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.

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.

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.

An Explicit-Implicit Predictor-Corrector Domain Decomposition Method for Time Dependent Multi-Dimensional Convection Diffusion Equations

The numerical solution of large scale multi-dimensional convection diffusion equations often requires efficient parallel algorithms.In this work,we consider the extension of a recently proposed non-overlapping domain decomposition method for two dimensional time dependent convection diffusion equations with variable coefficients. By combining predictor-corrector technique,modified upwind differences with explicitimplicit coupling,the method under consideration provides intrinsic parallelism while maintaining good stability and accuracy.Moreover,for multi-dimensional problems, the method can be readily implemented on a multi-processor system and does not have the limitation on the choice of subdomains required by some other similar predictor-corrector or stabilized schemes.These properties of the method are demonstrated in this work through both rigorous mathematical analysis and numerical experiments.

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.

Incremental Unknowns Method for Solving Three-Dimensional Convection-Diffusion Equations

We use the incremental unknowns method in conjunction with the iterative methods to approximate the solution of the nonsymmetric and positive-definite linear systems generated from a multilevel discretization of three-dimensional convection-diffusion equations. The condition numbers of incremental unknowns matrices associated with the convection-diffusion equations and the number of iterations needed to attain an acceptable accuracy are estimated. Numerical results are presented with two-level approximations, which demonstrate that the incremental unknowns method when combined with some iter- ative methods is very effcient.

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.

UPWIND SPLITTING SCHEME FOR CONVECTION-DIFFUSION EQUATIONS

WT5,5”BX] A new class of numerical schemes is proposed to solve convection diffusion equations by combining the upwind technique and the method of operator splitting. For every time step, the multi dimensional approximation is performed in several independent directions alternatively, while the upwind technique is applied to treat the convection term in every individual direction. This scheme possesses maximum principle. Stability and convergence are analysed by energy method.[WT5,5”HZ]

A Nearly Analytic Discrete Method for One-dimensional Unsteady Convection-dominated Diffusion Equations

In this paper, a nearly analytic discretization method for one-dimensional linear unsteady convection-dominated diffusion equations and viscous Burgers’ equation as one of the nonlinear equation is considered. In the case of linear equations, we find the local truncation error of the scheme is O(τ 2 + h4) and consider the stability analysis of the method on the basis of the classical von Neumann’s theory. In addition, the nearly analytic discretization method for the one-dimensional viscous Burgers’ equation is also constructed. The numerical experiments are performed for several benchmark problems presented in some literatures to illustrate the theoretical results. Theoretical and numerical results show that our method is to be higher accurate and nonoscillatory and might be helpful particularly in computations for the unsteady convection-dominated diffusion problems.

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.

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.

On the Behavior of Combination High-Order Compact Approximations with Preconditioned Methods in the Diffusion-Convection Equation

In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.

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.

STREAMLINE DIFFUSION F.E.M. FOR SOBOLEV EQUATIONS WITH CONVECTION DOMINATED TERM

In this paper,a streamline diffusion F.E.M. for linear Sobolev equations with convection dominated term is given.According to the range of space time F.E mesh parameter h ,two choices for artifical diffusion parameter δ are presented,and for the corresponding computation schemes the stability and error estimates in suitable norms are estabilished.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

SINGULAR SOLUTIONS FOR A CONVECTION DIFFUSION EQUATION WITH ABSORPTION

In this paper we discuss the existence and nonexistence of singular solutions for a porous medium equations with convection and absorption terms.

THE POINTWISE ESTIMATES OF SOLUTIONS FOR A NONLINEAR CONVECTION DIFFUSION REACTION EQUATION

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal Lp,1≤ p ≤ ＋∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

Numerical Simulation of Groundwater Pollution Problems Based on Convection Diffusion Equation

The analytical solution of the convection diffusion equation is considered by two-dimensional Fourier transform and the inverse Fourier transform. To get the numerical solution, the Crank-Nicolson finite difference method is constructed, which is second-order accurate in time and space. Numerical simulation shows excellent agreement with the analytical solution. The dynamic visualization of the simulating results is realized on ArcGIS platform. This work provides a quick and intuitive decision-making basis for water resources protection, especially in dealing with water pollution emergencies.

PERTURBATION FINITE VOLUME METHOD FOR CONVECTIVE-DIFFUSION INTEGRAL EQUATION

A perturbation finite volume(PFV)method for the convective-diffusion integral equa- tion is developed in this paper.The PFV scheme is an upwind and mixed scheme using any higher-order interpolation and second-order integration approximations,with the least nodes similar to the standard three-point schemes,that is,the number of the nodes needed is equal to unity plus the face-number of the control volume.For instance,in the two-dimensional(2-D)case,only four nodes for the triangle grids and five nodes for the Cartesian grids are utilized,respectively.The PFV scheme is applied on a number of 1-D linear and nonlinear problems,2-D and 3-D flow model equations.Comparing with other standard three-point schemes,the PFV scheme has much smaller numerical diffusion than the first-order upwind scheme(UDS).Its numerical accuracies are also higher than the second-order central scheme(CDS),the power-law scheme(PLS)and QUICK scheme.