Capturing Discontinuities with Precision: A Numerical Exploration of 3D Telegraph Interface Models via Multi-Resolution Technique

This study presents a hyperbolic three-dimensional telegraph interface model with regular interfaces, numerically solved using a hybrid scheme that integrates Haar wavelets and the finite difference method. Spatial derivatives are approximated via a truncated Haar wavelet series, while temporal derivatives are discretized using the finite difference method. For linear problems, the resulting algebraic system is solved using Gauss elimination; for nonlinear problems, Newton’s quasi-linearization technique is applied. The method’s accuracy and stability are evaluated through key performance metrics, including the maximum absolute error, root mean square error, and the computational convergence rate R c ( M ) , across various collocation point configurations. The numerical results confirm the proposed method’s efficiency, robustness, and capability to resolve sharp gradients and discontinuities with high precision.