Numerical Simulation of Hyperbolic Problems with Interface Discontinuities via Multi-Resolution Collocation Method

Hyperbolic interface problems are widely applied to model wave propagation and shock transmission across discontinuous media, such as acoustic waves in layered materials, seismic waves in the Earth’s crust, and stress or electromagnetic waves in composite structures. This study introduces a novel computational framework for hyperbolic interface problems, specifically designed to unify and extend the treatment of regular interfaces within partial differential equations. The proposed hybrid approach combined Haar wavelet-based spatial discretization with finite difference schemes for temporal integration. By employing truncated Haar series to approximate spatial derivatives and leveraging finite difference techniques for time evolution, the method delivers accurate solutions for both linear and nonlinear systems regardless of whether the governing coefficients are constant or spatially variable. In addressing linear problems, the resulting algebraic equations are solved efficiently using Gaussian elimination. For nonlinear formulations, the method incorporates a quasi-Newton linearization strategy, effectively transforming the system into a linear one. Extensive validation is performed through a suite of benchmark problems, with performance assessed via metrics including maximum absolute errors (MAEs), root mean square errors (RMSEs), and convergence behavior as a function of collocation point (CP) density. Numerical experiments highlight the method’s superior stability and accuracy, particularly in scenarios marked by discontinuities or sharp gradients in the solution. The approach proves especially effective in bridging inconsistencies between boundary and initial conditions, offering a robust alternative to existing techniques. Theoretical soundness, strong convergence properties, and comprehensive numerical validation collectively underscore the method’s reliability and adaptability across a broad spectrum of applications.