A stable meshless scaled local polynomial regression method for coupled advection–diffusion–reaction systems in complex domains

Simulating two-dimensional coupled nonlinear advection–diffusion–reaction (ADR) systems in complex domains presents significant computational challenges, particularly regarding grid generation and numerical stability. While meshless approaches offer flexibility for irregular geometries, high-degree polynomial approximations often lead to ill-conditioned matrices. To address this issue, we present a meshless numerical framework based on the scaled local polynomial regression (SLPR) method. By utilizing a scaling transformation, the proposed spatial discretization mitigates the ill-conditioning problem. For temporal discretization, we apply an implicit–explicit (IMEX) scheme, combining the Crank–Nicolson method for linear operators and the second-order Adams–Bashforth (AB2) method for nonlinear terms. A stability analysis of the numerical scheme is provided. Numerical simulations of benchmark problems, including the Brusselator and predator–prey systems in complex geometries, are presented to demonstrate the performance of the proposed approach. Comparisons with several existing meshless methods reported in the literature further confirm the superior accuracy of the proposed scheme.