Time adaptive numerical solution of a highly degenerate diffusion-reaction biofilm model based on regularisation

摘要

We consider a quasilinear degenerate diffusion-reaction system that describesbiofilm formation. The model exhibits two non-linear effects: a power lawdegeneracy as one of the dependent variables vanishes and a super diffusionsingularity as it approaches unity. Biologically relevant solutions arecharacterized by a moving interface and gradient blow-up there. Discretisationof the PDE in space by a standard Finite Volume scheme leads to a singularsystem of ordinary differential equations. We show that regularisation of thissystem allows the application of error controlled adaptive integrationtechniques to solve the underlying PDE. This overcomes the major limitation ofexisting methods for this type of problem which work with fixed time-steps. Weapply the resulting numerical method to study the effect of signal diffusion inthe aqueous phase on quorum sensing induction in a biofilm colony.