Estudo das propriedades de variação total para dois métodos de elementos finitos: Runge-Kutta Galerkin descontínuo e misto-híbrido

Abstract

The present work aims to study the properties Total Variation Diminishing (TVD), Total Variation Diminishing in the Means (TVDM) and Total Variation Bounded in the Means (TVBM) of a Strong-Stability-Preserving Runge-Kutta Discontinuous Galerkin (SSP-RKDG) method developed to solve differential problems with a convective nature written in the form of conservation laws. The TVD and Total Variation Bounded (TVB) properties of a mixed and hybrid finite element method built to solve convection and diffusion problems are also investigated. The motivation for this study is to verify the adequacy of the methods to the simulation of foam injection for enhanced oil recovery. The stability and convergence properties of the schemes were studied, concluding that it is necessary to implement flux limiters in the approximate solutions so that the SSPRKDG method satisfies the TVD, TVDM and/or TVBM properties. The investigation on the SSP-RKDG method was substantially developed from the works [16] and [25]. A general structure of the method is presented, going through the spatial discretization via the Discontinuous Galerkin method and the use of explicit and stable SSP Runge-Kutta methods for the treatment of the temporal variable; some demonstrations of results are made in detail in order to guarantee stability properties of the methods with and without flux limiters. For the mixed and hybrid method, the study was based on the works [55] and [30]. The analysis for this method was aimed to verify the stability property in specific cases.