Abstract
Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, usually a chemical one. Chemotaxis phenomenon plays an important role in bacteria/cell aggregation and pattern formation mechanisms, as well as in tumor growth. A common property of all chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in rapid growth of solutions in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. There is consequently a need for accurate and computationally efficient numerical methods for the chemotaxis models. In this work, we develop and study novel high-order hybrid finite-volume-finite-difference schemes for the Patlak-Keller-Segel chemotaxis system and related models. We demonstrate high-accuracy, stability and computational efficiency of the proposed schemes in a number of numerical examples.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Adler, A.: Chemotaxis in bacteria. Ann. Rev. Biochem. 44, 341–356 (1975)
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)
Ascher, U.M., Ruuth, S.J., Wetton, B.T.: Implicit-explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)
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)
Bonner, J.T.: The Cellular Slime Molds, 2nd edn. Princeton University Press, Princeton, New Jersey (1967)
Budrene, E.O., Berg, H.C.: Complex patterns formed by motile cells of escherichia coli. Nature 349, 630–633 (1991)
Budrene, E.O., Berg, H.C.: Dynamics of formation of symmetrical patterns by chemotactic bacteria. Nature 376, 49–53 (1995)
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)
Chertock, A., Kurganov, A.: A positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111, 169–205 (2008)
Childress, S., Percus, J.K.: Nonlinear aspects of chemotaxis. Math. Biosc. 56, 217–237 (1981)
Cohen, M.H., Robertson, A.: Wave propagation in the early stages of aggregation of cellular slime molds. J. Theor. Biol. 31, 101–118 (1971)
Conca, C., Espejo, E., Vilches, K.: Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in \(\mathbb {R}^{2}\). European J. Appl. Math. 22(6), 553–580 (2011)
Epshteyn, Y.: Upwind-difference potentials method for Patlak-Keller-Segel chemotaxis model. J. Sci. Comput. 53(3), 689–713 (2012)
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)
Epshteyn, Y., Kurganov, A.: New interior penalty discontinuous galerkin methods for the Keller-Segel chemotaxis model. SIAM J. Numer. Anal. 47, 386–408 (2008)
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)
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)
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)
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)
Fasano, A., Mancini, A., Primicerio, M.: Equilibrium of two populations subject to chemotaxis. Math. Models Methods Appl. Sci. 14, 503–533 (2004)
Filbet, F.: A finite volume scheme for the patlak-keller-segel chemotaxis model. Numer. Math. 104, 457–488 (2006)
Gottlieb, S., Ketcheson, D., Shu, C.W.: Strong stability preserving Runge-Kutta and multistep time discretizations. World Scientific Publishing Co. Pte. Ltd., Hackensack NJ (2011)
Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. 24, 633–683 (1997)
Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1-2), 183–217 (2009)
Horstmann, D.: From 1970 until now: The keller-segel model in chemotaxis and its consequences i. Jahresber. DMV 105, 103–165 (2003)
Horstmann, D.: From 1970 until now: The keller-segel model in chemotaxis and its consequences ii. Jahresber. DMV 106, 51–69 (2004)
Hundsdorfer, W., Verwer, J.G.: Numerical solution of time-dependent advection-diffusion-reaction equations, vol. 33 Springer Science & Business Media (2013)
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30, 225–234 (1971)
Keller, E.F., Segel, L.A.: Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 235–248 (1971)
Kurganov, A., Liu, Y.: New adaptive artificial viscosity method for hyperbolic systems of conservation laws. J. Comput. Phys. 231, 8114–8132 (2012)
Kurganov, A., Lukáčová-medviďová, M.: Numerical study of two-species chemotaxis models. Discrete Contin. Dyn. Syst. Ser. B 19, 131–152 (2014)
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)
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)
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)
Marrocco, A.: 2d simulation of chemotaxis bacteria aggregation. m2AN. Math. Model. Numer. Anal. 37, 617–630 (2003)
Nanjundiah, V.: Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42, 63–105 (1973)
Nessyahu, H., Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87(2), 408–463 (1990)
Patlak, C.S.: Random walk with persistence and external bias. Bull. Math: Biophys. 15, 311–338 (1953)
Perthame, B.: PDE models for chemotactic movements: parabolic, hyperbolic and kinetic. Appl. Math. 49, 539–564 (2004)
Perthame, B.: Transport Equations in Biology. Frontiers in Mathematics. Basel, Birkhäuser Verlag (2007)
Prescott, L.M., Harley, J.P., Klein, D.A.: Microbiology, 3rd edn. Wm. C. Brown Publishers, Chicago, London (1996)
Saito, N.: Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis. IMA J. Numer. Anal. 27(2), 332–365 (2007)
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)
Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21(5), 995–1011 (1984)
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)
Tyson, R., Stern, L.G., LeVeque, R.J.: Fractional step methods applied to a chemotaxis model. J. Math. Biol. 41, 455–475 (2000)
Wolansky, G.: Multi-components chemotactic system in the absence of conflicts. European J. Appl. Math. 13, 641–661 (2002)
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)
Acknowledgments
The work of A. Chertock was supported in part by NSF grant DMS-1521051. The work of A. Kurganov was supported in part by NSF grant DMS-1521009.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Carlos Garcia-Cervera
Rights and permissions
About this article
Cite this article
Chertock, A., Epshteyn, Y., Hu, H. et al. High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems. Adv Comput Math 44, 327–350 (2018). https://doi.org/10.1007/s10444-017-9545-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-017-9545-9
Keywords
- Patlak-Keller-Segel chemotaxis system
- Advection-diffusion-reaction systems
- High-order finite-difference
- Finite-volume methods
- Positivity-preserving algorithms