Local Discontinuous Galerkin Method Based On Upwind-biased Fluxes For Incompressible Miscible Displacement Problem

In recent years,the discontinuous galerkin(DG)finite element method has been widely used in the field of computational fluid.However,it is difficult to apply DG method directly in high order differential equations.In order to solve the problem better,the DG method is extended to obtain the LDG method,which has developed into an effective discontinuous finite element method for solving high order equations numerically.In this paper,one-dimensional and two-dimensional incompressible miscible displacement problems are studied by using local discontinuous galerkin method(LDG)with upwind-biased fluxes.Optimal error estimate of concentration and velocity is obtained by selecting appropriate numerical fluxes.This paper mainly analyzes the non-continuity,non-linearity and the jump of the numerical method.By selecting numerical fluxes of the upwind-biased fluxes,the boundary items are processed to reduce and avoid the numerical dispersion and the error estimation of the energy model are obtained.This paper is as follows:Firstly,this paper introduces the local discontinuous galerkin method based on upwind-biased fluxes(LDG)development present situation,background,research significance,through the full text of basic notation,projection and its properties,norm and its properties.Secondly,in view of the one-dimensional and two-dimensional incompressible miscible displacement problems on theory analysis,the introduction of auxiliary variables will be higher order to first-order system problems,through the variational form the LDG format problem is obtained,by choosing suitable upwind-biased fluxes processing unit the gap between value and error estimate,finally obtained the optimal error estimates of the problem.Finally,the one dimensional and two dimensional incompressible miscible displacement problem,numerical simulation for discrete time layer,using the third order Runge-Kutta method.Numerical experiments show the efficiency and feasibility of the LDG and problem of the optimal error estimates are obtained.