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High-Resolution Positivity and Asymptotic Preserving Numerical Methods for Chemotaxis and Related Models

  • Alina ChertockEmail author
  • Alexander Kurganov
Chapter
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)

Abstract

Many microorganisms exhibit a special pattern formation at the presence of a chemoattractant, food, light, or areas with high oxygen concentration. Collective cell movement can be described by a system of nonlinear PDEs on both macroscopic and microscopic levels. The classical PDE chemotaxis model is the Patlak-Keller-Segel system, which consists of a convection-diffusion equation for the cell density and a reaction-diffusion equation for the chemoattractant concentration. At the cellular (microscopic) level, a multiscale chemotaxis models can be used. These models are based on a combination of the macroscopic evolution equation for chemoattractant and microscopic models for cell evolution. The latter is governed by a Boltzmann-type kinetic equation with a local turning kernel operator that describes the velocity change of the cells.

A common property of the chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such singular solutions numerically is a challenging problem and the use of higher-order methods and/or adaptive strategies is often necessary. In addition, positivity preserving is an absolutely crucial property a good numerical method used to simulate chemotaxis should satisfy: this is the only way to guarantee a nonlinear stability of the method. For kinetic chemotaxis systems, it is also essential that numerical methods provide a consistent and stable discretization in certain asymptotic regimes.

In this paper, we review some of the recent advances in developing of high-resolution finite-volume and finite-difference numerical methods that possess the aforementioned properties of the chemotaxis-type systems.

Notes

Acknowledgements

A large portion of the material covered in this review is based on the work of the authors with Yekaterina Epshteyn, Hengrui Hu, Mária Lukáčová-Medvid’ ová, Mario Ricchiuto, Şeyma Nur Özcan, and Tong Wu, whose valuable contribution we would like to acknowledge here. The work of A. Chertock was supported in part by NSF grants DMS-1521051 and DMS-1818684. The work of A. Kurganov was supported in part by NSFC grant 11771201 and NSF grants DMS-1521009 and DMS-1818666.

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Authors and Affiliations

  1. 1.Department of Mathematics and Center for Research in Scientific ComputationNorth Carolina State UniversityRaleighUSA
  2. 2.Department of MathematicsSouthern University of Science and TechnologyShenzhenChina
  3. 3.Mathematics DepartmentTulane UniversityNew OrleansUSA

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