Abstracts of Posters 8-th European Conference on Mathematical ...

<strong>European</strong> <strong>Conference</strong> on Ma<strong>th</strong>ematical and Theoretical Biology 2011
Modelling bi<strong>of</strong>ilms: from gene regulation to large-scale structure and
function; Wednesday, June 29, 17:00
Hermann Eberl
University <strong>of</strong> Guelph
e-mail: heberl@uoguelph.ca
A numerical me<strong>th</strong>od for a doubly degenrate
diffusion-reaction model describing bi<strong>of</strong>ilm processes
Some bi<strong>of</strong>ilm systems and processes can be described by quasilinear parabolic equations
wi<strong>th</strong> two non-Fickian diffusion effects: (i) degeneracy <strong>of</strong> <strong>th</strong>e diffusion coefficients
for vanishing biomass density, and (ii) a super-diffusion singularity when <strong>th</strong>e
maximum biomass density is reached. Phenomenon (i) guarantees a well defined
interface between <strong>th</strong>e bi<strong>of</strong>ilm and <strong>th</strong>e surrounding aqueous phase <strong>th</strong>at moves at
finite speed, phenomenon (ii) ensures <strong>th</strong>at <strong>th</strong>e maximum biomass density is not exceeded.
In numerical simulations bo<strong>th</strong> <strong>th</strong>ese aspects are not easy to deal wi<strong>th</strong>. We
discuss a simple, yet relatively robust numerical me<strong>th</strong>od. We show <strong>th</strong>at under <strong>th</strong>is
numerical realisation <strong>th</strong>e effects <strong>of</strong> (i) and (ii) are maintained, we give a stability
result, show convergence numerically by grid refinement, and discuss <strong>th</strong>e parallel
speed-up gained on OpenMP platforms.
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