Convergence and stability of meshfree method based on radial basis function for a hyperbolic partial differential equation with piecewise constant arguments

Abstract

Meshfree method based on radial basis functions is a popular tool used to numerically solve partial differential equations. This paper deals with the convergence and stability of the proposed method for hyperbolic partial differential equations with piecewise constant arguments. Firstly, we apply central difference to the time variable and θ-weighted finite difference to the original equation simultaneously, then the numerical scheme is obtained by using the meshfree method based on radial basis functions to approximate the unknown function and its second-order spatial derivative. Secondly, the convergence is analysed in -norm and the sufficient conditions for asymptotic stability of the numerical solution are acquired under which the analytic solution is asymptotically stable. Finally, our study is supported by several numerical experiments.