Improved localized radial basis functions with fitting factor for dominated convection-diffusion differential equations

Ill-conditioning problem of Global Radial Basis Functions (GRBFs) is a fundamental limitation for approximation of differential equation using this method. Most recently, some researchers have applied Local Radial Basis Functions (LRBFs) to approximate singularly perturbed convention diffusion problems. In these kinds of problems, mostly appear boundary and (or) interior layers. Existence of these regions causes instability and oscillation solutions in numerical methods. To avoid these problems, upwind LRBFs are used. In other words, convective part and diffusion part of equations discretize through upwind LRBFs and central LRBFs method, respectively. Although this technique stabilizes the scheme but reduces the accuracy, by the way, the selection of upwind direction is more complicated. To overcome these problems, in this paper, we multiply an artificial diffusion to diffusion part and apply central scheme to discretize convective part. Since central method for these kinds of problems are unstable, adding artificial diffusion causes the central method is stabilized. This strategy significantly improve the accuracy. For verification, several numerical examples are considered and compared with other schemes. The comparisons show the superiority of the method to tradition approaches. Results of this study reveal that the proposed method can be successfully applied for singularly perturbed differential equations.