OpenMP Parallelization of a Mickens Time-Integration Scheme for a Mixed-Culture Biofilm Model and Its Performance on Multi-core and Multi-processor Computers

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.

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“…Lemma 3.5. Let V (t) be the solution of (30) and Z be the solution of the linear system (33). If ρ ≤ 1 − , then V (t) ≤ Z for all t > 0.…”

Section: Spatial Discretizationmentioning

confidence: 99%

“…[49], to show that the set Lemma 3.6. Let Z be the solution of the linear system (33). For small enough and ∆x, the inequality Z < 1 holds.…”

Section: Spatial Discretizationmentioning

confidence: 99%

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

Ghasemi

Eberl

2017

J Sci Comput

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“…To solve this linear system, we use the stabilised bi-conjugate gradient method [33]. More specifically, we use a routine from the SPARSKIT library [27], that is prepared for parallel execution using OpenMP as in [19].…”

Section: Time Integration: Embedded Rosenbrock-wanner Methodsmentioning

confidence: 99%

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

2018

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We propose a numerical method for the simulation of a quasi-linear parabolic biofilm model that exhibits three non-linear diffusion effects: (i) a power law degeneracy, (ii) a super diffusion singularity and (iii) non-linear cross-diffusion. The method is based on a spatial Finite Volume discretisation in which cross-diffusion terms are formally treated as convection terms. Time-integration of the resulting semi-discretised system is carried out using an error-controlled, time-adaptive, embedded Rosenbrock–Wanner method. We compare several variants of the method and two variants of the model to investigate how details such as the choice cross-diffusion coefficients, and specific variants of the time integrator affect simulation time.

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“…This semiimplicit method was first described in [8] for the class of density-dependent diffusion-reaction systems arising in mono-species biofilm models and then extended to multi-species systems in [17]. In every time step it requires the solution of four sparse linear systems, one for each dependent variable, for which the stabilized biconjugated gradient method is used.…”

Section: Computational Preliminariesmentioning

confidence: 99%

On the well-posedness of a mathematical model of quorum-sensing in patchy biofilm communities

Sonner

Efendiev

Eberl

2011

Math. Meth. Appl. Sci.

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We analyze a system of reaction-diffusion equations that models quorum-sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium-type degeneracy and super diffusion. We prove the wellposedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations.

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OpenMP Parallelization of a Mickens Time-Integration Scheme for a Mixed-Culture Biofilm Model and Its Performance on Multi-core and Multi-processor Computers

Cited by 15 publications

References 16 publications

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

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

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

On the well-posedness of a mathematical model of quorum-sensing in patchy biofilm communities

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