On the computational modeling of the behavior of a three-dimensional Brusselator system using a localized meshless method

Abstract

Nonlinear coupled reaction–diffusion equations form an important class of partial differential equations (PDEs) as they are central to the study of certain processes arising in chemical kinetics and biochemical reactions. The analytical solutions of such equations are difficult to establish and often require certain simplified assumptions, which demand for alternative solution procedures. Finite difference, finite element and spectral schemes are well-established methods to tackle such equations, yet they have the challenging issues of mesh generation, underlying integral evaluations, ill-conditioned dense system matrices, and are often restricted by domain geometry. This article presents an efficient and simple localized meshless approximation scheme to analyze the solution behavior of a three-dimensional reaction–diffusion system. The proposed scheme produces sparse (collocation) differentiation matrices for discretization of spatial differential operators which alleviates the problem of ill-conditioned and dense collocation matrices. The scheme is a truly meshless, background integration-free scheme and is equally effective for solving problems over non-rectangular domains with scattered data points. The convergence, stability, and positivity properties of the proposed scheme are theoretically established and numerically verified on some benchmark problems. The outcomes verify the reliability, accuracy, and simplicity of the proposed scheme in higher dimensions when compared to the available results in the literature.