-Guo Liu, Li Wang and Zhennan Zhou
Journal: Math. Comp. 87 (2018), 1165-1189
MSC (2010): Primary 65M06, 65M12, 35Q92
DOI: https://doi.org/10.1090/mcom/3250
Published electronically: September 19, 2017
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Abstract: We propose a semi-discrete scheme for 2D Keller-Segel equations based on a symmetrization reformation, which is equivalent to the convex splitting method and is free of any nonlinear solver. We show that, this new scheme is stable as long as the initial condition does not exceed certain threshold, and it asymptotically preserves the quasi-static limit in the transient regime. Furthermore, we show that the fully discrete scheme is conservative and positivity preserving, which makes it ideal for simulations. The analogical schemes for the radial symmetric cases and the subcritical degenerate cases are also presented and analyzed. With extensive numerical tests, we verify the claimed properties of the methods and demonstrate their superiority in various challenging applications.


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Additional Information

Jian-Guo Liu
Affiliation: Department of Mathematics and Department of Physics, Duke University, Box 90320, Durham, North Carolina 27708
Email: jliu@phy.duke.edu

Li Wang
Affiliation: Department of Mathematics and Computational and Data-Enabled Science and Engineering Program, State University of New York at Buffalo, 244 Mathematics Building, Buffalo, New York 14260
Email: lwang46@buffalo.edu

Zhennan Zhou
Affiliation: Department of Mathematics, Duke University, Box 90320, Durham, North Carolina 27708
Address at time of publication: Beijing International Center for Mathematical Research, Peking University, Beijing, People’s Republic of China 100871
Email: zhennan@bicmr.pku.edu.cn

DOI: https://doi.org/10.1090/mcom/3250
Received by editor(s): April 13, 2016
Received by editor(s) in revised form: April 28, 2016, October 17, 2016, and December 12, 2016
Published electronically: September 19, 2017
Additional Notes: The first author was partially supported by RNMS11-07444 (KI-Net) and NSF grant DMS 1514826
The second author was partially supported by a start-up fund from the State University of New York at Buffalo and NSF grant DMS 1620135
The third author was partially supported by a start-up fund from Peking University and RNMS11-07444 (KI-Net)
Article copyright: © Copyright 2017 American Mathematical Society

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