Open Access

ARTICLE

A Meshless Simulations for 2D Nonlinear Reaction-diffusion Brusselator System

by Ahmad Shirzadi¹, Vladimir Sladek², Jan Sladek³

1 Department of Mathematics, Persian Gulf University, Bushehr, Iran. Email: shirzadi@pgu.ac.ir, shirzadi.a@gmail.com
2 Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia. Email: vladimir.sladek@savba.sk
3 Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia. Email: jan.sladek@savba.sk

Computer Modeling in Engineering & Sciences 2013, 95(4), 259-282. https://doi.org/10.3970/cmes.2013.095.259

Abstract

This paper is concerned with the development of a numerical approach based on the Meshless Local Petrov-Galerkin (MLPG) method for the approximate solutions of the two dimensional nonlinear reaction-diffusion Brusselator systems. The method uses finite differences for discretizing the time variable and the moving least squares (MLS) approximation for field variables. The application of the weak formulation with the Heaviside type test functions supported on local subdomains (around the nodes used in MLS approximation) to semi-discretized partial differential equations yields the finite-volume local weak formulation. A predictor-corrector scheme is used to handle the nonlinearity of the problem within each time step. Numerical test problems are given to verify the accuracy of the proposed method. Under particular conditions, this system exhibits Turing instability which results in a pattern forming instability. This concept is studied and a test problem is given.

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