Simulación numérica para el parámetro óptimo de relajación del método sor aplicado a un esquema de diferencias finitas asociado a la ecuación de poisson

Abstract

The Numerical Analysis course carried out at ESPOCH focuses on classical methods for solving linear equations, but at the same time they are not efficient for solving large and complex systems. The lack of detailed information on iterative methods, such as SOR, limits its applicability in research and technologies; therefore, the aim of this work was to carry out a theoretical-practical documentary research on the optimal relaxation parameter of the SOR method applied to a finite difference scheme which was associated to Poisson’s equation. The methodology was based on a qualitative approach and descriptive level, focused on the bibliographic review of secondary sources. Particularly, documents concerning the optimization of the parameter in the SOR method were examined. This parameter was based on systems of equations derived from the finite difference discretization of the Poisson equation. The analyses and simulations caried out evidenced that the optimal value of the relaxation parameter of the SOR method increases as the number of subdivisions increases, thus it is important to mention that this value must guarantee the convergence of the method towards the solution of the system. Likewise, the need to increase the tolerance and the number of iterations in more complex systems to achieve accurate results is also highlighted. In conclusion, after performing a comparative analysis between the Jacobi, Gauss-Seidel and SOR iterative methods, it is established that the SOR method stands out due to its convergence speed and accuracy in the numerical simulation of the optimal relaxation parameter. The appropriate choice of this parameter is essential in the obtention of accurate solutions in large linear equation systems, improving the efficiency and accuracy of the method