Numerical study of local meshless collocation method in one-dimensional convection and diffusion problems

This paper proposes a local meshless collocation method based on global radial basis function (G-RBF) for one-dimensional convection-diffusion equations. The method transforms the equation into a steady-state form through time forward Euler discretization and uses local G-RBF interpolation with only three adjacent collocation points for spatial approximation. Numerical experiments have shown that this method has significant advantages in solving problems with different Péclet numbers. It not only has high accuracy, but also exhibits stable performance against numerical perturbations. It can effectively alleviate undesired numerical oscillations, notably reduce the condition number of discrete equations and cut down computational overhead, delivering a numerically reliable tool with competitive computational resource utilization for high-precision simulation of complex transport processes.