Multiquadric RBF-FD method for the convection-dominated diffusion problems base on Shishkin nodes

Abstract

In this paper, a new hybrid scheme based multiquadric radial basis function-generated finite difference (RBF-FD) method with Shishkin nodes is proposed to solve stationary convection-dominated diffusion problems, i.e., the boundary layer problems, which combines the midpoint upwind scheme on the coarse grid with a standard central scheme on the fine grid. Numerical results demonstrate the hybrid scheme with Shishkin nodes has an absolute advantage in accurate over standard upwind scheme with equidistant nodes. Moreover, by varying the shape parameter in multiquadric RBF-FD method, the convergence rates of numerical solutions have been further improved compared with the corresponding FD method.