Convergence of an implicit Voronoi finite volume method for reaction–diffusion problems. Issue 1 (27th July 2015)

• Record Type: Journal Article
• Authors: Fiebach, André; Glitzky, Annegret; Linke, Alexander
• Is Part Of: Numerical methods for partial differential equations. Volume 32:Issue 1(2016)
• Page Start: 141
• Page End: 174
• Publication Date: 2015-07-27
• Subjects: convergence -- finite volume method -- heterostructures -- long‐term simulation -- reaction–diffusion systems -- strong convergence; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21990
• Languages: English
• ISSNs: 0749-159X

Abstract

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 activities of the involved species as primary variables. The numerical scheme works with 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. 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 over several magnitudes of time, and obtain experimental orders of convergence for this quantity. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 141–174, 2016