A least-squares generalized finite difference method for solving nonlinear reaction–diffusion systems

Inspired by the benefits of the least-squares operation as introduced in the Kansa method (Chen and Ling, 2020), this paper introduces a least-squares framework predicated on the generalized finite difference method (GFDM). This method could be adeptly combined with a second-order semi-implicit backward differentiation formula (SBDF2), aiming to efficiently solve coupled nonlinear reaction–diffusion systems in both two-dimensional (2D) and three-dimensional (3D) spaces. The proposed Least-Squares Generalized Finite Difference Method (LS-GFDM) extends the conventional GFDM by incorporating the flexibility to collocate at arbitrary points, which brings out the advantages of both the function values approximation and the partial derivatives approximation at any given collocation point. Furthermore, this study provides an eigenvalue stability analysis of the LS-GFDM, employing the concept of variable separation. Finally, to underscore the efficacy of the LS-GFDM, several numerical examples are provided, including a benchmark test and applications to Turing pattern formations in both 2D and 3D contexts.