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High-accuracy positivity-preserving numerical method for Keller-Segel model.

Lin Zhang1,2, Yongbin Ge2, Xiaojia Yang3

  • 1School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China.

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|May 10, 2023
PubMed
Summary

This study introduces a high-precision, fourth-order accurate compact difference scheme for the Keller-Segel model, improving numerical simulations of biological processes like cell movement and chemoattractant concentration.

Keywords:
Keller-Segel modelfinite-difference methodfinite-time blow-uphigh-accuracypositivity-preserving

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Area of Science:

  • Mathematical Biology
  • Computational Science
  • Numerical Analysis

Background:

  • The Keller-Segel model describes cell density and chemoattractant concentration, crucial for simulating biological processes.
  • Existing numerical methods often lack temporal accuracy for this nonlinear partial differential system.

Purpose of the Study:

  • To develop a high-precision and stable compact difference scheme for the Keller-Segel model.
  • To achieve fourth-order accuracy in both space and time for numerical solutions.

Main Methods:

  • Utilized a fourth-order backward difference formula for temporal discretization.
  • Employed compact difference operators for spatial discretization.
  • Developed boundary conditions using Taylor series expansion for consistent accuracy.

Main Results:

  • Proposed a novel space-time fourth-order accurate compact difference scheme.
  • Established multigrid and positivity-preserving algorithms maintaining fourth-order accuracy.
  • Verified accuracy and reliability through numerical experiments, including finite-time blow-up and conservation laws.

Conclusions:

  • The new scheme offers a significant improvement in accuracy and stability for solving the Keller-Segel model.
  • The developed algorithms effectively handle complex phenomena like finite-time blow-up and preserve essential physical properties.