This thesis is focused on the physical phenomena appearing in the fabrication process steps and operation of interconnect lines. The investigated effects include diffusion and self-diffusion, field assisted diffusion, particle generation/annihilation processes and material conservation. The mathematical models of these phenomena consist of systems of nonlinear partial differential equations with specific structures. In this thesis various numerical methods for efficient and accurate solving of the model equations are presented, which enables reliable predictive simulation of the underlying physical phenomena. The simulation results are compared with the corresponding experimental results and checked on their physical soundness.