Mathematical and Computational Analysis in the Simulation of Iterative Algorithms for Solving Partial Differential Equations

Abstract

This research explores the use of iterative methods in conjunction with the Finite Difference Method (FDM) for solving partial differential equations (PDE). The central challenge addressed is the computational inefficiency and slow convergence that often arise when utilizing traditional numerical methods, particularly in large-scale systems. The study aims to develop a more efficient iterative approach to solve PDEs by minimizing computational time while ensuring the stability of the obtained solutions. The primary methods proposed include iterative solvers such as Gauss-Seidel and Successive Over-Relaxation (SOR), which are applied to numerical solutions derived from FDM. The research demonstrates that iterative methods, especially SOR, achieve faster convergence with fewer iterations compared to conventional methods like the Finite Element Method (FEM), which tends to be slower and more resource-intensive for large-scale problems. The study highlights the advantages of iterative solvers in efficiently handling large, sparse linear systems and reducing computational costs. In addition, it shows that these methods are capable of providing stable solutions, thereby maintaining accuracy with significantly lower computational effort. The results suggest that iterative methods, when applied in combination with FDM, offer a practical and scalable solution for solving complex PDEs. These methods are especially beneficial in engineering and theoretical physics applications where large-scale simulations are prevalent. The study concludes with recommendations for future research, which should focus on further optimizing solver parameters, exploring hybrid approaches, and extending the methods to more complex PDEs with non-linearities or irregular geometries. By doing so, these techniques could contribute to even more efficient and practical solutions for real-world applications.