Método de diferenças finitas para problemas de valor de contorno

Abstract

Boundary value problems arise in various fields of science and engineering, modeling everyday phenomena such as temperature distribution, structural deformation, and the behavior of electromagnetic fields. Mathematically, these problems are described by ordinary or partial differential equations, which often lack analytical solutions. This makes necessary the use of numerical methods to obtain approximate solutions. In this work, the finite difference method was chosen as the main approach. The main objective of this study is to introduce the finite difference method, focusing on its mathematical foundations and its application to solving second-order boundary value problems in one and two dimensions, subject to Dirichlet boundary conditions. Classical stationary iterative methods (Jacobi, Gauss-Seidel, and the Successive Over-relaxation method) for solving the linear systems resulting from the finite difference method are also discussed. In particular, it is shown that using second-order centered finite differences to approximate second-order derivatives ensures that the numerical solution converges with the same order of accuracy. Additionally, an analysis of the convergence of the solutions of linear systems solved by iterative methods is conducted, with a focus on the Poisson equation associated with the finite difference method. Numerical results are presented to verify and validate the discussed theory, demonstrating the effectiveness of the approach.