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2019 | OriginalPaper | Chapter

High-Resolution Positivity and Asymptotic Preserving Numerical Methods for Chemotaxis and Related Models

Authors : Alina Chertock, Alexander Kurganov

Published in: Active Particles, Volume 2

Publisher: Springer International Publishing

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Abstract

Many microorganisms exhibit a special pattern formation at the presence of a chemoattractant, food, light, or areas with high oxygen concentration. Collective cell movement can be described by a system of nonlinear PDEs on both macroscopic and microscopic levels. The classical PDE chemotaxis model is the Patlak-Keller-Segel system, which consists of a convection-diffusion equation for the cell density and a reaction-diffusion equation for the chemoattractant concentration. At the cellular (microscopic) level, a multiscale chemotaxis models can be used. These models are 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 that describes the velocity change of the cells.

A common property of the chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions rapidly growing in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. In either case, capturing such singular solutions numerically is a challenging problem and the use of higher-order methods and/or adaptive strategies is often necessary. In addition, positivity preserving is an absolutely crucial property a good numerical method used to simulate chemotaxis should satisfy: this is the only way to guarantee a nonlinear stability of the method. For kinetic chemotaxis systems, it is also essential that numerical methods provide a consistent and stable discretization in certain asymptotic regimes.

In this paper, we review some of the recent advances in developing of high-resolution finite-volume and finite-difference numerical methods that possess the aforementioned properties of the chemotaxis-type systems.

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Adler, A.: Chemotaxis in bacteria. Ann. Rev. Biochem. 44, 341–356 (1975)

Alt, W.: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9(2), 147–177 (1980)MathSciNetMATH

Arpaia, L., Ricchiuto, M.: r-adaptation for shallow water flows: conservation, well balancedness, efficiency. Comput. & Fluids 160, 175–203 (2018)

Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2-3), 151–167 (1997). Special issue on time integration (Amsterdam, 1996)

Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)MathSciNetMATH

Bialké, J., Löwen, H., Speck, T.: Microscopic theory for the phase separation of self-propelled repulsive disks. EPL (Europhysics Letters) 103(3), 30,008 (2013)

Bollermann, A., Noelle, S., Lukáčová-Medviďová, M.: Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10(2), 371–404 (2011)MathSciNetMATH

Bonner, J.T.: The cellular slime molds, 2nd edn. Princeton University Press, Princeton, New Jersey (1967)

Bournaveas, N., Calvez, V.: Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data 26(5), 1871–1895 (2009)

10.

Budrene, E.O., Berg, H.C.: Complex patterns formed by motile cells of Escherichia coli. Nature 349, 630–633 (1991)

11.

Budrene, E.O., Berg, H.C.: Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376, 49–53 (1995)

12.

Calvez, V., Carrillo, J.A.: Volume effects in the Keller-Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. (9) 86(2), 155–175 (2006)

13.

Calvez, V., Perthame, B., Sharifi Tabar, M.: Modified Keller-Segel system and critical mass for the log interaction kernel. In: Stochastic analysis and partial differential equations, Contemp. Math., vol. 429, pp. 45–62. Amer. Math. Soc., Providence, RI (2007)

14.

Carrillo, J.A., Yan, B.: An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis. Multiscale Model. Simul. 11(1), 336–361 (2013)MathSciNetMATH

15.

Chalub, F.A.C.C., Markowich, P.A., Perthame, B., Schmeiser, C.: Kinetic models for chemotaxis and their drift-diffusion limits. Monatsh. Math. 142(1-2), 123–141 (2004)MathSciNetMATH

16.

Chertock, A., Epshteyn, Y., Hu, H., Kurganov, A.: High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems. Adv. Comput. Math. 44(1), 327–350 (2018)MathSciNetMATH

17.

Chertock, A., Fellner, K., Kurganov, A., Lorz, A., Markowich, P.A.: Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: a high-resolution numerical approach. J. Fluid Mech. 694, 155–190 (2012)MathSciNetMATH

18.

Chertock, A., Kurganov, A.: A positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111, 169–205 (2008)MathSciNetMATH

19.

Chertock, A., Kurganov, A., Lukáčová-Medviďová, M., Özcan, c.N.: An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinet. Relat. Models 12, 195–216 (2019)MATH

20.

Chertock, A., Kurganov, A., Ricchiuto, M., Wu, T.: Adaptive moving mesh upwind scheme for the two-species chemotaxis model. Comput. Math. Appl. 77, 3172–3185 (2019)MathSciNetMATH

21.

Chertock, A., Kurganov, A., Wang, X., Wu, Y.: On a chemotaxis model with saturated chemotactic flux. Kinet. Relat. Models 5(1), 51–95 (2012)MathSciNetMATH

22.

Childress, S., Percus, J.K.: Nonlinear aspects of chemotaxis. Math. Biosc. 56, 217–237 (1981)MathSciNetMATH

23.

Cohen, M.H., Robertson, A.: Wave propagation in the early stages of aggregation of cellular slime molds. J. Theor. Biol. 31, 101–118 (1971)

24.

Conca, C., Espejo, E., Vilches, K.: Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in ℝ². European J. Appl. Math. 22(6), 553–580 (2011)MathSciNetMATH

25.

Eisenbach, M., Lengeler, J.W., Varon, M., Gutnick, D., Meili, R., Firtel, R.A., Segall, J.E., Omann, G.M., Tamada, A., Murakami, F.: Chemotaxis. Imperial College Press (2004)

26.

Epshteyn, Y.: Upwind-difference potentials method for Patlak-Keller-Segel chemotaxis model. J. Sci. Comput. 53(3), 689–713 (2012)MathSciNetMATH

27.

Epshteyn, Y., Izmirlioglu, A.: Fully discrete analysis of a discontinuous finite element method for the Keller-Segel chemotaxis model. J. Sci. Comput. 40(1-3), 211–256 (2009)MathSciNetMATH

28.

Epshteyn, Y., Kurganov, A.: New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model. SIAM J. Numer. Anal. 47, 386–408 (2008)MathSciNetMATH

29.

Espejo, E.E., Stevens, A., Suzuki, T.: Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species. Differential Integral Equations 25(3-4), 251–288 (2012)MathSciNetMATH

30.

Espejo, E.E., Stevens, A., Velázquez, J.J.L.: A note on non-simultaneous blow-up for a drift-diffusion model. Differential Integral Equations 23(5-6), 451–462 (2010)MathSciNetMATH

31.

Espejo, E.E., Vilches, K., Conca, C.: Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in \(\mathbb {R}^2\). European J. Appl. Math. 24, 297–313 (2013)

32.

Espejo Arenas, E.E., Stevens, A., Velázquez, J.J.L.: Simultaneous finite time blow-up in a two-species model for chemotaxis. Analysis (Munich) 29(3), 317–338 (2009)MathSciNetMATH

33.

Fasano, A., Mancini, A., Primicerio, M.: Equilibrium of two populations subject to chemotaxis. Math. Models Methods Appl. Sci. 14, 503–533 (2004)MathSciNetMATH

34.

Filbet, F.: A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. Numer. Math. 104, 457–488 (2006)MathSciNetMATH

35.

Filbet, F., Yang, C.: Numerical simulations of kinetic models for chemotaxis. SIAM J. Sci. Comput. 36(3), B348–B366 (2014)MathSciNetMATH

36.

Gajewski, H., Zacharias, K., Gröger, K.: Global behaviour of a reaction-diffusion system modelling chemotaxis. Mathematische Nachrichten 195(1), 77–114 (1998)MathSciNetMATH

37.

Herrero, M., Velázquez, J.: A blow-up mechanism for a chemotaxis model. Ann. Scuola Normale Superiore 24, 633–683 (1997)MathSciNetMATH

38.

Herrero, M.A., Medina, E., Velázquez, J.: Finite-time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10(6), 1739 (1997)MathSciNetMATH

39.

Herrero, M.A., Velázquez, J.J.: Chemotactic collapse for the iKeller-Segel model. J. Math. Biol. 35(2), 177–194 (1996)MathSciNetMATH

40.

Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Scuola Normale Superiore 24, 633–683 (1997)MathSciNetMATH

41.

Hillen, T., Othmer, H.G.: The diffusion limit of transport equations derived from velocity-jump processes. SIAM J. Appl. Math. 61(3), 751–775 (electronic) (2000)

42.

Hillen, T., Painter, K.: Global existence for a parabolic chemotaxis model with prevention of overcrowding. Adv. in Appl. Math. 26(4), 280–301 (2001)MathSciNetMATH

43.

Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1-2), 183–217 (2009)MathSciNetMATH

44.

Horstmann, D.: From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I. Jahresber. DMV 105, 103–165 (2003)MATH

45.

Horstmann, D.: From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. DMV 106, 51–69 (2004)MATH

46.

Huang, W., Russell, R.D.: Adaptive moving mesh methods, Applied Mathematical Sciences, vol. 174. Springer, New York (2011)

47.

Hundsdorfer, W., Ruuth, S.J.: IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 225(2), 2016–2042 (2007)MathSciNetMATH

48.

Hwang, H.J., Kang, K., Stevens, A.: Drift-diffusion limits of kinetic models for chemotaxis: a generalization. Discrete Contin. Dyn. Syst. Ser. B 5(2), 319–334 (2005)MathSciNetMATH

49.

Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 329(2), 819–824 (1992)MathSciNetMATH

50.

Jin, S., Pareschi, L., Toscani, G.: Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations. SIAM J. Numer. Anal. 35(6), 2405–2439 (electronic) (1998)

51.

Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)MathSciNetMATH

52.

Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)MATH

53.

Keller, E.F., Segel, L.A.: Traveling bands of chemotactic bacteria: A theoretical analysis. J. Theor. Biol. 30, 235–248 (1971)MATH

54.

Kurganov, A., Liu, Y.: New adaptive artificial viscosity method for hyperbolic systems of conservation laws. J. Comput. Phys. 231, 8114–8132 (2012)MathSciNetMATH

55.

Kurganov, A., Lukáčová-Medviďová, M.: Numerical study of two-species chemotaxis models. Discrete Contin. Dyn. Syst. Ser. B 19, 131–152 (2014)MathSciNetMATH

56.

Kurganov, A., Qu, Z., Rozanova, O., Wu, T.: Adaptive moving mesh central-upwind schemes for hyperbolic system of PDEs. Applications to compressible Euler equations and granular hydrodynamics Submitted

57.

Kurokiba, M., Ogawa, T.: Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type. Diff. Integral Eqns 4, 427–452 (2003)MathSciNetMATH

58.

van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32(1), 101–136 (1979)MATH

59.

Levy, D., Requeijo, T.: Modeling group dynamics of phototaxis: from particle systems to PDEs. Discrete Contin. Dyn. Syst. Ser. B 9(1), 103–128 (electronic) (2008)

60.

Lie, K.A., Noelle, S.: On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24(4), 1157–1174 (2003)MathSciNetMATH

61.

Liebchen, B., Löwen, H.: Modelling chemotaxis of microswimmers: from individual to collective behavior. arXiv preprint arXiv:1802.07933 (2018)

62.

Lin, C.S., Ni, W.M., Takagi, I.: Large amplitude stationary solutions to a chemotaxis system. J. Differential Equations 72(1), 1–27 (1988)MathSciNetMATH

63.

Marchuk, G.I.: Splitting and alternating direction methods. In: Handbook of numerical analysis, Vol. I, Handb. Numer. Anal., I, pp. 197–462. North-Holland, Amsterdam (1990)

64.

Marrocco, A.: 2d simulation of chemotaxis bacteria aggregation. M2AN Math. Model. Numer. Anal. 37, 617–630 (2003)

65.

Nagai, T.: Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. pp. 37–55

66.

Nagai, T., Senba, T., Yoshida, K.: Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis. Funkcial. Ekvac. 40(3), 411–433 (1997)MathSciNetMATH

67.

Nanjundiah, V.: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42, 63–105 (1973)

68.

Nessyahu, H., Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87(2), 408–463 (1990)MathSciNetMATH

69.

Ni, W.M.: Diffusion, cross-diffusion, and their spike-layer steady states. Notices Amer. Math. Soc. 45(1), 9–18 (1998)MathSciNetMATH

70.

Othmer, H.G., Dunbar, S.R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26(3), 263–298 (1988)MathSciNetMATH

71.

Othmer, H.G., Hillen, T.: The diffusion limit of transport equations. II. Chemotaxis equations. SIAM J. Appl. Math. 62(4), 1222–1250 (electronic) (2002)

72.

Pareschi, L., Russo, G.: Implicit-Explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25(1-2), 129–155 (2005)MathSciNetMATH

73.

Patlak, C.S.: Random walk with persistence and external bias. Bull. Math: Biophys. 15, 311–338 (1953)MathSciNetMATH

74.

Pedley, T.J., Kessler, J.O.: Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24(1), 313–358 (1992)MathSciNetMATH

75.

Perthame, B.: PDE models for chemotactic movements: parabolic, hyperbolic and kinetic. Appl. Math. 49, 539–564 (2004)MathSciNetMATH

76.

Perthame, B.: Transport equations in biology. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2007)

77.

Pohl, O., Stark, H.: Dynamic clustering and chemotactic collapse of self-phoretic active particles. Phys. Rev. Lett. 112(23), 238,303 (2014)

78.

Prescott, L.M., Harley, J.P., Klein, D.A.: Microbiology, 3rd edn. Wm. C. Brown Publishers, Chicago, London (1996)

79.

Saito, N.: Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis. IMA J. Numer. Anal. 27(2), 332–365 (2007)MathSciNetMATH

80.

Sleeman, B.D., Ward, M.J., Wei, J.C.: The existence and stability of spike patterns in a chemotaxis model. SIAM J. Appl. Math. 65(3), 790–817 (electronic) (2005)

81.

Stevens, A.: The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems. SIAM J. Appl. Math. 61(1), 183–212 (electronic) (2000)

82.

Stevens, A., Othmer, H.G.: Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math. 57(4), 1044–1081 (1997)MathSciNetMATH

83.

Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)MathSciNetMATH

84.

Strehl, R., Sokolov, A., Kuzmin, D., Turek, S.: A flux-corrected finite element method for chemotaxis problems. Computational Methods in Applied Mathematics 10(2), 219–232 (2010)MathSciNetMATH

85.

Stroock, D.W.: Some stochastic processes which arise from a model of the motion of a bacterium. Probab. Theory Relat. Fields 28(4), 305–315 (1974)MATH

86.

Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21(5), 995–1011 (1984)MathSciNetMATH

87.

Tang, H., Tang, T.: Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM J. Numer. Anal. 41(2), 487–515 (electronic) (2003)

88.

Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C.W., Kessler, J.O., Goldstein, R.E.: Bacterial swimming and oxygen transport near contact lines. PNAS 102, 2277–2282 (2005)MATH

89.

Tyson, R., Lubkin, S.R., Murray, J.D.: A minimal mechanism for bacterial pattern formation. Proc. Roy. Soc. Lond. B 266, 299–304 (1999)

90.

Tyson, R., Lubkin, S.R., Murray, J.D.: Model and analysis of chemotactic bacterial patterns in a liquid medium. J. Math. Biol. 38(4), 359–375 (1999)MathSciNetMATH

91.

Tyson, R., Stern, L.G., LeVeque, R.J.: Fractional step methods applied to a chemotaxis model. J. Math. Biol. 41, 455–475 (2000)MathSciNetMATH

92.

Vabishchevich, P.N.: Additive operator-difference schemes. De Gruyter, Berlin (2014). Splitting schemes

93.

Velázquez, J.J.L.: Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions. SIAM J. Appl. Math. 64(4), 1198–1223 (electronic) (2004)

94.

Velázquez, J.J.L.: Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions. SIAM J. Appl. Math. 64(4), 1224–1248 (electronic) (2004)

95.

Wang, X.: Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics. SIAM J. Math. Anal. 31(3), 535–560 (electronic) (2000)

96.

Wolansky, G.: Multi-components chemotactic system in the absence of conflicts. European J. Appl. Math. 13, 641–661 (2002)MathSciNetMATH

97.

Woodward, D., Tyson, R., Myerscough, M., Murray, J., Budrene, E., Berg, H.: Spatio-temporal patterns generated by S. typhimurium. Biophys. J. 68, 2181–2189 (1995)

98.

Yeomans, J.: The hydrodynamics of active systems. In: C.N. Likas, F. Sciortino, E. Zaccarelli, P. Ziherl (eds.) Proceedings of the International School of Physics “Enrico Fermi”, pp. 383–415. IOS, Amsterdam, SIF, Bologna (2016)

99.

Zhang, X., Shu, C.W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229(9), 3091–3120 (2010)MathSciNetMATH

Title: High-Resolution Positivity and Asymptotic Preserving Numerical Methods for Chemotaxis and Related Models
Authors: Alina Chertock
Alexander Kurganov
Publisher: Springer International Publishing
Book: Active Particles, Volume 2
Print ISBN: 978-3-030-20296-5

Electronic ISBN: 978-3-030-20297-2

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-20297-2_4

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