Comparison of a Semi-Implicit and a Fully-Implicit Time Integration Method for a Highly Degenerate Diffusion-Reaction Equation Coupled with an Ordinary Differential Equation

Thumbnail Image




Jalbert, Eric

Journal Title

Journal ISSN

Volume Title


University of Guelph


A certain class of highly degenerate diffusion equations arises when modelling biofilm growth and propagations. We focus on the cellulolytic Clostridium thermocellum and formulate a spatially explicit model coupled to an ODE for the non-diffusive growth promoting nutrient. A fully-implicit time integration method, using a fixed-point iteration at each time-step, is introduced to solve this problem with increased accuracy at the price of extra computational effort when compared against the semi-implicit method that it generalises. This is used to simulate C.thermocellum formation on cellulose sheets with the objective of (i) understanding formation patterns and (ii) understanding the effect of the spatial diffusion term. The results suggest (i) the existence of travelling wave solutions describing biofilm movement and (ii) if the spatial heterogeneity is important a two dimensional model that explicitly includes the diffusion should be used.



fully-implicit time integration method, PDE ODE coupled system, C.thermocellum simulation, travelling wave solution, spatially explicit model