Precise computations of chemotactic collapse using moving mesh methods

Authors:
C. J. Budd;R. Carretero-González;R. D. Russell
Affiliations:
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK;Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, USA;Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
Venue:
Journal of Computational Physics
Year:
2005

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Abstract

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 (2001) 061904]. 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 Herrero and Velazquez [Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996) 583-623]. 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.