Analysis of an Il’in Scheme for a System of Singularly Perturbed Convection-Diffusion Equations

In this paper, a numerical solution for a system of singularly perturbed convection-diffusion equations is studied. The system is discretized by the Il’in scheme on a uniform mesh. It is proved that the numerical scheme has first order accuracy, which is uniform with respect to the perturbation parameters. We show that the condition number of the discrete linear system obtained from applying the Il’in scheme for a system of singularly perturbed convection-diffusion equations is O(N) and the relevant coefficient matrix is well conditioned in comparison with the matrices obtained from applying upwind finite difference schemes on this problem. Numerical results confirm the theory of the method.

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.

An efficient locally one-dimensional finite-difference time-domain method based on the conformal scheme

An efficient conformal locally one-dimensional finite-difference time-domain（LOD-CFDTD） method is presented for solving two-dimensional（2D） electromagnetic（EM） scattering problems. The formulation for the 2D transverse-electric（TE） case is presented and its stability property and numerical dispersion relationship are theoretically investigated. It is shown that the introduction of irregular grids will not damage the numerical stability. Instead of the staircasing approximation, the conformal scheme is only employed to model the curve boundaries, whereas the standard Yee grids are used for the remaining regions. As the irregular grids account for a very small percentage of the total space grids, the conformal scheme has little effect on the numerical dispersion. Moreover, the proposed method, which requires fewer arithmetic operations than the alternating-direction-implicit（ADI） CFDTD method, leads to a further reduction of the CPU time. With the total-field/scattered-field（TF/SF） boundary and the perfectly matched layer（PML）, the radar cross section（RCS） of two2 D structures is calculated. The numerical examples verify the accuracy and efficiency of the proposed method.

LOCAL ONE-DIMENSIONAL ASE-I SCHEME FOR 2D DIFFUSION EQUATION

A local alternating segment explicit - implicit method for the solution of 2D diffusion equations is presented in this paper .The method is unconditionally stable and has the obvious property of parallelism. Some numerical experiments show the method is not only simple but also more accurate.

Atmosphere-Ocean Coupling Schemes in a One-Dimensional Climate Model

In this paper, the coupling schemes of atmosphere-ocean climate models are discussed with one-dimensional advection equations. The convergence and stability for synchronous and asynchronous schemes are demonstrated and compared.Conclusions inferred from the analysis are given below. The synchronous scheme as well as the asynchronous-implicit scheme in this model are stable for arbitrary integrating time intervals. The asynchronous explicit scheme is unstable under certain conditions, which depend upon advection velocities and heat exchange parameters in the atmosphere and oceans. With both synchronous and asynchronous stable schemes the discrete solutions converge to their unique exact ones. Advections in the atmosphere and ocean accelerate the rate of convergence of the asynchronous-implicit scheme. It is suggusted that the asynchronous-implicit coupling scheme is a stable and efficient method for most climatic simulations.

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.

The Direct Discontinuous Galerkin Methods with Implicit-Explicit Runge-Kutta Time Marching for Linear Convection-Diffusion Problems

In this paper,a fully discrete stability analysis is carried out for the direct discontinuous Galerkin(DDG)methods coupled with Runge-Kutta-type implicit-explicit time marching,for solving one-dimensional linear convection-diffusion problems.In the spatial discretization,both the original DDG methods and the refined DDG methods with interface corrections are considered.In the time discretization,the convection term is treated explicitly and the diffusion term implicitly.By the energy method,we show that the corresponding fully discrete schemes are unconditionally stable,in the sense that the time-step is only required to be upper bounded by a constant which is independent of the mesh size h.Opti-mal error estimate is also obtained by the aid of a special global projection.Numerical experiments are given to verify the stability and accuracy of the proposed schemes.

HARTEN SOLUTION FOR ONE-DIMENSIONAL UNSTEADY EQUATION

In order to use the second-order 5-point difference scheme mentioned to compute the solution of one dimension unsteady equations of the direct reflection of the strong plane detonation wave meeting a solid wall barrier,in this paper,we technically construct the difference schemes of the boundary and sub-boundary of the problem,and deduce the auto-analogue analytic solutions of the initial value problem,and at the same time,we present a method for the singular property of the initial value problem,from which we can get a satisfactory computation result of this difficult problem.The difference scheme used in this paper to deal with the discontinuity problems of the shock wave are valuable and worth generalization.

Computational analysis for fractional characterization of coupled convection-diffusion equations arising in MHD fows

The work is devoted to the fractional characterization of time-dependent coupled convection-diffusion systems arising in magnetohydrodynamics(MHD)flows.The time derivative is expressed by means of Caputo’s fractional derivative concept,while the model is solved via the full-spectral method(FSM)and the semi-spectral scheme(SSS).The FSM is based on the operational matrices of derivatives constructed by using higher-order orthogonal polynomials and collocation techniques.The SSS is developed by discretizing the time variable,and the space domain is collocated by using equal points.A detailed comparative analysis is made through graphs for various parameters and tables with existing literature.The contour graphs are made to show the behaviors of the velocity and magnetic fields.The proposed methods are reasonably efficient in examining the behavior of convection-diffusion equations arising in MHD flows,and the concept may be extended for variable order models arising in MHD flows.

PARAMETER-UNIFORM FINITE DIFFERENCE SCHEME FOR A SYSTEM OF COUPLED SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS

A coupled system of singularly perturbed convection-diffusion equations is considered. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solutions to the system have boundary layers that overlap and interact. The structure of these layers is analyzed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh an upwind difference scheme is proved to be almost first- order accurate, uniformly in both small parameters. We present the results of numerical experiments to confirm our theoretical results.

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.

An Interpolation-Free Cell-Centered Finite Volume Scheme for 3D Anisotropic Convection-Diffusion Equations on Arbitrary Polyhedral Meshes

Most existing cell-centered finite volume schemes need to introduce auxiliary unknowns in order to maintain the second-order accuracy when the mesh is distorted or the problem is discontinuous,so interpolation algorithms of auxiliary unknowns are required.Interpolation algorithms are not only difficult to construct,but also bring extra computation.In this paper,an interpolation-free cell-centered finite volume scheme is proposed for the heterogeneous and anisotropic convectiondiffusion problems on arbitrary polyhedral meshes.We propose a new interpolationfree discretization method for diffusion term,and two new second-order upwind algorithms for convection term.Most interestingly,the scheme can be adapted to any mesh topology and can handle any discontinuity strictly.Numerical experiments show that this new scheme is robust,possesses a small stencil,and has approximately secondorder accuracy for both diffusion-dominated and convection-dominated problems.

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 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.

A Family of Relaxation Schemes for Nonlinear Convection Diffusion Problems

In this work we present a family of relaxation schemes for nonlinear convection diffusion problems,which can tackle also the cases of degenerate diffusion and of convection dominated regimes.The schemes proposed can achieve any order of accuracy,give non-oscillatory solutions even in the presence of singularities and their structure depends only weakly on the particular PDE being integrated.One and two dimensional results are shown,and a nonlinear stability estimate is given.