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February  2019, 12(1): 195-216. doi: 10.3934/krm.2019009

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

Received  February 2018 Published  July 2018

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.

Keywords: Chemotaxis, kinetic equations, multiscale models, asymptotic preserving (AP) methods, operator splitting.
Mathematics Subject Classification: Primary: 92C17, 35Q20, 76M45; Secondary: 76P99, 65M06, 65M70.
Citation: Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic and Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
References:
[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. Bollermann, S. 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. Chalub, P. A. Markowich, B. 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.

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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.

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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. Filbet, P. 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.

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S. Gottlieb, C.-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. Guisset, S. Brull, E. 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. Herrero, E. 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. Hillen, K. 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. Hu, S. 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. Hu, Q. 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. Hwang, K. 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. Hwang, K. 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. Jang, F. Li, J.-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. Jin, L. 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. Othmer, S. 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.

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B. Perthame, PDE models for chemotactic movements: Parabolic, hyperbolic and kinetic, Appl. Math., 49 (2004), 539-564.  doi: 10.1007/s10492-004-6431-9.

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show all references

References:
[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. Bollermann, S. 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. Chalub, P. A. Markowich, B. 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. Filbet, P. 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. Gajewski, K. 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. Gottlieb, C.-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. Guisset, S. Brull, E. 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. Herrero, E. 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. Hillen, K. 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. Hu, S. 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. Hu, Q. 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. Hwang, K. 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. Hwang, K. 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. Jang, F. Li, J.-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. Jin, L. 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. Othmer, S. 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]

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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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$M=8$ $M=9.5$ $M=11$
$N$ $||e_{2N}||_\infty$ 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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