Journal of Scientific Computing

, Volume 78, Issue 3, pp 1387–1404 | Cite as

Energy Dissipative Local Discontinuous Galerkin Methods for Keller–Segel Chemotaxis Model

  • Li Guo
  • Xingjie Helen LiEmail author
  • Yang Yang


In this paper, we apply the local discontinuous Galerkin (LDG) method to solve the classical Keller–Segel (KS) chemotaxis model. The exact solution of the KS chemotaxis model may exhibit blow-up patterns with certain initial conditions, and is not easy to approximate numerically. Moreover, it has been proved that there exists a definition of free energy of the KS system which dissipates with respect to time. We will construct a consistent numerical energy and prove the energy dissipation with the LDG discretization. Several numerical experiments in one and two space dimensions will be given. Especially, for solutions with blow-up (converge to Dirac delta functions), the densities of KS model are computed to be strictly positive in the numerical experiments and the energies are also numerically observed to be strictly positive and decreasing as are seen in the figures. Therefore, the scheme is stable for the KS model with blow-up solutions.


Energy dissipation Local discontinuous Galerkin method Keller–Segel chemotaxis model Blow-up solutions 


  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller–Segel model: optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. 44, 1–33 (2006)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blanchet, A., Laurencot, P.: The Parabolic–Parabolic Keller-Segel System with Critical Diffusion as a Gradient Flow in \(R^d\), \(d\ge 3\). Commun. Partial Differ. Equ. 38, 658–686 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Blanchet, A., Carrillo, J., Kinderlehrer, D., Kowalczyk, M., Laurencot, P., Lisini, S.: A hybrid variational principle for the Keller–Segel system in \(R^2\). ESAIM:M2AN 49, 1553–1576 (2015)CrossRefzbMATHGoogle Scholar
  5. 5.
    Calvez, V., Corrias, L.: The parabolic–parabolic Keller–Segel model in \(R^2\). Commun. Math. Sci. 6, 417–447 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chertock, A., Kurganov, A.: A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models. Numer. Math. 111, 169–205 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Childress, S., Percus, J.: Nonlinear aspects of chemotaxis. Math. Biosci. 56, 217–237 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Cockburn, B., Shu, C.-W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin method for time dependent convection–diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cong, W., Liu, J.-G.: Uniform \(L^\infty \) boundedness for a degenerate parabolic–parabolic Keller–Segel model. Discrete Contin. Dyn. Syst. Ser. B 22, 307–338 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Epshteyn, Y.: Discontinuous Galerkin methods for the chemotaxis and haptotaxis models. J. Comput. Appl. Math. 224, 168–181 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Epshteyn, Y.: Upwind-difference potentials method for Patlak–Keller–Segel chemotaxis model. J. Sci. Comput. 53, 689–713 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Epshteyn, Y., Kurganov, A.: New interior penalty discontinuous Galerkin methods for the Keller–Segel chemotaxis model. SIAM J. Numer. Anal. 47, 368–408 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fatkullin, I.: A study of blow-ups in the Keller–Segel model of chemotaxis. Nonlinearity 26, 81–94 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Filbet, F.: A finite volume scheme for the Patlak–Keller–Segel chemotaxis model. Numer. Math. 104, 457–488 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gajewski, H., Zacharias, K.: Global behaviour of a reaction–diffusion system modelling chemotaxis. Math. Nachr. 195, 77–114 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hakovec, J., Schmeiser, C.: Stochastic particle approximation for measure valued solutions of the 2D Keller–Segel system. J. Stat. Phys. 135, 133–151 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Herrero, M.A., Velazquez, J.J.L.: Singularity patterns in a chemotaxis model. Math. Annu. 306, 583–623 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Horstman, D.: From 1970 until now: the Keller–Segel model in chemotaxis and its consequences I. Jahresber. DMV 105, 103–165 (2003)Google Scholar
  27. 27.
    Horstmann, D.: From 1970 until now: the Keller–Segel model in chemotaxis and its consequences II. Jahresber. DMV 106, 51–69 (2004)zbMATHGoogle Scholar
  28. 28.
    Ishida, S., Yokota, T.: Blow-up in finite or infinite time for quasilinear degenerate Keller–Segel systems of parabolic–parabolic type. Discrete Contin. Dyn. Syst. Ser. B 17, 2569–2596 (2013)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Keller, E.F., Segel, L.A.: Initiation on slime mold aggregation viewed as instability. J. Theor. Biol. 26, 399–415 (1970)CrossRefzbMATHGoogle Scholar
  30. 30.
    Li, X., Shu, C.-W., Yang, Y.: Local discontinuous Galerkin method for the Keller–Segel chemotaxis model. J. Sci. Comput. 73, 943–967 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Liu, J.-G., Wang, L., Zhou, Z.: Positivity-preserving and asymptotic preserving method for 2D Keller–Segel equations. Math. Comput. 87, 1165–1189 (2018)CrossRefzbMATHGoogle Scholar
  32. 32.
    Liu, J.-G., Wang, J.: Refined hyper-contractivity and uniqueness for the Keller–Segel equations. Appl. Math. Lett. 52, 212–219 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Marrocco, A.: 2D simulation of chemotaxis bacteria aggregation. ESAIM Math. Model. Numer. Anal. 37, 617–630 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 3, 581–601 (1995)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Nakaguchi, E., Yagi, Y.: Fully discrete approximation by Galerkin Runge–Kutta methods for quasilinear parabolic systems. Hokkaido Math. J. 31, 385–429 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Patlak, C.: Random walk with persistence and external bias. Bull. Math. Biophys. 15, 311–338 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    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)Google Scholar
  38. 38.
    Saito, N.: Conservative upwind finite-element method for a simplified Keller–Segel system modelling chemotaxis. IMA J. Numer. Anal. 27, 332–365 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Tyson, R., Stern, L.J., LeVeque, R.J.: Fractional step methods applied to a chemotaxis model. J. Math. Biol. 41, 455–475 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    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)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Data and Computer ScienceSun Yat-Sen UniversityGuangzhouPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsUNC CharlotteCharlotteUSA
  3. 3.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA

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