Local RBF-based differential quadrature collocation method for the boundary layer problems

In this article, the local RBF-based differential quadrature (LRBFDQ) collocation method is presented for the boundary layer problems, i.e., the singularly perturbed two-point boundary value problems. This novel method has an advantage over the globally supported RBF collocation method because it approximates the derivatives by RBF interpolation using a small set of nodes in the neighborhood of any collocation node. So it needs much less computational work than the globally supported RBF collocation method. It also could easily use the nodes in local support domain on the upwind side to obtain the non-oscillatory solution of boundary layer problems. Numerical examples are made by the multiquadric (MQ) RBF. Compared with the globally supported RBF collocation method and the finite difference method, numerical results demonstrate the accuracy and easy implementation of the LRBFDQ collocation method, even for the extremely thin layers in the boundary layer problems.