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

  • Alina ChertockEmail author
  • Alexander Kurganov
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


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.



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.


  1. 1.
    Adler, A.: Chemotaxis in bacteria. Ann. Rev. Biochem. 44, 341–356 (1975)CrossRefGoogle Scholar
  2. 2.
    Alt, W.: Biased random walk models for chemotaxis and related diffusion approximations. J. Math. Biol. 9(2), 147–177 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Arpaia, L., Ricchiuto, M.: r-adaptation for shallow water flows: conservation, well balancedness, efficiency. Comput. & Fluids 160, 175–203 (2018)Google Scholar
  4. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bonner, J.T.: The cellular slime molds, 2nd edn. Princeton University Press, Princeton, New Jersey (1967)CrossRefGoogle Scholar
  9. 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. 10.
    Budrene, E.O., Berg, H.C.: Complex patterns formed by motile cells of Escherichia coli. Nature 349, 630–633 (1991)CrossRefGoogle Scholar
  11. 11.
    Budrene, E.O., Berg, H.C.: Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376, 49–53 (1995)CrossRefGoogle Scholar
  12. 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. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Chertock, A., Kurganov, A.: A positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111, 169–205 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 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)zbMATHGoogle Scholar
  20. 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)MathSciNetCrossRefGoogle Scholar
  21. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Childress, S., Percus, J.K.: Nonlinear aspects of chemotaxis. Math. Biosc. 56, 217–237 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 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. 26.
    Epshteyn, Y.: Upwind-difference potentials method for Patlak-Keller-Segel chemotaxis model. J. Sci. Comput. 53(3), 689–713 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 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)MathSciNetzbMATHGoogle Scholar
  30. 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)MathSciNetzbMATHGoogle Scholar
  31. 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. 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)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Fasano, A., Mancini, A., Primicerio, M.: Equilibrium of two populations subject to chemotaxis. Math. Models Methods Appl. Sci. 14, 503–533 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Filbet, F.: A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. Numer. Math. 104, 457–488 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Filbet, F., Yang, C.: Numerical simulations of kinetic models for chemotaxis. SIAM J. Sci. Comput. 36(3), B348–B366 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Gajewski, H., Zacharias, K., Gröger, K.: Global behaviour of a reaction-diffusion system modelling chemotaxis. Mathematische Nachrichten 195(1), 77–114 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Herrero, M., Velázquez, J.: A blow-up mechanism for a chemotaxis model. Ann. Scuola Normale Superiore 24, 633–683 (1997)MathSciNetzbMATHGoogle Scholar
  38. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Herrero, M.A., Velázquez, J.J.: Chemotactic collapse for the iKeller-Segel model. J. Math. Biol. 35(2), 177–194 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Scuola Normale Superiore 24, 633–683 (1997)MathSciNetzbMATHGoogle Scholar
  41. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1-2), 183–217 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Horstmann, D.: From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I. Jahresber. DMV 105, 103–165 (2003)zbMATHMathSciNetGoogle Scholar
  45. 45.
    Horstmann, D.: From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. DMV 106, 51–69 (2004)zbMATHGoogle Scholar
  46. 46.
    Huang, W., Russell, R.D.: Adaptive moving mesh methods, Applied Mathematical Sciences, vol. 174. Springer, New York (2011)Google Scholar
  47. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 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. 51.
    Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)zbMATHCrossRefGoogle Scholar
  53. 53.
    Keller, E.F., Segel, L.A.: Traveling bands of chemotactic bacteria: A theoretical analysis. J. Theor. Biol. 30, 235–248 (1971)zbMATHCrossRefGoogle Scholar
  54. 54.
    Kurganov, A., Liu, Y.: New adaptive artificial viscosity method for hyperbolic systems of conservation laws. J. Comput. Phys. 231, 8114–8132 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Kurganov, A., Lukáčová-Medviďová, M.: Numerical study of two-species chemotaxis models. Discrete Contin. Dyn. Syst. Ser. B 19, 131–152 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 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 SubmittedGoogle Scholar
  57. 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)MathSciNetzbMATHGoogle Scholar
  58. 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)zbMATHGoogle Scholar
  59. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Liebchen, B., Löwen, H.: Modelling chemotaxis of microswimmers: from individual to collective behavior. arXiv preprint arXiv:1802.07933 (2018)Google Scholar
  62. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 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. 64.
    Marrocco, A.: 2d simulation of chemotaxis bacteria aggregation. M2AN Math. Model. Numer. Anal. 37, 617–630 (2003)Google Scholar
  65. 65.
    Nagai, T.: Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains. J. Inequal. Appl. pp. 37–55Google Scholar
  66. 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)MathSciNetzbMATHGoogle Scholar
  67. 67.
    Nanjundiah, V.: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42, 63–105 (1973)CrossRefGoogle Scholar
  68. 68.
    Nessyahu, H., Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87(2), 408–463 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Ni, W.M.: Diffusion, cross-diffusion, and their spike-layer steady states. Notices Amer. Math. Soc. 45(1), 9–18 (1998)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Othmer, H.G., Dunbar, S.R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26(3), 263–298 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Patlak, C.S.: Random walk with persistence and external bias. Bull. Math: Biophys. 15, 311–338 (1953)MathSciNetzbMATHGoogle Scholar
  74. 74.
    Pedley, T.J., Kessler, J.O.: Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24(1), 313–358 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Perthame, B.: PDE models for chemotactic movements: parabolic, hyperbolic and kinetic. Appl. Math. 49, 539–564 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Perthame, B.: Transport equations in biology. Frontiers in Mathematics. Birkhäuser Verlag, Basel (2007)Google Scholar
  77. 77.
    Pohl, O., Stark, H.: Dynamic clustering and chemotactic collapse of self-phoretic active particles. Phys. Rev. Lett. 112(23), 238,303 (2014)CrossRefGoogle Scholar
  78. 78.
    Prescott, L.M., Harley, J.P., Klein, D.A.: Microbiology, 3rd edn. Wm. C. Brown Publishers, Chicago, London (1996)Google Scholar
  79. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 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. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  84. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 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)zbMATHGoogle Scholar
  86. 86.
    Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21(5), 995–1011 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  87. 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. 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)zbMATHCrossRefGoogle Scholar
  89. 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)CrossRefGoogle Scholar
  90. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Tyson, R., Stern, L.G., LeVeque, R.J.: Fractional step methods applied to a chemotaxis model. J. Math. Biol. 41, 455–475 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    Vabishchevich, P.N.: Additive operator-difference schemes. De Gruyter, Berlin (2014). Splitting schemesGoogle Scholar
  93. 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. 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. 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. 96.
    Wolansky, G.: Multi-components chemotactic system in the absence of conflicts. European J. Appl. Math. 13, 641–661 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  97. 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. 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. 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)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Research in Scientific ComputationNorth Carolina State UniversityRaleighUSA
  2. 2.Department of MathematicsSouthern University of Science and TechnologyShenzhenChina
  3. 3.Mathematics DepartmentTulane UniversityNew OrleansUSA

Personalised recommendations