Sixth Mississippi State Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 15 (2007), pp. 77-95.

A finite difference scheme for a degenerated diffusion equation arising in microbial ecology

Hermann J. Eberl, Laurent Demaret

A finite difference scheme is presented for a density-dependent diffusion equation that arises in the mathematical modelling of bacterial biofilms. The peculiarity of the underlying model is that it shows degeneracy as the dependent variable vanishes, as well as a singularity as the dependent variable approaches its a priori known upper bound. The first property leads to a finite speed of interface propagation if the initial data have compact support, while the second one introduces counter-acting super diffusion. This squeezing property of this model leads to steep gradients at the interface. Moving interface problems of this kind are known to be problematic for classical numerical methods and introduce non-physical and non-mathematical solutions. The proposed method is developed to address this observation. The central idea is a non-local (in time) representation of the diffusion operator. It can be shown that the proposed method is free of oscillations at the interface, that the discrete interface satisfies a discrete version of the continuous interface condition and that the effect of interface smearing is quantitatively small.

Published February 28, 2007.
Math Subject Classifications: 35K65, 65M06, 92C17.
Key Words: Finite differences; nonlinear diffusion; non-local representation; non-standard discretisation; numerical simulation; biofilm.

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Hermann J. Eberl
Dept. Mathematics and Statistics
University of Guelph, Guelph, On, Canada
Laurent Demaret
Inst. Biomathematics and Biometry
GSF - National Research Centre for Environment and Health
Neuherberg, Germany

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