A hybrid meshless collocation approach for capturing solution and gradient jumps in 1D telegraph double-interface model with highly discontinuous coefficients

Abstract

Telegraph interface models accurately predict signal and wave behavior in layered media with multiple boundaries, providing a reliable basis for the design and analysis of electrical, communication, and geophysical systems. This article presents a numerical method for solving the one-dimensional telegraph equation with double interface conditions. The proposed scheme combines radial basis functions for spatial discretization with finite difference approximations in time, forming a unified and flexible computational framework. This hybrid approach is capable of handling both linear and nonlinear problems and can accommodate constant as well as variable coefficients. Linear algebraic systems arising from the discretization are solved using Gaussian elimination, while nonlinear problems are treated through a quasi-Newton linearization technique. The accuracy and efficiency of the method are evaluated by computing the maximum absolute error and the root mean square error for different spatial grid sizes and time step values. Numerical results confirm that the proposed scheme is easy to implement, exhibits fast convergence, and achieves high accuracy across a range of test problems. Therefore, the method provides a reliable and efficient computational tool for solving telegraph equations with interface discontinuities.