Skip to main content
SpringerLink
Log in

Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we apply the local discontinuous Galerkin (LDG) method to 2D Keller–Segel (KS) chemotaxis model. We improve the results upon (Epshteyn and Kurganov in SIAM J Numer Anal, 47:368–408, 2008) and give optimal rate of convergence under special finite element spaces before the blow-up occurs (the exact solutions are smooth). Moreover, to construct physically relevant numerical approximations, we consider \(P^1\) LDG scheme and develop a positivity-preserving limiter to the scheme, extending the idea in Zhang and Shu (J Comput Phys, 229:8918–8934, 2010). With this limiter, we can prove the \(L^1\)-stability of the numerical scheme. Numerical experiments are performed to demonstrate the good performance of the positivity-preserving LDG scheme. Moreover, it is known that the chemotaxis model will yield blow-up solutions under certain initial conditions. We numerically demonstrate how to find the approximate blow-up time by using the \(L^2\)-norm of the \(L^1\)-stable numerical solution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267–279 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Childress, S., Percus, J.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ciarlet, P.: Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  5. Cockburn, B., Dong, B.: An analysis of the minimal dissipation local discontinuous Galerkin method for convection-diffusion problems. J. Sci. Comput. 32, 233–262 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cockburn, B., Hou, S., Shu, C.-W.: The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math. Comput. 54, 545–581 (1990)

    MATH  MathSciNet  Google Scholar 

  7. Cockburn, B., Kanschat, G., Perugia, I., Schotzau, D.: Superconvergence of the local discontinuous Galerkin method for elliptic problems on cartesian grids. SIAM J. Numer. Anal. 39, 264–285 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cockburn, B., Lin, S.-Y., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84, 90–113 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MATH  MathSciNet  Google Scholar 

  10. Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time dependent convection-diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. Epshteyn, Y.: Discontinuous Galerkin methods for the chemotaxis and haptotaxis models. J. Comput. Appl. Math. 224, 168–181 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  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, 211–256 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Epshteyn, Y., Kurganov, A.: New interior penalty discontinuous Galerkin methods for the Keller-Segel Chemotaxis model. SIAM J. Numer. Anal. 47, 368–408 (2008)

    MATH  MathSciNet  Google Scholar 

  16. Fatkullin, I.: A study of blow-ups in the Keller-Segel model of chemotaxis. Nonlinearity 26, 81–94 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. Gajewski, H., Zacharias, K.: Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 195, 77–114 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gelfand, I.M.: Some questions of analysis and differential equations. Am. Math. Soc. Transl. 26, 201–219 (1963)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Guo, L., Yang, Y.: Positivity-preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions. J. Comput. Phys. 289, 181–195 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hakovec, J., Schmeiser, C.: Stochastic particle approximation for measure valued solutions of the 2d Keller-Segel system. J. Stat. Phys. 135, 133–151 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Herrero, M.A., Medina, E., Velázquez, J.J.L.: Finite-time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10, 1739–1754 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Herrero, M.A., Velazquez, J.J.L.: Singularity patterns in a chemotaxis model. Math. Annu. 306, 583–623 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

  27. Hurd, A.E., Sattinger, D.H.: Questions of existence and uniqueness for hyperbolic equations with discontinuous coefficients. Trans. Am. Math. Soc. 132, 159–174 (1968)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  30. Meng, X., Shu, C.-W., Zhang, Q., Wu, B.: Superconvergence of discontinuous Galerkin methods for scalar nonlinear conservation laws in one space dimension. SIAM J. Numer. Anal. 50, 2336–2356 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 3, 581–601 (1995)

    MATH  MathSciNet  Google Scholar 

  32. Nakaguchi, E., Yagi, Y.: Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems. Hokkaido Math. J. 31, 385–429 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  34. Qin, T., Shu, C.-W., Yang, Y.: Bound-preserving discontinuous Galerkin methods for relativistic hydrodynamics. J. Comput. Phys. 315, 323–347 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  35. Reed, W.H., Hill, T.R.: Triangular mesh methods for the Neutron transport equation. Los Alamos Scientific Laboratory Report LA-UR-73-479. Los Alamos, NM (1973)

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

    Article  MATH  MathSciNet  Google Scholar 

  37. Saito, N.: Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Commun. Pure Appl. Anal. 11, 339–364 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  38. Shu, C.-W.: Total-variation-diminishing time discretizations. SIAM J. Sci. Stat. Comput. 9, 1073–1084 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  39. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  40. Strehl, R., Sokolov, A., Kuzmin, D., Horstmann, D., Turek, S.: A positivity-preserving finite element method for chemotaxis problems in 3D. J. Comput. Appl. Math. 239, 290–303 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. Wang, H., Shu, C.-W., Zhang, Q.: Stability and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for advection-diffusion problems. SIAM J. Numer. Anal. 53, 206–227 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  43. Wang, H., Wang, S., Shu, C.-W., Zhang, Q.: Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems. ESAIM Math. Model. Numer. Anal. 50, 1083–1105 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  44. Yang, Y., Shu, C.-W.: Discontinuous Galerkin method for hyperbolic equations involving \(\delta \)-singularities: negative-order norm error estimates and applications. Numer. Math. 124, 753–781 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  45. Yang, Y., Wei, D., Shu, C.-W.: Discontinuous Galerkin method for Krause’s consensus models and pressureless Euler equations. J. Comput. Phys. 252, 109–127 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  46. Zhang, Q., Shu, C.-W.: Error estimates to smooth solutions of Runge-Kutta discontinuous Galerkin methods for scalar conservation laws. SIAM J. Numer. Anal. 42, 641–666 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  47. Zhang, Q., Shu, C.-W.: Stability analysis and a priori error estimates to the third order explicit Runge-Kutta discontinuous Galerkin method for scalar conservation laws. SIAM J. Numer. Anal. 48, 1038–1063 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  49. Zhang, X., Shu, C.-W.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)

  50. Zhang, X., Shu, C.-W.: Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. J. Comput. Phys. 230, 1238–1248 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  51. Zhang, Y., Zhang, X., Shu, C.-W.: Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes. J. Comput. Phys. 234, 295–316 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  52. Zhao, X., Yang, Y., Syler, C.: A positivity-preserving semi-implicit discontinuous Galerkin scheme for solving extended magnetohydrodynamics equations. J. Comput. Phys. 278, 400–415 (2014)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

Xingjie Helen Li would like to thank the Shanghai Centre for Mathematics Science (SCMS), Fudan University, for support during her visit.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, 28223, USA

    Xingjie Helen Li

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Chi-Wang Shu

  3. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Yang Yang

Authors
  1. Xingjie Helen Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chi-Wang Shu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yang Yang.

Additional information

Research supported by ARO Grant W911NF-15-1-0226, NSF Grant DMS-1418750, China National Natural Science Foundation (11571367 and 11601536), Michigan Technological University, Research Excellence Fund Scholarship and Creativity Grant.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, X.H., Shu, CW. & Yang, Y. Local Discontinuous Galerkin Method for the Keller-Segel Chemotaxis Model. J Sci Comput 73, 943–967 (2017). https://doi.org/10.1007/s10915-016-0354-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-016-0354-y

Keywords

Search

Navigation