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December  2013, 18(10): 2689-2704. doi: 10.3934/dcdsb.2013.18.2689

Numerical simulation of chemotaxis models on stationary surfaces

Andriy Sokolov 1, , Robert Strehl 1, and  Stefan Turek 1,

1. 

Institut für Angewandte Mathematik, TU Dortmund, 44227 Dortmund, Germany, Germany, Germany

Received  November 2012 Revised  April 2013 Published  October 2013

In this paper we present an implicit finite element method for a class of chemotaxis models, where a new linearized flux-corrected transport (FCT) algorithm is modified in such a way as to keep the density of on-surface living cells nonnegative. Level set techniques are adopted for an implicit description of the surface and for the numerical treatment of the corresponding system of partial differential equations. The presented scheme is able to deliver a robust and accurate solution for a large class of chemotaxis-driven models. The numerical behavior of the proposed scheme is tested on the blow-up model on a sphere and an ellipsoid and on the pattern-forming dynamics model of Escherichia coli on a sphere.
Keywords: FCT, Chemotaxis model, level set., FEM, pattern formation.
Mathematics Subject Classification: 58J47, 58J32, 35R01, 92C17, 65M6.
Citation: Andriy Sokolov, Robert Strehl, Stefan Turek. Numerical simulation of chemotaxis models on stationary surfaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2689-2704. doi: 10.3934/dcdsb.2013.18.2689
References:
[1]

J. London Math. Soc., 74 (2006), 453-474. doi: 10.1112/S0024610706023015.  Google Scholar

[2]

Scientiae Mathematicae Japonicae, 59 (2004), 577-590.  Google Scholar

[3]

J. Theor. Med., 6 (2005), 1-19. doi: 10.1080/1027366042000327098.  Google Scholar

[4]

Journal of Theoretical Medicine, 2 (2000), 129-154. doi: 10.1080/10273660008833042.  Google Scholar

[5]

J. Math. Biol., 61 (2010), 649-663. doi: 10.1007/s00285-009-0315-2.  Google Scholar

[6]

ACTA Biotheoretica, 43 (1995), 387-402. doi: 10.1007/BF00713561.  Google Scholar

[7]

Bulletin of Mathematical Biology, 60 (1998), 857-900. Google Scholar

[8]

Journal of Neuro-Oncology, 50 (2000), 37-51. Google Scholar

[9]

IMA Journal of Mathematics Applied in Medicine and Biology, 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149.  Google Scholar

[10]

Numer. Math., 111 (2008), 169-205. doi: 10.1007/s00211-008-0188-0.  Google Scholar

[11]

Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.  Google Scholar

[12]

Mathematical and Computer Modelling, 47 (2008), 755-764. doi: 10.1016/j.mcm.2007.06.005.  Google Scholar

[13]

Interfaces and Free Boundaries, 10 (2008), 119-138. doi: 10.4171/IFB/182.  Google Scholar

[14]

J. Comput. Appl. Math., 224 (2009), 168-181. doi: 10.1016/j.cam.2008.04.030.  Google Scholar

[15]

Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model,, SIAM J. Numer. Anal., 47 (): 386.  doi: 10.1137/07070423X.  Google Scholar

[16]

Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3.  Google Scholar

[17]

R-Report, Institut fuer Angewandte Analysis und Stochastik (IAAS), Berlin, 1993. Google Scholar

[18]

Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106.  Google Scholar

[19]

Phys. Rev. Lett., 90 118101 (2003) [4 pages]. doi: 10.1103/PhysRevLett.90.118101.  Google Scholar

[20]

J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.  Google Scholar

[21]

Journal für die Reine und angewandte Mathematik (Crelle's Journal), 654 (2011), 83-124. doi: 10.1515/CRELLE.2011.030.  Google Scholar

[22]

European Journal of Applied Mathematics, 12, (2001), 159-177. doi: 10.1017/S0956792501004363.  Google Scholar

[23]

Journal of Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[24]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[25]

J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[26]

Springer, 2nd edition, 2012. Google Scholar

[27]

in Flux-Corrected Transport: Principles, Algorithms, and Applications (Eds. D. Kuzmin, R. Löhner, S. Turek), Springer, Berlin (2005), 155-206. doi: 10.1007/3-540-27206-2_6.  Google Scholar

[28]

J. Comput. Phys., 175 (2002), 525-558. doi: 10.1006/jcph.2001.6955.  Google Scholar

[29]

J. Comput. Phys., 228 (2009), 2517-2534. doi: 10.1016/j.jcp.2008.12.011.  Google Scholar

[30]

Journal of Computational and Applied Mathematics, 236 (2012), 2317-2337. doi: 10.1016/j.cam.2011.11.019.  Google Scholar

[31]

J. Comput. Phys., 131 (1997), 327-353. Google Scholar

[32]

Physica A, 230 (1996), 499-543. Google Scholar

[33]

J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.  Google Scholar

[34]

Funkcialaj Ekvacioj, 44 (2001), 441-469.  Google Scholar

[35]

IMA J. Numer. Anal., 27 (2007), 332-365. doi: 10.1093/imanum/drl018.  Google Scholar

[36]

IMS Workshop on Reaction-Diffusion Systems (Shatin, 1999). Methods Appl. Anal., 8 (2001), 349-367.  Google Scholar

[37]

The EMBO Journal, 22 (2003), 1771-1779. doi: 10.1093/emboj/cdg176.  Google Scholar

[38]

Computational methods in applied mathematics, 10 (2010), 219-232. doi: 10.2478/cmam-2010-0013.  Google Scholar

[39]

Journal of Computational and Applied Mathematics, 239 (2013), 290-303. doi: 10.1016/j.cam.2012.09.041.  Google Scholar

[40]

Computers and Mathematics with Applications, 64 (2012), 175-189. doi: 10.1016/j.camwa.2011.12.040.  Google Scholar

[41]

Springer, Berlin, 1999. doi: 10.1007/3-540-48092-7.  Google Scholar

[42]

Proc. R. Soc. Lond. B, 266 (1999), 299-304. doi: 10.1098/rspb.1999.0637.  Google Scholar

[43]

J. Math. Biol., 38 (1999), 359-375. doi: 10.1007/s002850050153.  Google Scholar

[44]

J. Math. Biol., 41 (2000), 455-475. doi: 10.1007/s002850000038.  Google Scholar

[45]

J. R. Soc. Interface, 9 (2012), 3027-3044. doi: 10.1098/rsif.2012.0276.  Google Scholar

[46]

J. Biomech, 44 (2011), 359-364. Google Scholar

[47]

J. Comput. Phys., 31 (1979), 335-362. doi: 10.1016/0021-9991(79)90051-2.  Google Scholar

show all references

References:
[1]

J. London Math. Soc., 74 (2006), 453-474. doi: 10.1112/S0024610706023015.  Google Scholar

[2]

Scientiae Mathematicae Japonicae, 59 (2004), 577-590.  Google Scholar

[3]

J. Theor. Med., 6 (2005), 1-19. doi: 10.1080/1027366042000327098.  Google Scholar

[4]

Journal of Theoretical Medicine, 2 (2000), 129-154. doi: 10.1080/10273660008833042.  Google Scholar

[5]

J. Math. Biol., 61 (2010), 649-663. doi: 10.1007/s00285-009-0315-2.  Google Scholar

[6]

ACTA Biotheoretica, 43 (1995), 387-402. doi: 10.1007/BF00713561.  Google Scholar

[7]

Bulletin of Mathematical Biology, 60 (1998), 857-900. Google Scholar

[8]

Journal of Neuro-Oncology, 50 (2000), 37-51. Google Scholar

[9]

IMA Journal of Mathematics Applied in Medicine and Biology, 10 (1993), 149-168. doi: 10.1093/imammb/10.3.149.  Google Scholar

[10]

Numer. Math., 111 (2008), 169-205. doi: 10.1007/s00211-008-0188-0.  Google Scholar

[11]

Milan J. Math., 72 (2004), 1-28. doi: 10.1007/s00032-003-0026-x.  Google Scholar

[12]

Mathematical and Computer Modelling, 47 (2008), 755-764. doi: 10.1016/j.mcm.2007.06.005.  Google Scholar

[13]

Interfaces and Free Boundaries, 10 (2008), 119-138. doi: 10.4171/IFB/182.  Google Scholar

[14]

J. Comput. Appl. Math., 224 (2009), 168-181. doi: 10.1016/j.cam.2008.04.030.  Google Scholar

[15]

Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model,, SIAM J. Numer. Anal., 47 (): 386.  doi: 10.1137/07070423X.  Google Scholar

[16]

Numer. Math., 104 (2006), 457-488. doi: 10.1007/s00211-006-0024-3.  Google Scholar

[17]

R-Report, Institut fuer Angewandte Analysis und Stochastik (IAAS), Berlin, 1993. Google Scholar

[18]

Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106.  Google Scholar

[19]

Phys. Rev. Lett., 90 118101 (2003) [4 pages]. doi: 10.1103/PhysRevLett.90.118101.  Google Scholar

[20]

J. Nonlinear Sci., 21 (2011), 231-270. doi: 10.1007/s00332-010-9082-x.  Google Scholar

[21]

Journal für die Reine und angewandte Mathematik (Crelle's Journal), 654 (2011), 83-124. doi: 10.1515/CRELLE.2011.030.  Google Scholar

[22]

European Journal of Applied Mathematics, 12, (2001), 159-177. doi: 10.1017/S0956792501004363.  Google Scholar

[23]

Journal of Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022.  Google Scholar

[24]

J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5.  Google Scholar

[25]

J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6.  Google Scholar

[26]

Springer, 2nd edition, 2012. Google Scholar

[27]

in Flux-Corrected Transport: Principles, Algorithms, and Applications (Eds. D. Kuzmin, R. Löhner, S. Turek), Springer, Berlin (2005), 155-206. doi: 10.1007/3-540-27206-2_6.  Google Scholar

[28]

J. Comput. Phys., 175 (2002), 525-558. doi: 10.1006/jcph.2001.6955.  Google Scholar

[29]

J. Comput. Phys., 228 (2009), 2517-2534. doi: 10.1016/j.jcp.2008.12.011.  Google Scholar

[30]

Journal of Computational and Applied Mathematics, 236 (2012), 2317-2337. doi: 10.1016/j.cam.2011.11.019.  Google Scholar

[31]

J. Comput. Phys., 131 (1997), 327-353. Google Scholar

[32]

Physica A, 230 (1996), 499-543. Google Scholar

[33]

J. Inequal. Appl., 6 (2001), 37-55. doi: 10.1155/S1025583401000042.  Google Scholar

[34]

Funkcialaj Ekvacioj, 44 (2001), 441-469.  Google Scholar

[35]

IMA J. Numer. Anal., 27 (2007), 332-365. doi: 10.1093/imanum/drl018.  Google Scholar

[36]

IMS Workshop on Reaction-Diffusion Systems (Shatin, 1999). Methods Appl. Anal., 8 (2001), 349-367.  Google Scholar

[37]

The EMBO Journal, 22 (2003), 1771-1779. doi: 10.1093/emboj/cdg176.  Google Scholar

[38]

Computational methods in applied mathematics, 10 (2010), 219-232. doi: 10.2478/cmam-2010-0013.  Google Scholar

[39]

Journal of Computational and Applied Mathematics, 239 (2013), 290-303. doi: 10.1016/j.cam.2012.09.041.  Google Scholar

[40]

Computers and Mathematics with Applications, 64 (2012), 175-189. doi: 10.1016/j.camwa.2011.12.040.  Google Scholar

[41]

Springer, Berlin, 1999. doi: 10.1007/3-540-48092-7.  Google Scholar

[42]

Proc. R. Soc. Lond. B, 266 (1999), 299-304. doi: 10.1098/rspb.1999.0637.  Google Scholar

[43]

J. Math. Biol., 38 (1999), 359-375. doi: 10.1007/s002850050153.  Google Scholar

[44]

J. Math. Biol., 41 (2000), 455-475. doi: 10.1007/s002850000038.  Google Scholar

[45]

J. R. Soc. Interface, 9 (2012), 3027-3044. doi: 10.1098/rsif.2012.0276.  Google Scholar

[46]

J. Biomech, 44 (2011), 359-364. Google Scholar

[47]

J. Comput. Phys., 31 (1979), 335-362. doi: 10.1016/0021-9991(79)90051-2.  Google Scholar

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