| Abstract: |
This research examines the implementation and effectiveness of the Alternating Direction Implicit (ADI) iteration method for solving partial differential equations (PDEs) and nonlinear equation systems. The study provides a detailed analysis of ADI schemes, with particular emphasis on their application to parabolic and elliptic PDEs, as well as their extension to nonlinear problems. Through extensive numerical experiments, the results demonstrate that ADI methods offer higher computational efficiency than conventional explicit techniques by achieving faster convergence while requiring fewer computational resources. Furthermore, the modified ADI schemes developed for nonlinear equations exhibit improved convergence behavior, enhanced numerical stability, and greater reliability. Performance evaluation across multiple benchmark test cases confirms the effectiveness of ADI-based approaches in terms of solution accuracy, stability, convergence speed, and computational efficiency. Overall, the findings establish ADI techniques as powerful and reliable numerical methods for solving complex mathematical models encountered in engineering, physics, and computational science. |
