Accurate Treatment of Solution and Flux Discontinuities in 2D Time-Dependent Interface Problems via Meshless Method

Interface problems frequently arise in applications including heat conduction with composite materials, fluid flow in porous media, and diffusion–reaction processes across heterogeneous domains. This paper presents a numerical method that combines radial basis functions with a finite difference method to solve two-dimensional linear and nonlinear parabolic-type interface problems. In the proposed approach, spatial derivatives are approximated using multi-quadric radial basis functions, while the time derivative is discretized via a finite difference method. The method is applied to both linear and nonlinear problems. Gaussian elimination is used to solve the resulting algebraic equation in linear cases, while a quasi-Newton linearization technique is used to handle the nonlinear terms in nonlinear cases. Several numerical experiments are performed to assess the performance of the proposed method. The results are compared with the Haar wavelet collocation method, demonstrating the improved accuracy, computational efficiency, and straightforward implementation of the method.