AN ASYMPTOTIC PRESERVING SCHEME FOR KINETIC CHEMOTAXIS MODELS IN TWO SPACE DIMENSIONS

Chertock, Alina1; Kurganov, Alexander2,3; Lukacova-Medvidova, Maria4; Ozcan, Seyma Nur1

2019-02

10.3934/krm.2019009

Kinetic and Related Models   Impact Factor & Quartile

1937-5093

1937-5077

In this paper, we study two-dimensional multiscale chemotaxis models 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 which describes the velocity change of the cells. The parabolic scaling yields a non-dimensional kinetic model with a small parameter, which represents the mean free path of the cells. We propose a new asymptotic preserving numerical scheme that reflects the convergence of the studied micro-macro model to its macroscopic counterpart the Patlak-Keller-Segel system in the singular limit. The method is based on the operator splitting strategy and a suitable combination of the higher-order implicit and explicit time discretizations. In particular, we use the so-called even-odd decoupling and approximate the stiff terms arising in the singular limit implicitly. We prove that the resulting scheme satisfies the asymptotic preserving property. More precisely, it yields a consistent approximation of the Patlak-Keller-Segel system as the mean-free path tends to 0. The derived asymptotic preserving method is used to get better insight to the blowup behavior of two-dimensional kinetic chemotaxis model.

Chemotaxis kinetic equations multiscale models asymptotic preserving (AP) methods operator splitting

SCI

English

Others

NSF RNMS Grant[DMS-1107444]

Mathematics

Mathematics, Applied ; Mathematics

WOS:000440712000009

AMER INST MATHEMATICAL SCIENCES-AIMS

Web of Science

1.North Carolina State Univ, Dept Math, Raleigh, NC 27695 USA
2.Southern Univ Sci & Technol China, Dept Math, Shenzhen 518055, Peoples R China
3.Tulane Univ, Math Dept, New Orleans, LA 70118 USA
4.Johannes Gutenberg Univ Mainz, Inst Math, Staudingerweg 9, D-55099 Mainz, Germany

Chertock, Alina,Kurganov, Alexander,Lukacova-Medvidova, Maria,et al. AN ASYMPTOTIC PRESERVING SCHEME FOR KINETIC CHEMOTAXIS MODELS IN TWO SPACE DIMENSIONS[J]. Kinetic and Related Models,2019,12(1):195-216.

Chertock, Alina,Kurganov, Alexander,Lukacova-Medvidova, Maria,&Ozcan, Seyma Nur.(2019).AN ASYMPTOTIC PRESERVING SCHEME FOR KINETIC CHEMOTAXIS MODELS IN TWO SPACE DIMENSIONS.Kinetic and Related Models,12(1),195-216.

Chertock, Alina,et al."AN ASYMPTOTIC PRESERVING SCHEME FOR KINETIC CHEMOTAXIS MODELS IN TWO SPACE DIMENSIONS".Kinetic and Related Models 12.1(2019):195-216.