Numerical solution of elliptic equations by compact finite difference method

• Grant number: 16/17849-7
• Support type: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): October 01, 2016
• Effective date (End): January 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal researcher: Analice Costacurta Brandi
• Grantee: Rafael de Lima Sterza
• Home Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil

Abstract

The elliptic partial differential equations have two equations that highlight for being the best known, the Laplace equation and the Poisson equation. The first equation has several applications in fluid mechanics, electromagnetism and astronomy, since the second equation for theoretical physics, electrostatic and mechanical engineering. The resolution of an elliptic equation by numerical methods is given discretizing the derivatives by the finite difference method, for example, resulting in a linear equations system typically large and sparse, which requires iterative methods to solve them. In this context, this research project aims to study, implement, and compare the finite differences and compact finite differences methods applied to the two-dimensional Poisson equation for different auxiliary conditions. The obtained results will be compared to numerical results and analytical solutions existing in the literature, in order to analyze the convergence, and especially, the computational time spent in the numerical simulations, and it is necessary to investigate what iterative method is more suitable for the linear system resolution.