A time-stepping scheme based on numerical Green’s functions for the domain Boundary Element Method: Employing a temporal integral approach

This paper develops a comprehensive domain Boundary Element Method (BEM) approach for the effective solution of two-dimensional diffusion-advection problems. The first approach utilizes finite difference methods for time stepping, providing a systematic framework for temporal analysis. The second approach, designated BEM-T, introduces a sophisticated technique by incorporating temporal weighting into the fundamental BEM-D equation. BEM-T operates under the assumptions of linear and constant temporal variations for concentration and flux, respectively, enabling the robust capture of the system's dynamic behavior. The application of a constant temporal weighting function simplifies the computational process by reducing the order of the time derivative in the domain integral and facilitates the direct incorporation of initial conditions, thereby enhancing the overall accuracy and efficiency of the solution. The effectiveness and reliability of the proposed methodologies are demonstrated through two detailed case studies. In these studies, BEM-T results are validated by comparison with corresponding analytical solutions (where available) and standard BEM-D formulations, highlighting the potential of these approaches for solving complex diffusion-advection problems.