American Institute of Mathematical Sciences

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September  2019, 24(9): 4755-4782. doi: 10.3934/dcdsb.2019029

Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models

Ansgar Jüngel and  Oliver Leingang

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria

Received  September 2017 Revised  November 2017 Published  February 2019

Fund Project: The authors acknowledge partial support from the Austrian Science Fund (FWF), grants P27352, P30000, F65, and W1245

The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the implicit Euler, BDF, and Runge-Kutta methods, the same bounds for the blow-up time as in the continuous case are derived by discrete versions of the virial argument. The theoretical results are illustrated by numerical simulations using an upwind finite-element method combined with second-order time discretizations.

Keywords: Time discretization, chemotaxis, finite-time blow-up of solutions, existence of weak solutions, higher-order schemes, BDF scheme, Runge-Kutta scheme.
Mathematics Subject Classification: Primary: 35B44; Secondary: 35Q92, 65L06, 65M20, 92C17.
Citation: Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029
References:
[1]

M. Akhmouch and M. Amine, A time semi-exponentially fitted scheme for chemotaxis-growth models, Calcolo, 54 (2017), 609-641.  doi: 10.1007/s10092-016-0201-4.  Google Scholar

[2]

B. Andreianov, M. Bendahmane and M. Saad, Finite volume methods for degenerate model, J. Comput. Appl. Math., 235 (2011), 4015-4031.  doi: 10.1016/j.cam.2011.02.023.  Google Scholar

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M. Bessemoulin-Chatard and A. Jüngel, A finite volume scheme for a Keller-Segel model with additional cross-diffusion, IMA J. Numer. Anal., 34 (2014), 96-122.  doi: 10.1093/imanum/drs061.  Google Scholar

[4]

A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721.  doi: 10.1137/070683337.  Google Scholar

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A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $ {\mathbb R}^2$, Commun. Pure Appl. Math., 61 (2008), 1449-1481.  doi: 10.1002/cpa.20225.  Google Scholar

[6]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electr. J. Diff. Eqs., 2006 (2006), article 44, 32 pages.  Google Scholar

[7]

C. Bolley and M. Crouzeix, Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques, RAIRO Anal. Numér., 12 (1978), 237-245.  doi: 10.1051/m2an/1978120302371.  Google Scholar

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L. Bonaventura and A. Della Rocca, Unconditionally strong stability preserving extensions of the TR-BDF2 method, J. Sci. Comput., 70 (2017), 859-895.  doi: 10.1007/s10915-016-0267-9.  Google Scholar

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C. Budd, R. Carretero-González and R. Russell, Precise computations of chemotactic collapse using moving mesh methods, J. Comput. Phys., 202 (2005), 463-487.  doi: 10.1016/j.jcp.2004.07.010.  Google Scholar

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V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $ {\mathbb R}^2$, Commun. Math. Sci., 6 (2008), 417-447.  doi: 10.4310/CMS.2008.v6.n2.a8.  Google Scholar

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V. Calvez, L. Corrias and M. A. Ebde, Blow-up, concentration phenomena and global existence for the Keller-Segel model in high dimensions, Commun. Partial Diff. Eqs., 37 (2012), 561-584.  doi: 10.1080/03605302.2012.655824.  Google Scholar

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J. A. Carrillo, H. Ranetbauer and M.-T. Wolfram, Numerical simulation of continuity equations by evolving diffeomorphisms, J. Comput. Phys., 327 (2016), 186-202.  doi: 10.1016/j.jcp.2016.09.040.  Google Scholar

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M. Chapwanya, J. Lubuma and R. Mickens, Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences, Comput. Math. Appl., 68 (2014), 1071-1082.  doi: 10.1016/j.camwa.2014.04.021.  Google Scholar

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A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205.  doi: 10.1007/s00211-008-0188-0.  Google Scholar

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A. Chertock, Y. Epshteyn, H. Hu and A. Kurganov, High-order positivity-preserving hybrid finite-volume finite-difference methods for chemotaxis systems, Adv. Comput. Math., 44 (2018), 327-350.  doi: 10.1007/s10444-017-9545-9.  Google Scholar

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L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.  Google Scholar

[17]

S. Dejak, D. Egli, P. Lushnikov and I. Sigal, On blowup dynamics in the Keller-Segel model of chemotaxis, Algebra i Analiz, 25 (2013), 47-84.  doi: 10.1090/S1061-0022-2014-01306-4.  Google Scholar

[18]

Y. Epshtyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models, J. Comput. Appl. Math., 224 (2009), 168-181.  doi: 10.1016/j.cam.2008.04.030.  Google Scholar

[19]

M. A. Farina, M. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Dyn. Sys., Diff. Eqs. and Appl., AIMS Proceedings, (2015), 409–417. doi: 10.3934/proc.2015.0409.  Google Scholar

[20]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.  doi: 10.1007/s00211-006-0024-3.  Google Scholar

[21]

R. Garg and S. Spector, On regularity of solutions to Poisson's equation, Comptes Rendus Math., 353 (2015), 819-823.  doi: 10.1016/j.crma.2015.07.001.  Google Scholar

[22]

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

[23]

J. Haskovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151.  doi: 10.1007/s10955-009-9717-1.  Google Scholar

[24]

A. Jüngel and J.-P. Milišić, Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations, Numer. Meth. Partial Diff. Eqs., 31 (2015), 1119-1149.  doi: 10.1002/num.21938.  Google Scholar

[25]

A. Jüngel and S. Schuchnigg, A discrete Bakry-Emery method and its application to the porous-medium equation, Discrete Cont. Dyn. Sys., 37 (2017), 5541-5560.  doi: 10.3934/dcds.2017241.  Google Scholar

[26]

A. Jüngel and S. Schuchnigg, Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, Commun. Math. Sci., 15 (2017), 27-53.  doi: 10.4310/CMS.2017.v15.n1.a2.  Google Scholar

[27]

E. Keller and L. 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.  Google Scholar

[28]

D. Ketcheson, Step sizes for strong stability preservation with downwind-biased operators, SIAM J. Numer. Anal., 49 (2011), 1649-1660.  doi: 10.1137/100818674.  Google Scholar

[29]

H. Kozono and Y. Sugiyama, Local existence and finite time blow-up in the 2-D Keller-Segel system, J. Evol. Eqs., 8 (2008), 353-378.  doi: 10.1007/s00028-008-0375-6.  Google Scholar

[30]

N. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, Rhode Island, USA, 2008. doi: 10.1090/gsm/096.  Google Scholar

[31]

E. Lieb and M. Loss, Analysis, 2nd edition, Amer. Math. Soc., Providence, USA, 2001. doi: 10.1090/gsm/014.  Google Scholar

[32]

J.-G. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for the 2D Keller-Segel equations, Math. Comput., 87 (2018), 1165-1189.  doi: 10.1090/mcom/3250.  Google Scholar

[33]

T. Nagai and T. Ogawa, Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in $ {\mathbb R}^2$, Funkcialaj Ekvacioj, 59 (2016), 67-112.  doi: 10.1619/fesi.59.67.  Google Scholar

[34]

E. Nakaguchi and A. Yagi, Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems, Hokkaido Math. J., 31 (2002), 385-429.  doi: 10.14492/hokmj/1350911871.  Google Scholar

[35]

C. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[36]

B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, 2007.  Google Scholar

[37]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365.  doi: 10.1093/imanum/drl018.  Google Scholar

[38]

N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, B15 (2009), 125–146.  Google Scholar

[39]

N. Saito, Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis, Commun. Pure Appl. Anal., 11 (2012), 339-364.  doi: 10.3934/cpaa.2012.11.339.  Google Scholar

[40]

N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Appl. Math. Comput., 171 (2005), 72-90.  doi: 10.1016/j.amc.2005.01.037.  Google Scholar

[41]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Meth. Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.  Google Scholar

[42]

M. Smiley, A monotone conservative Eulerian-Lagrangian scheme for reaction-diffusion-convection equations modeling chemotaxis, Numer. Meth. Partial Diff. Eqs., 23 (2007), 553-586.  doi: 10.1002/num.20185.  Google Scholar

[43]

M. N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math., 42 (1983), 271-290.  doi: 10.1007/BF01389573.  Google Scholar

[44]

R. Strehl, A. Sokolov, D. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems, Comput. Meth. Appl. Math., 10 (2010), 219-232.  doi: 10.2478/cmam-2010-0013.  Google Scholar

[45]

R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D, J. Comput. Appl. Math., 239 (2013), 290-303.  doi: 10.1016/j.cam.2012.09.041.  Google Scholar

[46]

T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[47]

J. Valenciano and M. Chaplain, Computing highly accurate solutions of a tumour angiogenesis model, Math. Models Meth. Appl. Sci., 13 (2003), 747-766.  doi: 10.1142/S0218202503002702.  Google Scholar

[48]

R. Zhang, J. Zhu, A. Loula and X. Yu, Operator splitting combined with positivity-preserving discontinuous Galerkin method for the chemotaxis model, J. Comput. Appl. Math., 302 (2016), 312-326.  doi: 10.1016/j.cam.2016.02.018.  Google Scholar

[49]

G. Zhou and N. Saito, Finite volume methods for a Keller-Segel system: discrete energy, error estimates and numerical blow-up analysis, Numer. Math., 135 (2017), 265-311.  doi: 10.1007/s00211-016-0793-2.  Google Scholar

show all references

References:
[1]

M. Akhmouch and M. Amine, A time semi-exponentially fitted scheme for chemotaxis-growth models, Calcolo, 54 (2017), 609-641.  doi: 10.1007/s10092-016-0201-4.  Google Scholar

[2]

B. Andreianov, M. Bendahmane and M. Saad, Finite volume methods for degenerate model, J. Comput. Appl. Math., 235 (2011), 4015-4031.  doi: 10.1016/j.cam.2011.02.023.  Google Scholar

[3]

M. Bessemoulin-Chatard and A. Jüngel, A finite volume scheme for a Keller-Segel model with additional cross-diffusion, IMA J. Numer. Anal., 34 (2014), 96-122.  doi: 10.1093/imanum/drs061.  Google Scholar

[4]

A. Blanchet, V. Calvez and J. A. Carrillo, Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model, SIAM J. Numer. Anal., 46 (2008), 691-721.  doi: 10.1137/070683337.  Google Scholar

[5]

A. Blanchet, J. A. Carrillo and N. Masmoudi, Infinite time aggregation for the critical Patlak-Keller-Segel model in $ {\mathbb R}^2$, Commun. Pure Appl. Math., 61 (2008), 1449-1481.  doi: 10.1002/cpa.20225.  Google Scholar

[6]

A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electr. J. Diff. Eqs., 2006 (2006), article 44, 32 pages.  Google Scholar

[7]

C. Bolley and M. Crouzeix, Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques, RAIRO Anal. Numér., 12 (1978), 237-245.  doi: 10.1051/m2an/1978120302371.  Google Scholar

[8]

L. Bonaventura and A. Della Rocca, Unconditionally strong stability preserving extensions of the TR-BDF2 method, J. Sci. Comput., 70 (2017), 859-895.  doi: 10.1007/s10915-016-0267-9.  Google Scholar

[9]

C. Budd, R. Carretero-González and R. Russell, Precise computations of chemotactic collapse using moving mesh methods, J. Comput. Phys., 202 (2005), 463-487.  doi: 10.1016/j.jcp.2004.07.010.  Google Scholar

[10]

V. Calvez and L. Corrias, The parabolic-parabolic Keller-Segel model in $ {\mathbb R}^2$, Commun. Math. Sci., 6 (2008), 417-447.  doi: 10.4310/CMS.2008.v6.n2.a8.  Google Scholar

[11]

V. Calvez, L. Corrias and M. A. Ebde, Blow-up, concentration phenomena and global existence for the Keller-Segel model in high dimensions, Commun. Partial Diff. Eqs., 37 (2012), 561-584.  doi: 10.1080/03605302.2012.655824.  Google Scholar

[12]

J. A. Carrillo, H. Ranetbauer and M.-T. Wolfram, Numerical simulation of continuity equations by evolving diffeomorphisms, J. Comput. Phys., 327 (2016), 186-202.  doi: 10.1016/j.jcp.2016.09.040.  Google Scholar

[13]

M. Chapwanya, J. Lubuma and R. Mickens, Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences, Comput. Math. Appl., 68 (2014), 1071-1082.  doi: 10.1016/j.camwa.2014.04.021.  Google Scholar

[14]

A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205.  doi: 10.1007/s00211-008-0188-0.  Google Scholar

[15]

A. Chertock, Y. Epshteyn, H. Hu and A. Kurganov, High-order positivity-preserving hybrid finite-volume finite-difference methods for chemotaxis systems, Adv. Comput. Math., 44 (2018), 327-350.  doi: 10.1007/s10444-017-9545-9.  Google Scholar

[16]

L. Corrias, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.  doi: 10.1007/s00032-003-0026-x.  Google Scholar

[17]

S. Dejak, D. Egli, P. Lushnikov and I. Sigal, On blowup dynamics in the Keller-Segel model of chemotaxis, Algebra i Analiz, 25 (2013), 47-84.  doi: 10.1090/S1061-0022-2014-01306-4.  Google Scholar

[18]

Y. Epshtyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models, J. Comput. Appl. Math., 224 (2009), 168-181.  doi: 10.1016/j.cam.2008.04.030.  Google Scholar

[19]

M. A. Farina, M. Marras and G. Viglialoro, On explicit lower bounds and blow-up times in a model of chemotaxis, Dyn. Sys., Diff. Eqs. and Appl., AIMS Proceedings, (2015), 409–417. doi: 10.3934/proc.2015.0409.  Google Scholar

[20]

F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.  doi: 10.1007/s00211-006-0024-3.  Google Scholar

[21]

R. Garg and S. Spector, On regularity of solutions to Poisson's equation, Comptes Rendus Math., 353 (2015), 819-823.  doi: 10.1016/j.crma.2015.07.001.  Google Scholar

[22]

S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Review, 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

[23]

J. Haskovec and C. Schmeiser, Stochastic particle approximation for measure valued solutions of the 2D Keller-Segel system, J. Stat. Phys., 135 (2009), 133-151.  doi: 10.1007/s10955-009-9717-1.  Google Scholar

[24]

A. Jüngel and J.-P. Milišić, Entropy dissipative one-leg multistep time approximations of nonlinear diffusive equations, Numer. Meth. Partial Diff. Eqs., 31 (2015), 1119-1149.  doi: 10.1002/num.21938.  Google Scholar

[25]

A. Jüngel and S. Schuchnigg, A discrete Bakry-Emery method and its application to the porous-medium equation, Discrete Cont. Dyn. Sys., 37 (2017), 5541-5560.  doi: 10.3934/dcds.2017241.  Google Scholar

[26]

A. Jüngel and S. Schuchnigg, Entropy-dissipating semi-discrete Runge-Kutta schemes for nonlinear diffusion equations, Commun. Math. Sci., 15 (2017), 27-53.  doi: 10.4310/CMS.2017.v15.n1.a2.  Google Scholar

[27]

E. Keller and L. 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.  Google Scholar

[28]

D. Ketcheson, Step sizes for strong stability preservation with downwind-biased operators, SIAM J. Numer. Anal., 49 (2011), 1649-1660.  doi: 10.1137/100818674.  Google Scholar

[29]

H. Kozono and Y. Sugiyama, Local existence and finite time blow-up in the 2-D Keller-Segel system, J. Evol. Eqs., 8 (2008), 353-378.  doi: 10.1007/s00028-008-0375-6.  Google Scholar

[30]

N. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, Amer. Math. Soc., Providence, Rhode Island, USA, 2008. doi: 10.1090/gsm/096.  Google Scholar

[31]

E. Lieb and M. Loss, Analysis, 2nd edition, Amer. Math. Soc., Providence, USA, 2001. doi: 10.1090/gsm/014.  Google Scholar

[32]

J.-G. Liu, L. Wang and Z. Zhou, Positivity-preserving and asymptotic preserving method for the 2D Keller-Segel equations, Math. Comput., 87 (2018), 1165-1189.  doi: 10.1090/mcom/3250.  Google Scholar

[33]

T. Nagai and T. Ogawa, Global existence of solutions to a parabolic-elliptic system of drift-diffusion type in $ {\mathbb R}^2$, Funkcialaj Ekvacioj, 59 (2016), 67-112.  doi: 10.1619/fesi.59.67.  Google Scholar

[34]

E. Nakaguchi and A. Yagi, Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems, Hokkaido Math. J., 31 (2002), 385-429.  doi: 10.14492/hokmj/1350911871.  Google Scholar

[35]

C. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.  doi: 10.1007/BF02476407.  Google Scholar

[36]

B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, 2007.  Google Scholar

[37]

N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365.  doi: 10.1093/imanum/drl018.  Google Scholar

[38]

N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, B15 (2009), 125–146.  Google Scholar

[39]

N. Saito, Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis, Commun. Pure Appl. Anal., 11 (2012), 339-364.  doi: 10.3934/cpaa.2012.11.339.  Google Scholar

[40]

N. Saito and T. Suzuki, Notes on finite difference schemes to a parabolic-elliptic system modelling chemotaxis, Appl. Math. Comput., 171 (2005), 72-90.  doi: 10.1016/j.amc.2005.01.037.  Google Scholar

[41]

T. Senba and T. Suzuki, Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Meth. Appl. Anal., 8 (2001), 349-367.  doi: 10.4310/MAA.2001.v8.n2.a9.  Google Scholar

[42]

M. Smiley, A monotone conservative Eulerian-Lagrangian scheme for reaction-diffusion-convection equations modeling chemotaxis, Numer. Meth. Partial Diff. Eqs., 23 (2007), 553-586.  doi: 10.1002/num.20185.  Google Scholar

[43]

M. N. Spijker, Contractivity in the numerical solution of initial value problems, Numer. Math., 42 (1983), 271-290.  doi: 10.1007/BF01389573.  Google Scholar

[44]

R. Strehl, A. Sokolov, D. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems, Comput. Meth. Appl. Math., 10 (2010), 219-232.  doi: 10.2478/cmam-2010-0013.  Google Scholar

[45]

R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D, J. Comput. Appl. Math., 239 (2013), 290-303.  doi: 10.1016/j.cam.2012.09.041.  Google Scholar

[46]

T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005. doi: 10.1007/0-8176-4436-9.  Google Scholar

[47]

J. Valenciano and M. Chaplain, Computing highly accurate solutions of a tumour angiogenesis model, Math. Models Meth. Appl. Sci., 13 (2003), 747-766.  doi: 10.1142/S0218202503002702.  Google Scholar

[48]

R. Zhang, J. Zhu, A. Loula and X. Yu, Operator splitting combined with positivity-preserving discontinuous Galerkin method for the chemotaxis model, J. Comput. Appl. Math., 302 (2016), 312-326.  doi: 10.1016/j.cam.2016.02.018.  Google Scholar

[49]

G. Zhou and N. Saito, Finite volume methods for a Keller-Segel system: discrete energy, error estimates and numerical blow-up analysis, Numer. Math., 135 (2017), 265-311.  doi: 10.1007/s00211-016-0793-2.  Google Scholar

Figure 1.  Cell density computed from the BDF-2 scheme at times $ t = 0 $ (top left), $ t = 0.005 $ (top right), $ t = 0.007 $ (bottom left), $ t = 0.02 $ (bottom right)
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Figure 2.  Cell density computed from the BDF-2 scheme at times $ t = 0 $ (top left), $ t = 0.005 $ (top right), $ t = 0.02 $ (bottom left), $ t = 0.1001 $ (bottom right)
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Figure 3.  $ L^p $ error $ e_p $ for $ p = 1,2,4,\infty $ at time $ T = 0.01 $ for various time step sizes $ \tau_k = \tau $ (left: BDF-2 discretization; right: midpoint discretization)
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Figure 4.  Cell density at time step $ k = 0 $ (left) and $ k = k_{\rm max} = 44 $. The mesh size is $ h = 0.02 $
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Figure 5.  The residuum $ R_k $ for time steps 1 to 81 (left) and time steps 30 to 81 (right) versus time steps k
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Figure 6.  $ L^\infty $ norm $ \|n_k\|_{L^\infty(\Omega)} $ (left) and second moment $ I_k $ (right) versus time. The vertical line marks the upper bound $ k_{\rm max} $ defined in (18)
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Figure 7.  Cell density computed from the BDF-2 scheme with a coarse mesh at times $ t = 0 $ (top left), $ t = 0.006 $ (top right), $ t = 0.021 $ (bottom left) and the $ L^1 $ norm of $ n_k $ (bottom right)
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