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High-order positivity-preserving hybrid finite-volume-finite-difference methods for chemotaxis systems

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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.

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References

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

    Article  Google Scholar 

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonner, J.T.: The Cellular Slime Molds, 2nd edn. Princeton University Press, Princeton, New Jersey (1967)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. 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 

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Filbet, F.: A finite volume scheme for the patlak-keller-segel chemotaxis model. Numer. Math. 104, 457–488 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

  23. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Herrero, M.A., Velázquez, J.J.L.: A blow-up mechanism for a chemotaxis model. Ann. Sc. Norm. Super. 24, 633–683 (1997)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Horstmann, D.: From 1970 until now: The keller-segel model in chemotaxis and its consequences i. Jahresber. DMV 105, 103–165 (2003)

    MathSciNet  MATH  Google Scholar 

  27. 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 

  28. Hundsdorfer, W., Verwer, J.G.: Numerical solution of time-dependent advection-diffusion-reaction equations, vol. 33 Springer Science & Business Media (2013)

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. 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 

  35. 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)

    Article  MATH  Google Scholar 

  36. 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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Marrocco, A.: 2d simulation of chemotaxis bacteria aggregation. m2AN. Math. Model. Numer. Anal. 37, 617–630 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Perthame, B.: Transport Equations in Biology. Frontiers in Mathematics. Basel, Birkhäuser Verlag (2007)

    MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  47. 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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  50. 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)

    Article  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Alina Chertock & Hengrui Hu

  2. Department of Mathematics, The University of Utah, Salt Lake City, UT, 84112, USA

    Yekaterina Epshteyn

  3. Department of Mathematics, Southern University of Science and Technology of China, Shenzhen, 518055, China

    Alexander Kurganov

  4. Mathematics Department, Tulane University, New Orleans, LA, 70118, USA

    Alexander Kurganov

Authors
  1. Alina Chertock
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  2. Yekaterina Epshteyn
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  3. Hengrui Hu
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  4. Alexander Kurganov
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Corresponding author

Correspondence to Alina Chertock.

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Communicated by: Carlos Garcia-Cervera

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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

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  • DOI: https://doi.org/10.1007/s10444-017-9545-9

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