[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, 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.
|