Tarcísio Arantes de Moraes Pernambuco
Resumo
O metodo Galerkin descontnuo (Gd) se constituiu ao longo dos anos em uma bem estabelecida ferramenta de soluc~ao numerica em variados modelos matematicos. Inicialmente concebido para a soluc~ao numerica de equac~oes do tipo hiperbolico, o metodo se desenvolveu para aplicac~oes em equac~oes elpticas e de convecc~ao-difus~aorea c~ao. A utilizac~ao de bases de elementos nitos descontnuas torna o metodo especialmente adaptado ao emprego em modelos matematicos cujas equac~oes apresentam descontinuidade nos par^ametros ou nas condic~oes iniciais, como ocorre em inumeros problemas da fsica matematica e da engenharia. Em raz~ao destas caractersticas o metodo Gd tem atrado muita atenc~ao na mec^anica dos uidos e na industria, como e o caso da simulac~ao numerica de escoamentos em meios porosos presentes na explora c~ao de petroleo. No calculo de reservatorios petrolferos, os quais apresentam uma complexa formac~ao geologica com variac~ao de permeabilidade aleatoria e descontínua e onde os escoamentos se estabelecem em regime ora convectivo dominante ora convectivo-difusivo e que podem ocorrer associados a efeitos reativos, o metodo Gd tem demonstrado ser bastante efetivo. Neste trabalho s~ao propostas novas formulac~oes variacionais para o tratamento numerico das equac~oes puramente hiperbolicas associadas a problemas convectivos e um tratamento numerico upwind para o termo difusivo. Estas formulac~oes se reunem
nas equac~oes de convecc~ao-difus~ao trazendo benefcios a modelac~ao numerica.
Abstract
The discontinuous Galerkin method (Gd) has constituted along the years in one well established tools for numerical solution in varied mathematical models. Initially conceived for the numerical solution of hyperbolic type equations, the method was developed for applications in elliptical equations and in convection-di usion-reaction equations. The use of discontinuous bases of nite elements makes the method specially well-adjusted for its employment in the mathematical models whose equations present discontinuity in the parameters or in the initial conditions, as it takes place in countless problems of the mathematical physics and of the engineering. On account of these characteristics the method Gd has been attracting much attention in the uid mechanics and in the industry as it is the case of the numerical simulation of the uid ow in porous medium present in the oil exploration activity. In the calculation of oil producing reservoirs, which present a complex geological formation with random and broken permeability and where the drainages are established in convective dominant regime or convective-di usive regime and where reactive e ects can take place, the method Gd has been demonstrating very good e ective being. In this work we propose new formulations for numerical solution of the purely hyperbolic equations which are associated with convective problems and a numerical upwind approach for the di usive term. These formulations are re-joined in the convection-di usion equation bringing bene ts to the numerical model.
