A stable RBF method for nonlinear sinh-Gordon equation in two dimensions

Abstract. This paper presents a locally stabilized radial basis functions (RBF) meshless method based on the QR algorithm for finding the approximate solutions of the two dimensional nonlinear sinh-Gordon (ShG) equation. The proposed method consists of two phases. First, a semi-discrete approach with second order accuracy is performed by means of the central finite difference (FD) and θ-weighted methods. Second, a local RBF based on a FD is employed to discretize the space variables. The QR algorithm is used for numerically stable computations with RBFs for all values of the free shape parameter. Numerical example confirms the feasibility of the proposed method.