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.
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.
MSC:
| 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 |
Keywords:
adaptive mesh methods; numerical examples; reaction-diffusion equation; blowup solutions; chemotaxis systems; chemotactic collapseCitations:
Zbl 0864.35008References:
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