Desenvolvimento de um modelo computacional que resolve a equação de Darcy pelo método das diferenças finitas utilizando a fórmula de extensão de 9 pontos

Abstract

In fluid dynamics and hydrology, Darcy’s law is a phenomenological constitutive equa- tion that describes the flow of a fluid through a porous medium. The equation that governs the

flow of fluids in porous media is the Laplace equation that represents an elliptic equation in partial derivatives. The finite difference method is a method of solving differential equations that is based on approximating derivatives by finite differences. The approximation formula is obtained from the Taylor series of the derived function. In traditional formulations, in 2D problems, the central node and the anterior and posterior nodes are used, both in the x and y directions, using a total of 5 nodes. In the proposal presented in this work, the central node and the 8 nodes that surround it are used, thus totaling 9 nodes. The fact of using a larger number of nodes allows to obtain a better approximation of the results. Application examples will be presented and the approximations with 5 knots will be compared with the results obtained with the extension to 9 knots and the analytical solution.