Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion

Authors:
Zhigui Lin;Ricardo Ruiz-Baier;Canrong Tian
Affiliations:
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China;Modeling and Scientific Computing, MATHICSE-SB, ícole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland;Department of Basic Sciences, Yancheng Institute of Technology, Yancheng 224003, China
Venue:
Journal of Computational Physics
Year:
2014

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Abstract

This paper is concerned with the study of pattern formation for an inhomogeneous Brusselator model with cross-diffusion, modeling an autocatalytic chemical reaction taking place in a three-dimensional domain. For the spatial discretization of the problem we develop a novel finite volume element (FVE) method associated to a piecewise linear finite element approximation of the cross-diffusion system. We study the main properties of the unique equilibrium of the related dynamical system. A rigorous linear stability analysis around the spatially homogeneous steady state is provided and we address in detail the formation of Turing patterns driven by the cross-diffusion effect. In addition we focus on the spatial accuracy of the FVE method, and a series of numerical simulations confirm the expected behavior of the solutions. In particular we show that, depending on the spatial dimension, the magnitude of the cross-diffusion influences the selection of spatial patterns.