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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Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria |
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, doi: 10.3934/dcdsb.2019029
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References:
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
E. Lieb and M. Loss, Analysis, 2nd edition, Amer. Math. Soc., Providence, USA, 2001.
doi: 10.1090/gsm/014. |
| [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. |
| [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. |
| [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. |
| [35] |
C. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
| [36] |
B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, 2007. |
| [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. |
| [38] |
N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, B15 (2009), 125–146. |
| [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. |
| [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. |
| [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. |
| [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. |
| [43] |
M. N. Spijker,
Contractivity in the numerical solution of initial value problems, Numer. Math., 42 (1983), 271-290.
doi: 10.1007/BF01389573. |
| [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. |
| [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. |
| [46] |
T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005.
doi: 10.1007/0-8176-4436-9. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [31] |
E. Lieb and M. Loss, Analysis, 2nd edition, Amer. Math. Soc., Providence, USA, 2001.
doi: 10.1090/gsm/014. |
| [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. |
| [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. |
| [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. |
| [35] |
C. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
| [36] |
B. Perthame, Transport Equations in Biology, Birkhäuser, Basel, 2007. |
| [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. |
| [38] |
N. Saito, Conservative numerical schemes for the Keller-Segel system and numerical results, RIMS Kôkyûroku Bessatsu, B15 (2009), 125–146. |
| [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. |
| [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. |
| [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. |
| [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. |
| [43] |
M. N. Spijker,
Contractivity in the numerical solution of initial value problems, Numer. Math., 42 (1983), 271-290.
doi: 10.1007/BF01389573. |
| [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. |
| [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. |
| [46] |
T. Suzuki, Free Energy and Self-Interacting Particles, Birkhäuser, Boston, 2005.
doi: 10.1007/0-8176-4436-9. |
| [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. |
| [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. |
| [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. |
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)
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)
Figure 4.
Cell density at time step $ k = 0 $ (left) and $ k = k_{\rm max} = 44 $ . The mesh size is $ h = 0.02 $
Figure 5.
The residuum $ R_k $ for time steps 1 to 81 (left) and time steps 30 to 81 (right) versus time steps k
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)
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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