Numerical Treatment of Hyperbolic-Type Problems with Single and Double Interfaces via Meshless Method

Hyperbolic interface problems frequently arise in a wide range of scientific and engineering applications, particularly in scenarios involving wave propagation or transport phenomena across media with discontinuous properties. These problems are characterized by abrupt changes in material coefficients or domain features, which pose significant challenges for numerical approximation. In this study, we propose an efficient and robust computational framework for solving one-dimensional hyperbolic interface problems with both single and double interfaces. The methodology combines the finite difference method (FDM) for time discretization with meshless radial basis functions (RBFs) for spatial approximation, enabling accurate resolution of interface discontinuities. This hybrid approach is adaptable to both linear and nonlinear models and is capable of handling constant as well as variable coefficients. Linear systems are solved using Gaussian elimination, while nonlinear systems are addressed through a quasi-Newton linearization method. To validate the performance of the proposed method, we compute the maximum absolute errors (MAEs) and root mean square errors (RMSEs) over various spatial and temporal discretizations. Numerical experiments demonstrate that the approach exhibits fast convergence, excellent accuracy, and ease of implementation, making it a practical tool for solving complex hyperbolic problems with interface conditions. Overall, the method provides a reliable and scalable solution for a class of problems where traditional numerical techniques often discontinuties.