| Abstract: |
This research investigates the implementation and performance of the Alternating Direction Implicit (ADI) iteration method for solving partial differential equations (PDEs) and nonlinear equation systems. The study presents a comprehensive evaluation of ADI schemes, with particular emphasis on their application to parabolic and elliptic partial differential equations, as well as their extension to nonlinear mathematical problems. The theoretical formulation of the ADI method is supported by extensive numerical experiments conducted on representative benchmark problems to evaluate its computational performance. The results indicate that ADI-based approaches significantly outperform conventional explicit numerical methods by providing faster convergence, improved numerical stability, and reduced computational cost. In addition, the modified ADI schemes developed for nonlinear equation systems demonstrate enhanced convergence characteristics, greater robustness, and improved solution reliability under varying computational conditions. Comparative analyses performed across multiple benchmark test cases confirm that the proposed ADI methods consistently achieve high solution accuracy while maintaining computational efficiency and stability. |
