Convergence of an implicit Voronoi finite volume method for reaction-diffusion problems

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Date
2013
Volume
1827
Issue
Journal
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Publisher
Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik
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Abstract

We investigate the convergence of an implicit Voronoi finite volume method for reaction- diffusion problems including nonlinear diffusion in two space dimensions. The model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. The numerical scheme uses boundary conforming Delaunay meshes and preserves positivity and the dissipative property of the continuous system. Starting from a result on the global stability of the scheme (uniform, mesh-independent global upper and lower bounds), we prove strong convergence of the chemical activities and their gradients to a weak solution of the continuous problem. In order to illustrate the preservation of qualitative properties by the numerical scheme, we present a long-term simulation of the Michaelis-Menten-Henri system. Especially, we investigate the decay properties of the relative free energy and the evolution of the dissipation rate over several magnitudes of time, and obtain experimental orders of convergence for these quantities.

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Keywords
Reaction-diffusion systems, heterostructures, finite volume method, convergence, long-term simulation
Citation
Fiebach, A., Glitzky, A., & Linke, A. (2013). Convergence of an implicit Voronoi finite volume method for reaction-diffusion problems (Vol. 1827). Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik.
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