Sixth Mississippi State Conference on Differential Equations and
Computational Simulations.
Electronic Journal of Differential Equations,
Conference 15 (2007), pp. 77-95.
Title: A finite difference scheme for a degenerated diffusion equation
arising in microbial ecology
Authors: Hermann J. Eberl (National Research Centre, Neuherberg, Germany)
Laurent Demaret (Univ. of Guelph, ON, Canada)
Abstract:
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.