An efficient localized meshless technique for approximating nonlinear sinh-Gordon equation arising in surface theory

Abstract

This paper adopts an efficient localized meshless technique for computing the solution of the nonlinear sinh-Gordon equation (NShGE). The NShGE is one useful description for many natural processes such as solid state physics, surface theory, fluid dynamics, nonlinear optics and dislocation in materials. In the proposed method, at the first step, a second-order accurate formulation is implemented to obtain the temporal discretization. At the second step, a localized collocation meshless technique based on the radial basis function partition of unity is proposed to derive the spatial discretization. A major drawback associated with global collocation techniques is the computational cost due to resulting dense algebraic system. The localized technique tackles the ill-conditioning inherent in global collocation techniques and reduces the associated computational cost. It is shown that the proposed method is stable and second-order convergent with respect to the time variable. Numerical results and comparisons illustrate the high accuracy of the proposed method.