Simulating natural patterns via space-time accurate barycentric collocation methods for nonlinear PDEs

This paper develops a barycentric interpolation collocation method (BICM) for simulating nonlinear reaction–diffusion systems. The framework integrates barycentric spatial collocation with Fourier spectral and finite difference strategies to efficiently handle nonlinearities. Numerical experiments demonstrate that BICM achieves superior accuracy with fewer grid points compared to classical finite difference and existing collocation methods. Rigorous convergence analysis and empirical studies confirm the reliability and scalability of the scheme. Applications to two- and three-dimensional pattern formation further validate its effectiveness, establishing BICM as a powerful tool for advancing numerical solutions of nonlinear PDEs in science and engineering.