Local Petrov-Galerkin meshfree method based on radial point interpolation for the numerical solution of 2D linear hyperbolic equations with variable coefficients

Abstract

In this work, we apply the local Petrov-Galerkin method based on radial basis functions to solving the two dimensional linear hyperbolic equations with variable coefficients subject to given appropriate initial and boundary conditions. Due to the presence of variable coefficients of the differential operator, special treatment is carried out in order to apply Green's theorem and derive the variational formulation. We use the radial point interpolation method to construct shape functions and a Crank-Nicolson finite difference scheme is employed to approximate the time derivatives. The stability, convergence and error analysis of the method are also discussed and theoretically proven. Some numerical examples are presented to examine the efficiency and accuracy of the proposed method.