Estudo e implementação de métodos numéricos para resolução de problemas estacionários.

Abstract

This work studies the numerical solution of steady-state problems described by partial differential equations (PDEs), which can be classified as elliptic, parabolic or hyperbolic. These equations can model several physical phenomena of interest in Engineering, where steady-state problems often are modelled from elliptic PDEs. Thus, the main objective of this study was to solve elliptic PDE using a second order finite difference method (FDM) to discretize this equation. Considering a serie of assumptions, Navier-Stokes's equations have been reduced to a steady-state problem which is modelled by the elliptic PDE. This equation was discretized via second order FDM on a computational domain, resulting in a sparse linear system. At last, the content discussed is then illustrated by numerical applications. The tool used to solve the linear system obtained was the Conjugate Gradient Method (CGM), to estimate the velocity and shear stress profiles in fully developed flows on pipes of different cross-sections. Implementation was performed using C language.