An automatic node-adaptive scheme applied with a RBF-collocation meshless method

Abstract

We propose an automatic node adaptive algorithm designed to be used in conjunction with the radial basis function collocation method. The proposed scheme identifies areas in the domain where more nodes are needed by taking into consideration both the sudden change in solution, and differences in the local/global collocation approximation. This proposed means of error identification is normalized to provide bounds of refinement thresholds; upper θlow, and lower θup. It then performs a node refinement or coarsening by focusing on the centroid point of each group of three neighboring nodes of the node marked. The proposed algorithm was tested on two well-known challenging PDEs models namely, the convection-dominated, and Burgers PDEs at high Reynolds number (Re ≥ 500). The results obtained reveal the effectiveness of the algorithm in terms of the accuracy, the number of nodes required, and the matrix condition number. The instability phenomenon normally encountered and acknowledged in literature was found to be reduced by the algorithm where the numerical results obtained revealed good approximation of the exact solution.