An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions

Alina Chertock; Alexander Kurganov; Mária Lukáčová-Medvi${\rm{\check{d}}}$ová; Șeyma Nur Özcan

doi:10.3934/krm.2019009

Article Contents

2019, Volume 12, Issue 1: 195-216. Doi: 10.3934/krm.2019009

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An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions

Alina Chertock^1, ,,
Alexander Kurganov^2,3,,
Mária Lukáčová-Medvi${\rm{\check{d}}}$ová^4, and
Șeyma Nur Özcan^1,

1.
Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA
2.
Department of Mathematics, Southern University of Science and Technology of China, Shenzhen 518055, China
3.
Mathematics Department, Tulane University, New Orleans, LA 70118, USA
4.
Institute of Mathematics, University of Mainz, Staudingerweg 9, 55099 Mainz, Germany

* Corresponding author: Alina Chertock
* Corresponding author: Alina Chertock

Received: January 31, 2018

Published: July 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 92C17, 35Q20, 76M45; Secondary: 76P99, 65M06, 65M70.

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Figure 1. $ \verb"Example 1: Behavior of"$ $||\rho||_\infty/M$ $ \verb"in time for varying values of"$ $M$; $N_x = N_y = 128$

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Figure 2. $ \verb"Example 1: Behavior of"$ $||\rho||_\infty$ $ \verb"in time for"$ $M = 7$ $ \verb"(left) and"$ $M = 9$ $ \verb"(right) on three consecutive meshes"$

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Figure 3. $ \verb"Example 2a: Behavior of"$ $||\rho||_\infty$ $ \verb"in time for"$ $M = 1$ $ \verb"(left)"$, $M = 8$ $ \verb"(middle) and"$ $M = 11$ $ \verb"(right) on four consecutive meshes"$

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Figure 4. $ \verb"Example 2a: The density"$ $\rho(x,y,T = 0.0005)$ $ \verb"for"$ $M = 11$ $ \verb"computed on the meshes with:"$ $N_x = N_y = 128$ $ \verb"(left) and"$ $N_x = N_y = 256$ $ \verb"(right)"$

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Figure 5. $ \verb"Example 2b: Behavior of"$ $||\rho||_\infty$ $ \verb"in time for"$ $M = 8$ $ \verb"(left)"$, $M = 9.5$ $ \verb"(middle) and"$ $M = 11$ $ \verb"(right) on three consecutive meshes"$

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Figure 6. $ \verb"Example 3: The displacement of the density for"$ $M = 3$

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Figure 7. $ \verb"Example 3: The displacement of the density for"$ $M = 7$

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Figure 8. $ \verb"Example 3: The displacement of the density for"$ $M = 11$

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Table 1. $ \verb"Example 2b:"$ $L^\infty$- $ \verb"errors for"$ $M = 8, 9.5$ $ \verb"and"$ $11$ $ \verb"(from left to right)"$

	$M=8$		$M=9.5$		$M=11$
$N$	$\|\|e_{2N}\|\|_\infty$	rate$_{4N}$	$\|\|e_{2N}\|\|_\infty$	rate$_{4N}$	$\|\|e_{2N}\|\|_\infty$	rate$_{4N}$
32	1.5680E-02	-	2.9056E-02	-	4.9613E-02	-
64	3.2150E-03	2.2860	8.1861E-03	1.8276	1.6752E-02	1.5663
128	8.4486E-04	1.9280	2.1204E-03	1.9488	4.5867E-03	1.8688
256	2.0985E-04	2.0093	5.3892E-04	1.9761	1.1662E-03	1.9756

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