THE FINITE DIFFERENCE STREAMLINE DIFFUSION METHODS FOR TIME-DEPENDENT CONVECTION-DIFFUSION EQUATIONS

In this paper, two finite difference streamline diffusion (FDSD) schemes for solving two-dimensional time-dependent convection-diffusion equations are constructed. Stability and optimal order error estimati-ions for considered schemes are derived in the norm stronger than L^2-norm.

Multidomain pseudospectral methods for nonlinear convection-diffusion equations

Multidomain pseudospectral approximations to nonlinear convection-diffusion equations are considered. The schemes are formulated with the Legendre-Galerkin method but the nonlinear term is collocated at the Legendre/Chebyshev-Gauss-Lobatto points inside each subinterval. Appropriate base functions are introduced so that the matrix of the system is sparse, and the method can be implemented efficiently and in parallel. The stability and the optimal rate of convergence of the methods are proved. Numerical results are given for both the single domain and the multidomain methods to make a comparison.

CHARACTERISTIC GALERKIN METHOD FOR CONVECTION-DIFFUSION EQUATIONS AND IMPLICIT ALGORITHM USING PRECISE INTEGRATION

This paper presents a finite element procedure for solving transient, multidimensional convection-diffusion equations. The procedure is based on the characteristic Galerkin method with an implicit algorithm using precise integration method. With the operator splitting procedure, the precise integration method is introduced to determine the material derivative in the convection-diffusion equation, consequently, the physical quantities of material points. An implicit algorithm with a combination of both the precise and the traditional numerical integration procedures in time domain in the Lagrange coordinates for the characteristic Galerkin method is formulated. The stability analysis of the algorithm shows that the unconditional stability of present implicit algorithm is enhanced as compared with that of the traditional implicit numerical integration procedure. The numerical results validate the presented method in solving convection-diffusion equations. As compared with SUPG method and explicit characteristic Galerkin method, the present method gives the results with higher accuracy and better stability.

Exponential B-Spline Solution of Convection-Diffusion Equations

We present an exponential B-spline collocation method for solving convection-diffusion equation with Dirichlet’s type boundary conditions. The method is based on the Crank-Nicolson formulation for time integration and exponential B-spline functions for space integration. Using the Von Neumann method, the proposed method is shown to be unconditionally stable. Numerical experiments have been conducted to demonstrate the accuracy of the current algorithm with relatively minimal computational effort. The results showed that use of the present approach in the simulation is very applicable for the solution of convection-diffusion equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithm is seen to be very good alternatives to existing approaches for such physical applications.

H^1 space-time discontinuous finite element method for convection-diffusion equations

An H1 space-time discontinuous Galerkin （STDG） scheme for convection- diffusion equations in one spatial dimension is constructed and analyzed. This method is formulated by combining the H1 Galerkin method and the space-time discontinuous finite element method that is discontinuous in time and continuous in space. The existence and the uniqueness of the approximate solution are proved. The convergence of the scheme is analyzed by using the techniques in the finite difference and finite element methods. An optimal a-priori error estimate in the L∞ （H1） norm is derived. The numerical exper- iments are presented to verify the theoretical results.

Two-grid method for characteristic mixed finite-element solutions of nonlinear convection-diffusion equations

A two-grid method for solving nonlinear convection-dominated diffusion equations is presented. The method use discretizations based on a characteristic mixed finite-element method and give the linearization for nonlinear systems by two steps. The error analysis shows that the two-grid scheme combined with the characteristic mixed finite-element method can decrease numerical oscillation caused by dominated convections and solve nonlinear advection-dominated diffusion problems efficiently.

An approach for choosing discretization schemes and grid size based on the convection-diffusion equation

A new approach for selecting proper discretization schemes and grid size is presented. This method is based on the convection-diffusion equation and can provide insight for the Navier-Stokes equation. The approach mainly addresses two aspects, i.e., the practical accuracy of diffusion term discretization and the behavior of high wavenum- ber disturbances. Two criteria are included in this approach. First, numerical diffusion should not affect the theoretical diffusion accuracy near the length scales of interest. This is achieved by requiring numerical diffusion to be smaller than the diffusion discretization error. Second, high wavenumber modes that are.much smaller than the length scales of interest should not be amplified. These two criteria provide a range of suitable scheme combinations for convective flux and diffusive flux and an ideal interval for grid spacing. The effects of time discretization on these criteria are briefly discussed.

A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation(PIDE)with a weakly singular kernel.Cubic trigonometric B-spline(CTBS)functions are used for interpolation in both methods.The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations.The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values.An efficient tridiagonal solver is used for the solution of the linear system obtained in the first method as well as for determination of weighting coefficients in the second method.An explicit scheme is employed as time integrator to solve the system of ODEs obtained in the second method.The methods are tested with three nonhomogeneous problems for their validation.Stability,computational efficiency and numerical convergence of the methods are analyzed.Comparison of errors in approximations produced by the present methods versus different values of discretization parameters and convection-diffusion coefficients are made.Convection and diffusion dominant cases are discussed in terms of Peclet number.The results are also compared with cubic B-spline collocation method.

A Spectral Method for Convection-Diffusion Equations

In the practical problems such as nuclear waste pollution and seawater intrusion etc., many problems are reduced to solving the convection-diffusion equation, so the research of convection-diffusion equation is of great value. In this work, a spectral method is presented for solving one and two dimensional convection-diffusion equation with source term. The finite difference method is also used to solve the convection diffusion equation. The numerical experiments show that the spectral method is more efficient than other methods for solving the convection-diffusion equation.

Comparison of Fixed Point Methods and Krylov Subspace Methods Solving Convection-Diffusion Equations

The paper first introduces two-dimensional convection-diffusion equation with boundary value condition, later uses the finite difference method to discretize the equation and analyzes positive definite, diagonally dominant and symmetric properties of the discretization matrix. Finally, the paper uses fixed point methods and Krylov subspace methods to solve the linear system and compare the convergence speed of these two methods.

DECAY RATE FOR DEGENERATE CONVECTION DIFFUSION EQUATIONS IN BOTH ONE AND SEVERAL SPACE DIMENSIONS

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.

STRONG COMPACTNESS OF APPROXIMATE SOLUTIONS TO DEGENERATE ELLIPTIC-HYPERBOLIC EQUATIONS WITH DISCONTINUOUS FLUX FUNCTION

Under a non-degeneracy condition on the nonlinearities we show that sequences of approximate entropy solutions of mixed elliptic-hyperbolic equations are strongly precompact in the general case of a Caratheodory flux vector. The proofs are based on deriving localization principles for H-measures associated to sequences of measurevalued functions. This main result implies existence of solutions to degenerate parabolic convection-diffusion equations with discontinuous flux. Moreover, it provides a framework in which one can prove convergence of various types of approximate solutions, such as those generated by the vanishing viscosity method and numerical schemes.

FINITE PROXIMATE METHOD FOR CONVECTION-DIFFUSION EQUATION

A finite proximate method was presented to solve the convection-diffusion equation in curvilinear grids. The method has characteristics of automatic upwind effect and the good stability. It was verified through exact solution and other calculation results of two-dimensional dam-break flow in a frictionless, horizontal channel. The calculation results are in good agreement with the exact solution and other calculation results, which show that the finite proximate method can be applied to solve the convection-diffusion equation directly not only in the rectangular grids, but also in the curvilinear grids.

A PERTURBATIONAL FOURTH-ORDER UPWIND FINITE DIFFERENCE SCHEME FOR THE CONVECTION-DIFFUSION EQUATION

In this study,a compact fourth-order upwind finite difference scheme for the con- vection-diffusion equation is developed,by the scheme perturbation technique and the compact second-order upwind scheme proposed by the authors.The basic fourth-order scheme,which like the classical upwind scheme is free of cell Reynolds-number limitation in terms of spurious oscil- lation and involves only immediate neighbouring nodal points,is presented for the one-dimen- sional equation,and subsequently generalized to multi-dimensional cases.Numerical examples including one-to three-dimensional model equations,with available analytical solutions,of fluid flow and a problem,with benchmark solutions,of natural convective heat transfer are given to illustrate the excellent behavior in such aspects as accuracy,resolution to‘shock wave’-and ‘boundary layer’-effects in convection dominant cases,of the present scheme.Besides,the fourth-order accuracy is specially verified using double precision arithmetic.

Global Existence and Decay Estimate of Solution to One Dimensional Convection-Diffusion Equation

We study the global existence of solution to one di- mensional convection-diffusion equation. Through constructing a Cauchy sequence in a Banach space, we get the local existence of solution to the equation, t3ased on the global bounds of the solu- tion, we extend the local one to a global one that decays in Hl space.

A MORE ACCURATE ALGORITHM OF CONVECTION-DIFFUSION EQUATIONS

Convection-diffusion equation is one of the basic differential equations of motion, describing transport process of momentum, vorticity, heat and energy. The convectional algorithm is either central or upwind difference scheme being accurate in first or second order. Many researchers have been trying to devise a more accurate and stable finite-difference scheme. Lin Qun and Lü Tao have proposed a predictor-corrector difference scheme. Basing on the higher order scheme of Poisson equation, we develop a new finite-difference scheme to solve convection-diffusion equations.

ERROR ESTIMATES OF THE FINITE ELEMENT METHOD WITH WEIGHTED BASIS FUNCTIONS FOR A SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATION

In this paper, we establish a convergence theory for a finite element method with weighted basis functions for solving singularly perturbed convection-diffusion equations. The stability of this finite element method is proved and an upper bound O（h｜lnε｜3/2） for errors in the approximate solutions in the energy norm is obtained on the triangular Bakhvalov-type mesh. Numerical results are presented to verify the stability and the convergent rate of this finite element method.

The random walking method for the steady linear convection-diffusion equation with axisymmetric disc boundary

In this article,the random walking method is used to solve the steady linear convection-diffusion equation(CDE)with disc boundary condition.The integral solution corresponding to the random walking method is deduced and the relationship between the diffusion coefficient of CDE and the intensity of the random diffusion motion is obtained.The random number generator for arbitrary axisymmetric disc boundary is deduced through the polynomial fitting and inverse transform sampling method.The proposed method is tested through two numerical cases.The results show that the random walking method can solve the steady linear CDE effectively.The influence of the parameters on the results is also studied.It is found that the error of the solution can be decreased by increasing the particle releasing rate and the total walking time.

THE COMPACT SECOND-ORDER UPWIND FINITE DIFFERENCE SCHEMES FOR THE CONVECTION-DIFFUSION EQUATIONS

The compact second-order upwind finite difference schemes free of ceil Reynolds number limitation are developed in this paper for the one-to three-dimensional steady convection- diffusion equations,using a perturbational technique applied to the classical first-order upwind schemes.The present second-order schemes take essentially the same form as those of the first- order schemes,but involve a simple modification to the diffusive coefficients.Numerical exam- ples including one-to three-dimensional model equations of fluid flow and a problem of natural convection with boundary-layer effect are given to illustrate the excellent behavior of the present schemes.

A posteriori error estimates for optimal control problems constrained by convection-diffusion equations

We propose a characteristic finite element evolutionary type convection-diffusion optimal control discretization of problems. Non- divergence-free velocity fields and bilateral inequality control constraints are handled. Then some residual type a posteriori error estimates are analyzed for the approximations of the control, the state, and the adjoint state. Based on the derived error estimators, we use them as error indicators in developing efficient multi-set adaptive meshes characteristic finite element algorithm for such optimal control problems. Finally, one numerical example is given to check the feasibility and validity of multi-set adaptive meshes refinements.