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Active Particles, Volume 2 pp 109–148Cite as

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

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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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References

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  4. 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)

    Google Scholar 

  5. 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)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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  Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  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)

    Google Scholar 

  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)

    Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

A large portion of the material covered in this review is based on the work of the authors with Yekaterina Epshteyn, Hengrui Hu, Mária Lukáčová-Medvid’ ová, Mario Ricchiuto, Şeyma Nur Özcan, and Tong Wu, whose valuable contribution we would like to acknowledge here. The work of A. Chertock was supported in part by NSF grants DMS-1521051 and DMS-1818684. The work of A. Kurganov was supported in part by NSFC grant 11771201 and NSF grants DMS-1521009 and DMS-1818666.

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Authors and Affiliations

  1. Department of Mathematics and Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, USA

    Alina Chertock

  2. Department of Mathematics, Southern University of Science and Technology, Shenzhen, China

    Alexander Kurganov

  3. Mathematics Department, Tulane University, New Orleans, LA, USA

    Alexander Kurganov

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  1. Alina Chertock
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Correspondence to Alina Chertock .

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  1. Department of Mathematical Sciences, Politecnico di Torino, Torino, Italy

    Nicola Bellomo

  2. Department of Mathematics, Imperial College London, London, UK

    Pierre Degond

  3. CSCAMM, CSIC Bldg 406, University of Maryland, College Park, USA

    Eitan Tadmor

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Chertock, A., Kurganov, A. (2019). High-Resolution Positivity and Asymptotic Preserving Numerical Methods for Chemotaxis and Related Models. In: Bellomo, N., Degond, P., Tadmor, E. (eds) Active Particles, Volume 2. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-20297-2_4

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