Precise computations of chemotactic collapse using moving mesh methods

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 Velázquez [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.