Article Contents
Article Contents

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

• * Corresponding author: Alina Chertock
• 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.

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

 Citation:

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

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

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

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

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

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

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

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

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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