Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



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
Published electronically: September 19, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Shen Bian and Jian-Guo Liu, Dynamic and steady states for multi-dimensional Keller-Segel model with diffusion exponent $ m>0$, Comm. Math. Phys. 323 (2013), no. 3, 1017-1070. MR 3106502,
  • [2] Adrien Blanchet, Jean Dolbeault, and Benoît Perthame, Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations (2006), No. 44, 32. MR 2226917
  • [3] Adrien Blanchet, José Antonio Carrillo, David Kinderlehrer, Michał Kowalczyk, Philippe Laurençot, and Stefano Lisini, A hybrid variational principle for the Keller-Segel system in $ \mathbb{R}^2$, ESAIM Math. Model. Numer. Anal. 49 (2015), no. 6, 1553-1576. MR 3423264,
  • [4] Michael P. Brenner, Peter Constantin, Leo P. Kadanoff, Alain Schenkel, and Shankar C. Venkataramani, Diffusion, attraction and collapse, Nonlinearity 12 (1999), no. 4, 1071-1098. MR 1709861,
  • [5] Vincent Calvez and Lucilla Corrias, The parabolic-parabolic Keller-Segel model in $ \mathbb{R}^2$, Commun. Math. Sci. 6 (2008), no. 2, 417-447. MR 2433703
  • [6] José A. Carrillo, Alina Chertock, and Yanghong Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys. 17 (2015), no. 1, 233-258. MR 3372289,
  • [7] José A. Carrillo and Bokai Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Model. Simul. 11 (2013), no. 1, 336-361. MR 3032835,
  • [8] Yingda Cheng and Irene M. Gamba, Numerical study of one-dimensional Vlasov-Poisson equations for infinite homogeneous stellar systems, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 5, 2052-2061. MR 2863075,
  • [9] Alina Chertock and Alexander Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math. 111 (2008), no. 2, 169-205. MR 2456829,
  • [10] Alina Chertock, Alexander Kurganov, Xuefeng Wang, and Yaping Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models 5 (2012), no. 1, 51-95. MR 2875735,
  • [11] Wenting Cong and Jian-Guo Liu, Uniform $ L^\infty$ boundedness for a degenerate parabolic-parabolic Keller-Segel model, Discrete Contin. Dyn. Syst. Ser. B 22 (2017), no. 2, 307-338. MR 3639118
  • [12] K. Craig, I. Kim, and Y. Yao, Congested aggregation via newtonian interaction, arXiv:1603.03790 (2016).
  • [13] Yekaterina Epshteyn and Alexander Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model, SIAM J. Numer. Anal. 47 (2008/09), no. 1, 386-408. MR 2475945,
  • [14] Elio Espejo, Karina Vilches, and Carlos Conca, Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $ \mathbb{R}^2$, European J. Appl. Math. 24 (2013), no. 2, 297-313. MR 3031781,
  • [15] Francis Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math. 104 (2006), no. 4, 457-488. MR 2249674,
  • [16] Zhen Guan, John S. Lowengrub, Cheng Wang, and Steven M. Wise, Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations, J. Comput. Phys. 277 (2014), 48-71. MR 3254224,
  • [17] Shi Jin and Li Wang, An asymptotic preserving scheme for the Vlasov-Poisson-Fokker-Planck system in the high field regime, Acta Math. Sci. Ser. B Engl. Ed. 31 (2011), no. 6, [November 2010 on cover], 2219-2232. MR 2931501,
  • [18] Shi Jin and Bokai Yan, A class of asymptotic-preserving schemes for the Fokker-Planck-Landau equation, J. Comput. Phys. 230 (2011), no. 17, 6420-6437. MR 2818606,
  • [19] E. Keller and L. Segel, Traveling bands of chemotactic bacteria: a theoretical analysis, J. Theoret. Biol. 30 (1971), no. 2, 6420-6437.
  • [20] Alexander Kurganov and Mária Lukáčová-Medviďová, Numerical study of two-species chemotaxis models, Discrete Contin. Dyn. Syst. Ser. B 19 (2014), no. 1, 131-152. MR 3245085,
  • [21] X. Li, C.-W. Shu, and Y. Yang, Local discontinuous Galerkin method for the Keller-Segel chemotaxis model, IMA J. Numer. Anal. submittted.
  • [22] Jian-Guo Liu and Jinhuan Wang, Refined hyper-contractivity and uniqueness for the Keller-Segel equations, Appl. Math. Lett. 52 (2016), 212-219. MR 3416408,
  • [23] Clifford S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15 (1953), 311-338. MR 0081586
  • [24] Benoît Perthame, Transport equations in biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. MR 2270822

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M06, 65M12, 35Q92

Retrieve articles in all journals with MSC (2010): 65M06, 65M12, 35Q92

Additional Information

Jian-Guo Liu
Affiliation: Department of Mathematics and Department of Physics, Duke University, Box 90320, Durham, North Carolina 27708

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

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

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

American Mathematical Society