Numerical Solution of a Degenerate, Diffusion Reaction Based Biofilm Growth Model on Structured Non-Orthogonal Grids (Communications in Computational Physics, 2018)
A previously developed semi-implicit method to solve a density dependentdiffusion-reaction biofilm growth model on uniform Cartesian grids is extended toaccommodate non-orthogonal grids in order to allow simulation on more complicateddomains. The model shows two non-linear diffusion effects: it degenerates where thedependent solution vanishes, and a super-diffusion singularity where it approaches itsupper bound. The governing equation is transformed to a general non-orthogonal ξ−ηcurvilinear coordinate system and then discretized spatially using a cell centered finitevolume method. The nonlinear biomass fluxes at the faces of the control volume cellare split into orthogonal and non-orthogonal components. The orthogonal componentis handled in a conventional manner, while the non-orthogonal component is treatedas a part of the source term. Extensive tests showed that this treatment of the non-orthogonalflux component on the control volume face works well if the maximumdeviation from orthogonality in the region of the grid where the biomass is growingis within 15-20 degrees. This range of validity is smaller than the one obtained withthe same method for the simpler porous medium equation which is the standard testproblem for degenerate diffusion equation but does not have all of the features of thebiofilm model.
Keywords: Biofilms; degenerate diffusion; non-orthogonal grids.