The localized meshless method of lines for the approximation of two-dimensional reaction-diffusion system

Abstract

Nonlinear coupled reaction-diffusion systems often arise in cooperative processes of chemical kinetics and biochemical reactions. Owing to these potential applications, this article presents an efficient and simple meshless approximation scheme to analyze the solution behavior of a two-dimensional coupled Brusselator system. On considering radial basis functions in the localized settings, meshless shape functions owing Kronecker delta function property are constructed to discretize the spatial derivatives in the time-dependent partial differential equation (PDE). A system of first-order ordinary differential equations (ODEs), obtained after spatial discretization, is then integrated in time via a high-order ODE solver. The proposed scheme’s convergence, stability, and efficiency are theoretically established and numerically verified on several benchmark problems. The outcomes verify reliability, accuracy, and simplicity of the proposed scheme against the available methods in the literature. Some recommendations are made regarding time-step size under different node distributions and RBFs.