MAC Methods For Several Flow Problems And Its Stability And Super-convergence Analysis

Mark and Cell(MAC),a class of finite difference methods based on the stag-gered grids,has been one of the simplest and most effective method among the many numerical methods that are available for solving the Navier-Stokes equa-tions.This point has been shared by Girault and Lopez in[35],Han and Wu in[40].The MAC scheme was introduced by Lebedev[48]and Daly et al.[21]in the 1960s.But its theoretical analysis was not carried out until 1992 by Xicolaides[60]and Nicolaides and Wu[61].The difficulty is due to the fact that only a few mathematical tools are available for finite volume methods.In 1996 Girault and Lopez[35]performed analysis of the MAC scheme by interpreting it as a mixed finite element method with special numerical quadrature.Then in 1998,Han and Wu[40]showed that the MAC scheme can be obtained from a new mixed finite element method.In 2008,Kanschat[63]showed that the MAC scheme is alge-braically equivalent to the first order local discontinuous Galerkin method with a proper quadrature.Inspired by the work of Kanschat[63],very soon Minev[58]demonstrated that the MAC scheme can be obtained by using the first order Nedelec edge element on rectangular cells.We would like to remark thar all these above papers[35,40,60,63]proved that the MAC scheme has first order convergent for both the velocity(in the H1 norm)and the pressure(in the L2 norm)on uniform rectangular meshes.But Nicolaides[60]pointed out that,uumerical results suggest that the velocity is second order convergent without any proof.In 2015,Li and Sun[50]constructed a MAC scheme for stationary Stokes equations on non-uniform grids and proved the stability of velocity.To our knowledge,[50]is the first paper to analyze the second order convergence for the velocity.But in their paper there are some questions,for example,the convergence analysis for velocity is based on the assumption that the pressure has second order convergence,and this assumption has not been proved.Then in 2017.Rui and Li[71]established the discrete Ladyzhenskaya-Babuska-Brezzi(LBB)condition to proof the stability of pressure.They obtained the second order super-convergence in L2 norm for both velocity and pressure.They also obtained the second order super-convergence for some terms of H1 norm of the velocity,and the other terms of H1 norm are second order super-convergence on uniform grids.In this paper,we will use the similar technique in[71]:Firstly,establish the discrete LBB condition to obtain the stability.Then use a discrete auxiliary velocity function to obtain the super-convergence.In the paper,we consider the MAC scheme for the Darcy-Stokes-Brinkman problems,the Stokes problems with damping and the coupled Stokes-Darcy prob-lems.We analyze the stability and convergence of the scheme.The paper is organized as follows:In chapter 1,we define some notation,including the space discretization,the discrete difference quotients,the discrete inner products and the discrete norms.In chapter 2,we consider the MAC methods for the Darcy-Stokes-Brinkman equations and analyze the stability and convergence of the method on non-uniform grids.The Darcy-Stokes-Brinkman problems.or Brinkman problems for short,will change between the Darcy flow and the Stokes flow as the pertur-bation parameter ? varies.Therefor it has advantage to use the MAC method which can be used to solve the Stokes problems to solve the Brinkman problems.In this chapter,firstly,to obtain the stability for both velocity and pressure,we establish the discrete LBB condition and transfer the scheme to the mixed finite element method.Then we introduce an auxiliary function depending on the velocity and discretizing parameters to analyze the super-convergence.We obtain that the velocity of the MAC scheme is second-order convergence to the auxiliary function in the norm ||·||?.According to the definition of the auxiliary function,we can obtain the results as follows:1.We obtain the second-order convergence in L2 norm for both velocity and pressure for the MAC scheme on the non-uniform grids;2.When ? is not approaching 0,we can obtain the second-order convergence for the difference quotient of x-component of velocity along x-direction and for the difference quotient of y-component of velocity along y-direction in discrete L2 norms;3.Furthermore,we have the difference quo-tient of x-eomponent of velocity along y-direction and the difference quotient of y-component of velocity along x-direetion converge with first-order accuracy on non-uniform grids in discrete L2 norms,and converge with second-order accuracy on uniform grids if the terms near the boundary are not included in the norms and converge with one-and a-half-order accuracy otherwise.Finally,numerical experiments are carried out to verify the theoretical results.In chapter 3,we consider the Stokes equations with damping and apply the MAC method for disereting the problems on the non-uniform grids.How to deal with the nonlinear term,called the damping term(or the Forchheimer term),is very important in this chapter.We will use the techniques in[36,70].To analyze the stability of the method,we firstly establish the discrete LBB condition and transfer the scheme to the mixed finite element method.To analyze the convergence,we introduce the auxiliary function and obtain that the velocity of the MAC scheze is second-order conwrgeizce to the auxiliary function in discrete H1 norm.According to the definit.ion of the auxiliary function,we can obtain the results as follows:1.On the non-uniform grids,we obtain the second-order convergence in L2 norm for both velocity and pressure for the MAC scheme;2.We also obtain the secold-order convergelce for the difference quotient of x-component of velocity along x-direction and for the difference quotient of y-component of velocity alolg y-direction in discrete L2 norms;3.Furthermore,the difference quotient of x-component of velocity along y-direction and the difference quotient of y-component of velocity along x-direction converge are first-order convergence on non-uniform grids in discrete L2 norms,aud second-order accuracy on uniform grids if the terms near the boundary are not induded in the norms,and one and a half order convergence otherwise.Finally,we carry out the numerical experiments to verify the theoretical results,where we use the Picard iteration to deal with the nonlinear term.In the Chapter 4,we construct the IAC scheme for the Coupled Stokes-Darcy problems on the staggered grids.As we know,the MAC scheme is one of the simplest and most effective numerical schemes for solving the Stokes equa-tions.The block-centered finite difference methods treating the Darcy and Darcy-Forchheimer flow have been proposed in[70,83]respectively,in which the second-order convergence for pressure and veloeity in L2 norms are obtained.Both the block-centered finite difference methods and MAC methods using the same stag-gered grids for velocity and pressure so the block-centered finite difference meth-ods can be thought as a kind of MAC scheme for porous media flow.But the problem is difficult to approximate because the Stokes and Darcy solutions have very different regularity properties and the tangential velocity may be discontinu-ous on the interface between the two regions.This chapter is organized as follows:Firstly,we construct the MAC scheme on non-uniform grids using the covolume integration on the interface.Then to obtain the stability for both the Stokes flows and the Darcy flows,we establish the discrete LBB condition.For analyzing the convergence of the scheme,we introduce the auxiliary function and obtain that:1.On the non-uniform grid,the velocity and the pressure of both Stokes and Darcy flows are second-order convergence in discrete L2 norms;2.For the velocity of the Stokes flow,we also obtain the second-order convergence for the difference quotient of x-component along x-direction and for the difference quotient of y-component along y-direction in discrete L2 norms;3.Furthermore,the difference quotient of x-component of velocity along y-direction and the difference quotient of y-component of velocity along x-direction converge are first-order convergence on non-uniform grids in discrete L2 norms,and second-order accuracy on uniform grids if the terms near the boundary are nor included in the norms,and one and a half order convergence otherwise.Finally,numerical experiments are also carried out to verifv the theoretical results.In the Chapter 5,we make a conclusion of the paper and look forward to the future work.