Time Adaptive Numerical Solution of a Highly Degenerate Diffusion–Reaction Biofilm Model Based on Regularisation

Abstract

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

Time adaptive numerical solution of a highly non-linear degenerate cross-diffusion system arising in multi-species biofilm modelling

“…We have shown in Section 2 that the solution of the PDE system (2.1)-(2.3) is separated from unity i.e., the singularity in the diffusion coefficients is not attained. This, together with the simulation experiments in [9,10], suggests that using error controlled adaptive time-stepping methods for (3.26) should prevent the numerical solution from reaching or overshooting the singularity, which is a breakdown scenario for fixed time-step methods, such as the semi-implicit method [21]. Among the error-controlled time adaptive initial value problem solvers, we use embedded ROW methods.…”

“…This is in particular a problem for methods with fixed time-steps, and also for explicit time-adaptive methods because of the stiffness induced by the singularity, cf. [10] and the discussion therein for the single-species case.…”

“…In [10], using regularisation techniques, a semi-discrete approximation of the underlying single-species biofilm model was analysed, which is obtained by spatial discretisation with a Finite Volume method. It was shown that adaptive, error controlled implicit solvers for the resulting initial value problem reliably keep the numerical solution separated from the singularity, i.e., prevent its overshooting.…”

“…In order to obtain an improved numerical method for the cross-diffusion biofilm model (2.1)-(2.3), we propose to combine the spatial discretisation strategy of [21] with the time-integration method of [10]. To this end, we rewrite the model as…”

“…In [10], based on a regularisation approach, it was shown that a semi-discrete approximation of the single-species biofilm model of [6] with properties (i) and (ii) [but not (iii)] has sufficient regularity to allow the application of higher order, error controlled time adaptive schemes, such as embedded Rosenbrock-Wanner (ROW) methods. In [9], this was generalized to a multi-species biofilm model without cross-diffusion.…”