Local Discontinuous Galerkin Method For Miscible Displacement Problem In Porous Media

Discontinuous Galerkin methods have been very attractive for numerical simulations in computational fluid dynamics recently.In this paper,we develop LDG(local discontinuous Galerkin)method for the two different two-dimensional coupled system in flow displacement problems in porous reservoirs by choosing the numerical fluxes reasonably,and we obtain the optimal error estimates for concentration c and velocity u,some numerical examples are presented to show the efficiency of the present scheme.The main work is that we introduce the LDG method to the coupled system of flow and transport equations in two different coupled systems,and we discuss the stability bounds and the inter-element jump terms which arise from the discontinuous nature of the numerical method,the non-linearity,and the coupling of the models,we choose the reasonable fluxes and treat the inter-element discontinuities of two independent solution variables(one from the flow equation and the other from the transport equation)at cell interfaces.Finally,some numerical experiments are shown to demonstrate the theoretical results.We also give the numerical simulations with some actual conditions and the results are according to the expectations.The main results of this paper are outlined as follows:In chapter one,the paper briefly introduces the development process of the discontinuous finite element method,the background of the problem.In chapter two,we present some preliminaries,including the basic notations,norms,projections and the characteristics of the finite element space we use throughout the paper.In chapter three,we give the LDG spatial discretization,the error equations and the details of the optimal error estimates for incompressible miscible displacement problem.In chapter four,the LDG spatial discretization,the error equations and the details of the optimal error estimates for compressible miscible displacement problem is given.In chapter five,we discuss the incompressible miscible displacement problem and the compressible miscible displacement problem by using LDG method.The theoretical results are illustrated by numerical examples.