Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Alina Chertock

    * Corresponding author: Alina Chertock 
Abstract Full Text(HTML) Figure(8) / Table(1) Related Papers Cited by
  • 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.


    \begin{equation} \\ \end{equation}
  • 加载中
  • 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
     | Show Table
    DownLoad: CSV
  • [1] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.
    [2] A. BollermannS. Noelle and M. Lukáčová-Medviďová, Finite volume evolution Galerkin methods for the shallow water equations with dry beds, Commun. Comput. Phys., 10 (2011), 371-404.  doi: 10.4208/cicp.220210.020710a.
    [3] N. Bournaveas and V. Calvez, Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 26, Elsevier, 2009, 1871–1895. doi: 10.1016/j.anihpc.2009.02.001.
    [4] V. Calvez and J. A. Carrillo, Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175.  doi: 10.1016/j.matpur.2006.04.002.
    [5] V. Calvez, B. Perthame and M. Sharifi Tabar, Modified Keller-Segel system and critical mass for the log interaction kernel, in Stochastic Analysis and Partial Differential Equations, vol. 429 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2007, 45–62. doi: 10.1090/conm/429/08229.
    [6] J. A. Carrillo and B. Yan, An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis, Multiscale Modeling & Simulation, 11 (2013), 336-361.  doi: 10.1137/110851687.
    [7] F. A. C. C. ChalubP. A. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.
    [8] A. Chertock and A. Kurganov, A positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205.  doi: 10.1007/s00211-008-0188-0.
    [9] A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinetic and Related Models, 5 (2012), 51–95, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=6915. doi: 10.3934/krm.2012.5.51.
    [10] S. Childress and J. Percus, Nonlinear aspects of chemotaxis, Math. Biosc., 56 (1981), 217-237.  doi: 10.1016/0025-5564(81)90055-9.
    [11] A. Crestetto, N. Crouseilles and M. Lemou, Asymptotic-preserving scheme based on a finite volume/particle-in-cell coupling for Boltzmann-BGK-like equations in the diffusion scaling, in Finite Volumes for Complex Applications. VII. Elliptic, Parabolic and Hyperbolic Problems, vol. 78 of Springer Proc. Math. Stat., Springer, Cham, 2014,827–835. doi: 10.1007/978-3-319-05591-6_83.
    [12] N. Crouseilles and M. Lemou, An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits, Kinet. Relat. Models, 4 (2011), 441-477.  doi: 10.3934/krm.2011.4.441.
    [13] G. Dimarco and L. Pareschi, Asymptotic preserving implicit-explicit Runge-Kutta methods for nonlinear kinetic equations, SIAM J. Numer. Anal., 51 (2013), 1064-1087.  doi: 10.1137/12087606X.
    [14] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229 (2010), 7625-7648.  doi: 10.1016/j.jcp.2010.06.017.
    [15] F. FilbetP. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement, Journal of Mathematical Biology, 50 (2005), 189-207.  doi: 10.1007/s00285-004-0286-2.
    [16] H. GajewskiK. Zacharias and K. Gröger, Global behaviour of a reaction-diffusion system modelling chemotaxis, Mathematische Nachrichten, 195 (1998), 77-114.  doi: 10.1002/mana.19981950106.
    [17] S. GottliebC.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.
    [18] S. Gottlieb, D. I. Ketcheson and C.-W. Shu, Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. doi: 10.1142/7498.
    [19] S. GuissetS. BrullE. D'Humières and B. Dubroca, Asymptotic-preserving well-balanced scheme for the electronic $ M_1$ model in the diffusive limit: particular cases, ESAIM Math. Model. Numer. Anal., 51 (2017), 1805-1826.  doi: 10.1051/m2an/2016079.
    [20] M. Herrero and J. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683. 
    [21] M. A. HerreroE. Medina and J. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system, Nonlinearity, 10 (1997), 1739-1754.  doi: 10.1088/0951-7715/10/6/016.
    [22] M. A. Herrero and J. J. Velázquez, Chemotactic collapse for the Keller-Segel model, Journal of Mathematical Biology, 35 (1996), 177-194.  doi: 10.1007/s002850050049.
    [23] T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math., 61 (2000), 751-775 (electronic).  doi: 10.1137/S0036139999358167.
    [24] T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.  doi: 10.1006/aama.2001.0721.
    [25] T. HillenK. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125-144.  doi: 10.3934/dcdsb.2007.7.125.
    [26] T. Hillen and K. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.
    [27] D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences ⅰ, Jahresber. DMV, 105 (2003), 103-165. 
    [28] D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences ⅱ, Jahresber. DMV, 106 (2004), 51-69. 
    [29] J. HuS. Jin and L. Wang, An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: a splitting approach, Kinet. Relat. Models, 8 (2015), 707-723.  doi: 10.3934/krm.2015.8.707.
    [30] J. HuQ. Li and L. Pareschi, Asymptotic-preserving exponential methods for the quantum Boltzmann equation with high-order accuracy, J. Sci. Comput., 62 (2015), 555-574.  doi: 10.1007/s10915-014-9869-2.
    [31] H. J. HwangK. Kang and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 319-334.  doi: 10.3934/dcdsb.2005.5.319.
    [32] H. J. HwangK. Kang and A. Stevens, Global solutions of nonlinear transport equations for chemosensitive movement, SIAM Journal on Mathematical Analysis, 36 (2005), 1177-1199.  doi: 10.1137/S0036141003431888.
    [33] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Transactions of the American Mathematical Society, 329 (1992), 819-824.  doi: 10.1090/S0002-9947-1992-1046835-6.
    [34] J. JangF. LiJ.-M. Qiu and T. Xiong, Analysis of asymptotic preserving DG-IMEX schemes for linear kinetic transport equations in a diffusive scaling, SIAM J. Numer. Anal., 52 (2014), 2048-2072.  doi: 10.1137/130938955.
    [35] S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: A review, Riv. Mat. Univ. Parma, 3 (2012), 177-216. 
    [36] S. JinL. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal., 35 (1998), 2405-2439 (electronic).  doi: 10.1137/S0036142997315962.
    [37] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM Journal on Scientific Computing, 21 (1999), 441-454.  doi: 10.1137/S1064827598334599.
    [38] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.
    [39] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.  doi: 10.1016/0022-5193(71)90050-6.
    [40] E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.  doi: 10.1016/0022-5193(71)90051-8.
    [41] A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift diffusion semiconductor equations, SIAM J. Sci. Comput., 19 (1998), 2032-2050.  doi: 10.1137/S1064827595286177.
    [42] A. Kurganov and M. Lukáčová-Medviďová, Numerical study of two-species chemotaxis models, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 131-152.  doi: 10.3934/dcdsb.2014.19.131.
    [43] M. Lemou and L. Mieussens, A new asymptotic preserving scheme based on micro-macro formulation for linear kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 31 (2008), 334-368.  doi: 10.1137/07069479X.
    [44] G. I. Marchuk, Metody Rasshchepleniya, (Russian) [Splitting Methods] "Nauka", Moscow, 1988.
    [45] G. I. Marchuk, Splitting and alternating direction methods, in Handbook of numerical analysis, Vol. I, Handb. Numer. Anal., I, North-Holland, Amsterdam, 1990,197–462.
    [46] T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis intwo-dimensional domains, J. Inequal. Appl., 6 (2001), 37–55, URL http://www.emis.ams.org/journals/HOA/JIA/Volume6_1/55.pdf. doi: 10.1155/S1025583401000042.
    [47] H. G. Othmer and T. Hillen, The diffusion limit of transport equations. Ⅱ. Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250 (electronic).  doi: 10.1137/S0036139900382772.
    [48] H. OthmerS. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.  doi: 10.1007/BF00277392.
    [49] C. Patlak, Random walk with persistence and external bias, Bull. Math: Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.
    [50] B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.  doi: 10.1007/s10492-004-6431-9.
    [51] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471.  doi: 10.1016/0021-9991(88)90177-5.
    [52] A. Stevens and H. G. Othmer, Aggregation, blowup, and collapse: the ABC's of taxis in reinforced random walks, SIAM Journal on Applied Mathematics, 57 (1997), 1044-1081.  doi: 10.1137/S0036139995288976.
    [53] G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), 506-517.  doi: 10.1137/0705041.
    [54] D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Probability Theory and Related Fields, 28 (1974), 305-315.  doi: 10.1007/BF00532948.
  • 加载中




Article Metrics

HTML views(563) PDF downloads(257) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint