Robust numerical techniques for modeling telegraph equations in multi-scale and heterogeneous environments

The article presents an innovative concept called the hyperbolic telegraph interface model, which effectively integrates regular interfaces. This hybrid method leverages Haar wavelets in conjunction with the finite difference method to provide robust numerical solutions. It is expertly designed for both linear and nonlinear models, adeptly handling constant or variable coefficients across regular interfaces. At the heart of this technique is the approximation of spatial derivatives using truncated Haar series, while time derivatives are efficiently processed through the finite difference method. The methodology has been rigorously tested across a variety of linear and nonlinear models, demonstrating its effectiveness. In linear problems, the algebraic system is solved with precision using the Gauss elimination method. For nonlinear challenges, the Quasi-Newton linearization formula is applied to successfully eliminate non-linearity from the model. To evaluate the technique's performance, we analyze key metrics such as maximum absolute errors, root mean square errors, and computational convergence rates with varying numbers of collocation points. The proposed approach consistently outperforms existing methods, particularly in situations involving abrupt changes in the solution space or discontinuities between boundary and initial conditions, delivering stable solutions in these critical scenarios. The combination of strong theoretical foundations and computational stability, along with excellent convergence rates and comprehensive numerical studies, firmly validates the accuracy and versatility of this method, confirming its wide range of applications.