Precise computations of chemotactic collapse using moving mesh methods. (English) Zbl 1063.65096

Summary: We consider the problem of computing blow up solutions of chemotaxis systems, or the so-called chemotactic collapse. In two spatial dimensions, such solutions can have approximate self-similar behaviour, which can be very challenging to verify in numerical simulations [cf. Betterton and Brenner, Collapsing bacterial cylinders, Phys. Rev. E 64, 061904 (2001)]. We analyse a dynamic (scale-invariant) remeshing method which performs spatial mesh movement based upon equidistribution.
Using a suitably chosen monitor function, the numerical solution resolves the fine detail in the asymptotic solution structure, such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by M. A. Herrero and J. J. L. Velázquez [Math. Ann. 306, No. 3, 583–623 (1996; Zbl 0864.35008)]. We believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected.


65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
92E20 Classical flows, reactions, etc. in chemistry


Zbl 0864.35008


Full Text: DOI


[1] Betterton, M.D.; Brenner, M.P., Collapsing bacterial cylinders, Phys. rev. E, 64, 61904, (2001)
[2] Herrero, M.A.; Velázquez, J.J.L., Singularity patterns in a chemotaxis model, Math. ann., 306, 583-623, (1996) · Zbl 0864.35008
[3] Keller, E.F.; Segel, L.A., Initiation of slime mold aggregation viewed as an instability, J. theo. biol., 6, 399-415, (1970) · Zbl 1170.92306
[4] Keller, E.F.; Segel, L.A., Model for chemotaxis, J. theo. biol., 26, 225-234, (1971) · Zbl 1170.92307
[5] Keller, E.F.; Segel, L.A., Traveling bands of chemotactic bacteria: a theoretical analysis, J. theo. biol., 30, 235-248, (1971) · Zbl 1170.92308
[6] M.P. Brenner, T.P. Witelski, On Spherically Symmetric Gravitational Collapse, preprint, 1998 · Zbl 0933.35160
[7] P. Biler, T. Nadzieja, Global and exploding solutions in a model of self-gravitating systems, preprint · Zbl 1043.85001
[8] Hanawa, T.; Nakayama, K., Stability of similarity solutions for a gravitationally contracting isothermal sphere: convergence to the larson-penston solution, The astrophys. J., 484, 238-244, (1997)
[9] I.A. Guerra, Stabilization and Blow-up for some Multidimensional Nonlinear PDEs, PhD Thesis, TU Eindhoven, 2003
[10] Levine, H.A.; Sleeman, B.D., A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. appl. math., 57, 3, 683-730, (1997) · Zbl 0874.35047
[11] Velázquez, J.J.L., Stability of some mechanisms of chemotactic aggregation, SIAM J. appl. math., 62, 5, 1581-1633, (2002) · Zbl 1013.35004
[12] Budd, C.J.; Huang, W.; Russell, R.D., Moving mesh methods for problems with blow-up, SIAM J. sci. comput., 17, 2, 305-327, (1996) · Zbl 0860.35050
[13] Huang, W.; Russell, R.D., A moving collocation method for the numerical solution of time dependent partial differential equations, Appl. numer. math., 20, 101-116, (1996) · Zbl 0859.65112
[14] Huang, W.; Ren, Y.; Russell, R.D., Moving mesh partial differential equations based on the equidistribution principle, SIAM J. numer. anal., 31, 709-730, (1994) · Zbl 0806.65092
[15] Huang, W.; Ren, Y.; Russell, R.D., Moving mesh methods based on moving mesh partial differential equations, J. comput. phys., 113, 279-290, (1994) · Zbl 0807.65101
[16] C.J. Budd, R.D. Russell, in preparation
[17] Othmer, H.G.; Stevens, A., Aggregation, blowup, and collapse: the ABC’s of t-axis in reinforced random walks, SIAM J. appl. math., 57, 4, 1044-1081, (1997) · Zbl 0990.35128
[18] Jäger, W.; Luckhaus, S., On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. amer. math. soc., 329, 819-824, (1992) · Zbl 0746.35002
[19] Nagai, T., Behavior of solutions to a parabolic-elliptic system modelling chemotaxis, J. Korean math. soc., 37, 721-733, (2000) · Zbl 0962.35026
[20] Nagai, T., Blowup of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. inequal. appl., 6, 37-55, (2001) · Zbl 0990.35024
[21] Lin, C.-S.; Ni, W.-M.; Takagi, I., Large amplitude stationary solutions to a chemotaxis system, J. diff. eq., 72, 1-27, (1988) · Zbl 0676.35030
[22] Ni, W.-M., Diffusion, cross-diffusion, and their spike layer steady states, Notices amer. math. soc., 45, 9-18, (1998) · Zbl 0917.35047
[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) · Zbl 0909.35071
[24] Herrero, M.A.; Medina, E.; Velázquez, J.J.L., Self-similar blow-up for a reaction diffusion-system, J. comput. appl. math., 97, 99-119, (1998) · Zbl 0934.35066
[25] Brenner, M.P.; Constantin, P.; Kadanoff, L.P.; Schenkel, A.; Venkataramani, S.C., Diffusion, attraction and collapse, Nonlinearity, 12, 1071-1098, (1999) · Zbl 0942.35018
[26] Petzold, L.R., A description of DASSL: a differential/algebrasystem system solver, SAND 82-8637, (1982), Sandia Labs Livermore, CA
[27] Dongarra, J.; Bunch, J.; Moler, C.; Stewart, G.W., Linpack users’ guide, (1979), SIAM Philadelphia · Zbl 0476.68025
[28] Hairer, E.; Wanner, G., Solving ordinary differential equations II: stiff and differential algebraic problems, Springer series in computational mathematics, 14, (1996), Springer-Verlag Berlin
[29] Childress, S.; Percus, J.K., Nonlinear aspects of chemotaxis, Math. biosci., 56, 217-237, (1981) · Zbl 0481.92010
[30] Childress, S., Chemotactic collapse in two dimensions, Lecture notes in biomathematics, 55, (1984), Springer-Verlag Berlin, pp. 61-66
[31] C.J. Budd, R. Carretero-González, R.D. Russell. Blow-up rings in chemotactic collapse, in preparation
[32] Beckett, G.; Mackenzie, J.A., On a uniform accurate finite difference approximation of a singularly perturbed reaction-diffusion problem using grid equidistribution, J. comput. appl. math., 131, 1-2, 381-405, (2001) · Zbl 0984.65076
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