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Volume 35, Issue 5
Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations

Yubin Lu, Chi-An Chen, Xiaofan Li & Chun Liu

DOI: 10.4208/cicp.OA-2023-0284

Commun. Comput. Phys., 35 (2024), pp. 1352-1386.

Published online: 2024-06

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  • Abstract

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.

  • Keywords

Keller-Segel equations, energy dissipation, positive preserving, ADI scheme.

  • AMS Subject Headings

65M06, 35K61,35K55, 65Z05

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-35-1352, author = {Lu , YubinChen , Chi-AnLi , Xiaofan and Liu , Chun}, title = {Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations}, journal = {Communications in Computational Physics}, year = {2024}, volume = {35}, number = {5}, pages = {1352--1386}, abstract = {

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0284}, url = {http://global-sci.org/intro/article_detail/cicp/23195.html} }
TY - JOUR T1 - Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations AU - Lu , Yubin AU - Chen , Chi-An AU - Li , Xiaofan AU - Liu , Chun JO - Communications in Computational Physics VL - 5 SP - 1352 EP - 1386 PY - 2024 DA - 2024/06 SN - 35 DO - http://doi.org/10.4208/cicp.OA-2023-0284 UR - https://global-sci.org/intro/article_detail/cicp/23195.html KW - Keller-Segel equations, energy dissipation, positive preserving, ADI scheme. AB -

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.

Yubin Lu, Chi-An Chen, Xiaofan Li & Chun Liu. (2024). Finite Difference Approximation with ADI Scheme for Two-Dimensional Keller-Segel Equations. Communications in Computational Physics. 35 (5). 1352-1386. doi:10.4208/cicp.OA-2023-0284
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