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Kinetic models and intrinsic timescales: Simulation comparison for a 2nd order queueing model
An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions
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 |
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.
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. |
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C. Patlak,
Random walk with persistence and external bias, Bull. Math: Biophys., 15 (1953), 311-338.
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[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. |
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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.
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G. Strang,
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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.
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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] |
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,
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rate |
rate |
rate |
||||
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 |
rate |
rate |
rate |
||||
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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