Superconvergence Analysis And Its Applications Of Finite Difference Methods On Staggered Grids

Both the marker and cell(MAC)method and the block-centered finite dif-ference method are based on the finite difference method on staggered grids.The MAC method,a class of finite volume scheme based on staggered grids,has been one of the simplest and most effective numerical schemes for solving the Stokes and Navier-Stokes equations.The MAC method was introduced by Lebedev and Daly et al.in 1960s,and has been widely used in engineering applications as evidenced being the basis of many flow packages.It has the ability to enforce the incompressibility constraint of the velocity field point-wisely.Moreover,it has been shown to locally conserve the mass,momentum and kinetic energy.The MAC method is a class of finite volume method on rectangular cells with pres-sure approximated at the cell center,the x-component of velocity approximated at the midpoint of vertical edges of the cell,and the y-component of velocity approximated at the midpoint of horizontal edges of the cell.Block-centered finite difference method,also called cell-centered finite difference method,can be thought as the lowest order Raviart-Thomas mixed element method,with proper quadrature formulation.The applications of the block-centered finite dif-ference methods enable us to approximate variables with second-order accuracy on non-uniform grids,which is the superconvergence.Also the block-centered finite difference methods can guarantee local mass conservation.Besides,the ap-plications of the block-centered finite difference methods enable us to transfer the saddle point problem to symmetric positive definite problem.The thesis aims to discuss the superconvergence analysis and its applications of finite difference methods on staggered grids.More specifically,this dissertation respectively studies MAC finite difference methods for Stokes and Navier-Stokes equations,block-centered finite difference methods for compressible wormhole propagation,nonlinear time-fractional parabolic equation,and gradient flows re-spectively.The outline of the dissertation is as follows:In chapter 1,some preliminaries are demonstrated,including time and s-pace discretizations,the continuous function space with its norm definition,and discrete inner product and norm definition.In chapter 2,we consider the stability and superconvergence of MAC scheme for Stokes equations on non-uniform grids.MAC finite difference method has been one of the simplest and most efficient numerical schemes for the Stokes and Navier-Stokes equations.Its numerical super convergence on uniform grids has been observed since 1992 by Nicolaides and his collaborators by transforming the MAC scheme into a finite volume method.The difficulty is due to the fact that there exists a few mathematical tools to solve finite volume methods.In this chapter,we first establish the LBB condition and obtain the stabil-ity for both the velocity and pressure of the MAC scheme for stationary Stokes equations on non-uniform grids.Then we analyze the superconvergence of MAC scheme for Stokes problems by constructing an auxiliary function depending on the velocity and discretizing parameters.Our results include the following parts:1.We find a discrete auxiliary velocity function depending on the exact velocity and discretizing parameters and prove that the approximate velocity of the MAC method converges to the auxiliary velocity with second order accuracy in discrete H1 norms;2.We obtain the secoind-order superconvergence for the cdifference quotient of x-component of velocity along x-direction and for the di:fference quo?tient of y-component of velocity along u-direction in discrete L2 norms;3.The di:fference quotient of x-component of velocity along y-direction and the difference quotient of y-component of velocity along x-direction converge with first-order ac-curacy on non-uniform grids,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.4.We obtain the second-order superconvergence for velocity u and pressure p in discrete L2 norms.All these convergence results are verified by numerical examples.In chapter 3,a characteristics marker and cell(C-MAC)scheme is established for the Navier-Stokes equations on non-uniform grids.Compared with the work in chapter 2,there are at least three issues in theoretical analysis of the proposed C-MAC scheme.First,to obtain superconvergence for MAC scheme of Stokes equations,an auxiliary function was introduced.But in this paper,due to the convection term,this technique does not work.The second problem is to estimate the extra error terms introduced by discretizing the material derivative,which easily reduce the accuracy.The last problem is to overcome the nonlinearity of the Navier-Stokes equations.To resolve the first issue,we introduce an auxiliary problem to preserve the accuracy.And we use the technique of summation-by-parts formulae in space and time to solve the second problem perfectly.For the last issue,the mathematical induction is applied to overcome nonlinearity of the Navier-Stokes equations.The key problem for the proof of superconvergence is how to estimate the essential supremum norm of the numerical velocity and its first derivatives.We obtain the second order superconvergence in the L2 norm for both velocity and pressure and the second order superconvergence for some terms of the H1 norm of the velocity for the C-MAC scheme on non-uniform grids.To our knowledge,there have been no paper to analyze the second order superconvergence for the velocity and pressure until now.Finally,some numerical experiments are presented to show the correctness and accuracy of the C-MAC scheme and the robustness and efficiency of the overall solution technique have been demonstrated using the lid-driven cavity model:In chapter 4,we consider the compressible wormhole model.A wormhole[1—6]is a simulation in which a worm-like hole is generated and propagated in the subsurface formations due to the injection of acids into a supercritical acid dissolution system.To enhance oil production rate,matrix acidization technique was introduced and applied widely.In this technique,we inject acid into matrix to dissolve the rocks,thus a channel called wormhole is formed.Usually such channel is established with high porosities.Though this channel,we can easily push oil and gas components in the reservoir to the surface.It is crucial to point out that the advancement of the chemical reaction front does not propagate uni-formly along the injection direction.Actually,the heterogeneity of porosity and permeability in the subsurface formations plays an significant role in promoting the non-uniformity of the chemical reaction front.All in all,the chemical reaction front tends to advance in certain directions more than other directions,thus a wormhole pattern is formed.In this chapter,we construct block-centered finite difference method for com-pressible wormhole propagation on non-uniform grids.Before analyzing the con-vergence of our block-centered finite difference method,we develop estimates of the mixed finite element method with quadrature applied to linear parabolic e-quations.Using these estimates,we obtain the superconvergence of the pressure,velocity,porosity,concentration and its flux in different discrete norms.In fact,in our proposed scheme,there are at least three issues in theoretical analysis for the compressible wormhole propagation:the first one is to estimate and bound the porosity which can change during time evolution,the second one is the fully coupling relation of multi-variables and the last one is the complication resulted from the introduced auxiliary flux variable.To resolve these issues,we intro-duce some useful lemmas and consider the coupled analysis method.Recently,block-centered finite difference method has been applied to the incompressible wormhole propagation in[7].But for compressible wormhole propagation,due to the appearance of the term ap/at in the mass conservation equation,the same technique can not be used.Compared with the error analyses in[7],the error es-timates in this paper are more complex which should be taken the time difference of approximate velocity and auxiliary flux into consideration.Moreover,stability results are proven rigorously which are not given in[7].The error estimates are deduced carefully in this paper,and we carry out some numerical experiments.The numerical results are consistent with the theoretical analysis.Besides,those examples vividly show that the block-centered finite difference method is capable of effectively simulating wormhole propagation.In chapter 5,we describe the problem of two dimension nonlinear time-fractional parabolic equation with the Neumann condition and a nonlinear re-action term.In recent years,many problems in physical science,electromag-netism,electrochemistry,diffusion and general transport theory can be solved by the fractional calculus approach,which provides attractive applications as a new modeling tool in a variety of scientific and engineering fields.In this chapter our target is to present the block-centered finite difference method to solve the nonlinear time-fractional parabolic equation with the Neumann condition and a nonlinear reaction term[8,9].And specially,second-order error estimates in spa-cial mesh-size both for pressLre and velocity in discrete Lw norms axe established on both uniform and non-uniform rectangular grids.The unconditionally stable result,which just depends on initial value and source item,is derived.Some a priori estimates of L2 norm with optimal order of convergence O(?t2-?+h2)with pressure and velocity are obtained,where ?t is the time step,h is the maximal space step.Finally,some numerical experiments using the block-centered finite difference schemes are carried out.In chapter 6,we consider the two-grid block-centered finite difference method for nonlinear time-fractional parabolic equation with the Neumann condition.The idea for two-grid scheme was taken from Xu[10].Besides,Dawson and his coworkers[11]extencded this technique to the block-centered finite difference method.It is well known that for solving nonlinear problems,two-grid method is an extremely accurate and efficient method.We can transform the nonlinear problem on the fine grid into a small nonlinear problem on the coarse grid and a linear problem on the fine grid.More specifically,we try to obtain a rough approximation on the coarse grid of size H and then use this approximation solution as the initial guess to solve a linear problem on the fine grid of size h.Recently,a two-grid block-centered finite difference method has been studied for nonlinear non-fickian flow model in[12].By constructing an auxiliary problem,we obtain that the convergence rates of velocity and pressure are both O(?t +h2 + H3).But for the nonlinear time-fractional parabolic equation,we would like to obtain(2-?)order temporal convergence rate.The same technique can not be used.Then we introduce an auxiliary function to construct a proper relation between the velocity u and the difference quotient of the pressure p(ref.[13]).Compared with the error analyses in[12],the error estimates in this paper are more complex.Besides,the term of discrete time derivative is different from that in[12]and the case n = 1 in this paper should be coped with separately to obtain(2-?)order temporal convergence rate.Moreover,stability results are proven rigorously and carefully which are not given in[12].Error estimates are established on non-uniform rectangular grids which show that the discrete L?(L2)and L2(H1)errors are 0(At2-? + h2 + H3).Finally,some numerical experiments are presented to show the efficiency of the two-grid method and verify that the convergence rates are in agreement with the theoretical analysis.Moreover,we also give the numerical examples of the nonlinear implicit scheme to illustrate the efficiency of the two-grid block-centered finite difference method.In chapter 7,we consider the scalar auxiliary variable block-centered finite difference schemes based on Crank-Nicolson(SAV/CN-BCFD)for gradient flows.Gradient flows are widely applied in mathematical models for problems in many fields of science and engineering,particularly in materials science and fluid dy-namics,see for instance,[14-17]and the references therein.There are so many physical problems taking the form of gradient flows,which are dynamics driv-en by a free energy.Therefore for algorithms design of any gradient flows,an important goal is to guarantee the energy stable property at the discrete level irrespectively of the coarseness of the discretization in time and space.In this chapter,we construct the scalar auxiliary variable block-centered fi-nite difference schemes based on Crank-Nicolson(SAV/CN-BCFD)for gradient flows.In view of the scarce of error analysis for SAV/CN-BCFD scheme,the pri-mary contribution of our paper is that we carry out the energy stability and error analyses to obtain second-order accuracy both in time and space directions in different discrete norms.Actually compared with the very recently work in[18],which established the first-order convergence and error estimates for the semi-discrete scheme of the space-continuous case,there are at least three important parts for the proposed SAV/CN-BCFD schemes.Firstly,to estimate the extra er-ror terms introduced by proposing the scalar auxiliary variable,we give some rea-sonable sufficient conditions about boundedness and continuity for the nonlinear functionals but without the Lipschitz assumption,and apply the mathematical induction to overcome nonlinearity and use the technique of summation-by-parts formulae both in space and time to solve the problem perfectly.Secondly,to an-alyze the error of the chemical potential,one must test the scheme with the error of phase function instead of the error of chemical potential and combine with the obtained inequality which is derived by the traditional argument for gradient flows,and then we have derived second-order convergence both in time and space increments.Lastly,for the constructed unconditional energy stable schemes,an adaptive time stepping strategy is implemented successfully so that the time step is only dictated by accuracy rather than by stability.To the best of the authors'knowledge,there have been no paper to analyze second-order convergence both in time and space of the SAV/CN-BCFD method for gradient flows until now.Finally,some numerical experiments of typical solutions to the Allen-Calm and Cahn-Hilliard equations are demonstrated to show the robustness and accuracy of the SAV/CN-BCFD schemes for gradient flows.