An accurate localized meshfree collocation technique for the telegraph equation in propagation of electrical signals

Abstract

This paper presents an accurate localized meshfree collocation technique for the approximate solution of the second-order two-dimensional telegraph model. This model is an useful description of the propagation of electrical signals in a transmission line as well as wave phenomena. The proposed algorithm approximates the unknown solution in two steps. First, the discretization of time variable is accomplished by the Crank–Nicolson finite difference. Additionally, the unconditional stability and the convergence of the temporal semi-discretization approach are analysed with the help of the energy method in an appropriate Sobolev space. Second, the discretization of the spatial variable and its partial derivatives is obtained by the localized radial basis function partition of unity collocation method. The global collocation methods pose a considerable computational burden due to the calculation of the dense algebraic system. With the proposed approach, the domain is decomposed into several subdomains via a kernel approximation on every local domain. Therefore, it is possible to make the algebraic system more sparse and, consequently, to achieve a small condition number and a limited computational cost. Three numerical examples support the theoretical study and highlight the effectiveness of the method.