A novel local meshless collocation method with partial upwind scheme for solving convection-dominated diffusion problems

In this study, a novel upwind local backward substitution method (upwind-BSM) is developed for the simulation of two-dimensional convection-dominated diffusion problems. The high-order Pascal polynomial expansion method with a characteristic length factor is adopted to approximate the known boundary condition for transforming the original problem into a problem with a homogeneous boundary condition. Then the transformed problem is solved by the radial basis function-finite difference method, which generates a sparse interpolation matrix to improve the computation efficiency and avoid the ill-conditioned problem of the traditional global backward substitution method. A partial upwind point-taking scheme is introduced to eliminate the numerical oscillation resulting from the convection term in the governing equation. Several numerical examples are considered to illustrate the accuracy and efficiency of the newly proposed method. The results demonstrate that the proposed local collocation method is an accurate, efficient, and oscillation-free method for the two-dimensional convection-dominated diffusion problems.