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

Part of: Parabolic equations and systems Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  10 September 2018

MARYAM GHASEMI ,
STEFANIE SONNER  and
HERMANN J. EBERL
MARYAM GHASEMI
Affiliation:
Department Applied Mathematics, University of Waterloo, Waterloo Ontario, Canada email: m23ghase@uwaterloo.ca
STEFANIE SONNER
Affiliation:
Department Mathematics, Radboud University, Nijmegen, the Netherlands email: stefanie.sonner@uni-graz.at
HERMANN J. EBERL
Affiliation:
Department Mathematics and Statistics, University of Guelph, Guelph Ontario, Canada email: heberl@uoguelph.ca
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Abstract

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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.

Keywords

biofilmcross-diffusiontime-adaptivity

MSC classification

Primary: 35K65: Degenerate parabolic equations
Secondary: 65L05: Initial value problems 68U20: Simulation 92D25: Population dynamics (general)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 6: Biofilms , December 2018 , pp. 1035 - 1061
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This studies was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) with a PGS-D scholarship awarded to MG, as well as a Discovery Grant and a Research Tools and Infrastructure Grant awarded to HJE.

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