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 compactsecond-order upwind scheme proposed by the authors.The basic fourth-order scheme,which likethe 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 examplesincluding one-to three-dimensional model equations,with available analytical solutions,of fluidflow and a problem,with benchmark solutions,of natural convective heat transfer are given toillustrate 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,thefourth-order accuracy is specially verified using double precision arithmetic.     

PERTURBATIONAL FINITE DIFFERENCE SCHEME OF CONVECTION-DIFFUSION EQUATION

The Perturbational Finite Difference (PFD) method is a kind of high-order-accurate compact difference method, But its idea is different from the normal compact method and the multi-nodes method. This method can get a Perturbational Exact Numerical Solution (PENS) scheme for locally linearlized Convection-Diffusion (CD) equation. The PENS scheme is similar to the Finite Analytical (FA) scheme and Exact Difference Solution (EDS) scheme, which are all exponential schemes, but PENS scheme is simpler and uses only 3, 5 and 7 nodes for 1-, 2- and 3-dimensional problems, respectively. The various approximate schemes of PENS scheme are also called Perturbational-High-order-accurate Difference (PHD) scheme. The PHD schemes can be got by expanding the exponential terms in the PENS scheme into power series of grid Renold number, and they are all upwind schemes and remain the concise structure form of first-order upwind scheme. For 1-dimensional (1-D) CD equation and 2-D incompressible Navier-Stokes equation, their PENS and PHD schemes were constituted in this paper, they all gave highly accurate results for the numerical examples of three 1-D CD equations and an incompressible 2-D flow in a square cavity.     

MODIFIED INTEGRAL-FACTOR METHOD FOR STEADY CONVECTION-DIFFUSION EQUATIONS

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A HIGH-ORDER FINITE DIFFERENCE METHOD FOR UNSTEADY CONVECTION-DIFFUSION PROBLEMS WITH SOURCE TERM

1.1NTRODUCTIONTheconvectionanddiffusionarethebasicprocessesinfluidflowandheatormasstransfer.Asafundamentalwaytoimprovethereliabilityofnumericalmedeling,muchat-tentionhasbeendrawntothehigh-accuracynumericalmethodsforsolvingtheconvection-diffusionequations(Rai,l987;RogerandKwak,l99oIYangetal-,199l;Hirsh,l975FDennisandHundson,I989;ChenandYang,199l;Chenetal.,19911Yangetal.,1991;Yangeta1.,l991),suchasthecompactdifferencemethodwithtimesavingandhighaccuracy(Hirsh,1975;DennisandHundson,1989…     

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

The compact second-order upwind finite difference schemes free of ceil Reynoldsnumber 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 upwindschemes.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 naturalconvection with boundary-layer effect are given to illustrate the excellent behavior of the presentschemes.     

ISOPARAMETRIC ELEMENT TRANSFORMATION FOR FINITE ANALYTIC METHOD

The Finite Analytic Method (FAM) is a new numerical method for solvingNavier-Stokes equations. However, on complex geometric domains this method must employ theboundary-fitted coordinate transformation, which becomes very difficult for some flow regions.In this paper, the Isoparametric Element, which has been widely used in the Finite ElementMethod, is incorporated into the FAM. The flow region is subdivided into a number of small ar-bitary quadrilaterals, and each element is transformed into a square element by the shape func-tions of Isoparametric Element. Then the partial differential equations on the square element canbe solved by the FAM. As an example, the flow in tube bundles has been calculated by thismethod.     

A NEW FINITE VOLUME METHOD FOR SOLVING SHALLOW WATER EQUATIONS

This paper presents a new numerical model using the finite volume method witharbitrary polygonal elements,It is suitable for simulating the shallow water problem with com-plex topography and boundary conditions.The calculated results show that this kind of methodhas the advantages of good stability,convenience of solving,easy programming,less,computingtime and storage,high accuracy,ect.     

实际地形下溃坝波的有限近似解

采用有限近似法5点格式离散二维浅水波方程组,建立平面二维溃坝洪水波模型。通过与平底无摩擦二维非对称部分溃坝问题其他数模的结果比较,结果表明该模型能够较好地模拟自由面的变化规律,具有较强的间断捕捉能力。进而对实际地形下的二维溃坝问题进行了计算模拟,给出了精细的数值结果,揭示了复杂的运动特性,进一步显示了有限近似法模拟溃坝洪水波间断面的形状和位置的优良性能,是求解溃坝流动的有效方法之一。     

用有限近似法求解二维溃坝波的演进

为了对水库和堤防的失事影响作出定量评价,以便合理确定水库或堤防的防洪标准以及采取恰当的避险措施。采用有限近似法5点格式离散二维浅水波方程组,建立平面二维溃坝洪水波模型。通过与平底无摩擦二维非对称部分溃坝问题其他数模结果的比较,表明该模型能够较好地模拟自由面的变化规律,具有较强的间断捕捉能力。进而模拟有障碍物水槽中的二维溃坝波绕流现象,给出了精细的数值结果,揭示了水流复杂的运动特性。研究结果表明了有限近似法在模拟溃坝洪水波间断的形状和位置方面的优良性能,是求解溃坝流动的有效方法之一。     

AN OPERATOR-SPLITTING ALGORITHM FOR THREE-DIMEN-SIONAL CONVECTION-DIFFUSION PROBLEMS

An operator-splitting algorithm for the three-dimensional convection-diffusion equa-tion is presented.The flow region is discretized into tetrahedronal elements which are fixed in time.The transport equation is split into two successive initial value problems:a pure convection problemand a pure diffusion problem.For the pure convection problem,solutions are found by the method ofcharacteristiCS.The solution algorithm involves tracing the characteristic lines backwards in time froma vertex of an element to an interior point.A cubic polynomial is used to interpolate the concentrationand its derivatives within each element.For the diffusion problem,an explicit finite element algorithmis employed.Numerical examples are given which agree well with the analytical solutions.     

CHEBYSHEV FINITE SPECTRAL METHOD FOR 2-D EXTENDED BOUSSINESQ EQUATIONS

In this article,an accurate Chebyshev finite spectral method for the 2-D extended Boussinesq equations is proposed.The method combines the advantages of both the finite difference and spectral methods.The Adams-Bashforth predictor and the fourth-order Adams-Moulton corrector are adopted for the numerical solution of the governing differential equations.An efficient wave absorption strategy is also proposed to effectively absorb waves at outgoing wave boundaries and reflected waves from the interior of the computational domain due to barriers and bottom slopes at the incident wave boundary to avoid contamination of the specified incident wave conditions.The proposed method is verified by a case where experimental data are available for comparison for both regular and irregular waves.The case is wave diffraction over a shoal reported by Vincent and Briggs.Numerical results agree very well with the corresponding experimental data.     

LATTICE BGK MODEL FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATION

1 .　INTRODUCTIONInthelastdecadeorso ,theLatticeBoltz mann (LB)methodhasemergedasanewandef fectivenumericaltechniqueofComputationalFluidDynamics (CFD) [1 3] ,especially ,theLBGKmodelhasachieved greatsuccessinsimulationsoffluidflowsandinmodeling physicsinfluids[4 9] .ButLBGKmodelshouldbeviewedasanarticicialcompressibleschemetosimulateincompressiblefluid[10 ] .Inotherwords,throughmultiscalingex pansionandChapman Enskog procedure[11] ,thecompressibleNavier Stokesequationscanberecov …     

SEMI-ADAPTIVE GRID METHOD FOR CONVECTION-DOMINATED FLOW

A simple and efficient adaptive grid method (so-called "semi-adaptive grid method" ,SAGM) is presented in the paper. An obvious speciality of this method is that all computations are carried out on the physical plane. This semi-adaptive grid method has been successfully applied to solve the problems of steady and unsteady convection-diffusion equations with convection-dominated. A various examples have been tested. The numerical results show that numerical solutions are monotonic (near boundary or shock layer). The accuracy is higher than that on the uniform meshes.     

HYBRID FINITE ANALYTIC SOLUTION FOR THREE-DIMENSIONAL TIDAL FLOW WITH SIGMA COORDINATE SYSTEM

1.　INTRODUCTIONTounderstandthecirculationandpollutanttransportinestuarineandcoastalwaters,numericalmodelshavebeenextensivelydevelopedduringthelasttwodecades.Thetransportofsalinity,heat,sedimentandwaterqualityconstituentwasdescribedbyadvectionanddiffusioninnumericalsimulations.Toconsidertheverticalvariation,inmanythree-dimensionaltransportmodels(SwansonandSpaulding1985;BlumbergandMllor1987;andSheng1987)thewell-knownσ-transformation(Philips1957)wasemployedinthepresenceofirrgularbathymtr…     

BLOCK ITERATIVE METHODS FOR LINEAR ALGEBRAIC EQUATION AND DOMAIN DECOMPOSITION METHOD FOR INCOMPRESSIBLE VISCOUS FLOW

It was proved numerically that the Domain Decomposition Method (DDM) with one layer overlapping grids is identical to the block iterative method of linear algebra equations. The results obtained using DDM could be in reasonable aggeement with the results of full-domain simulation. With the three dimensional solver developed by the authors, the flow field in a pipe was simulated using the full-domain DDM with one layer overlapping grids and with patched grids respectively. Both of the two cases led to the convergent solution. Further research shows the superiority of the DDM with one layer overlapping grids to the DDM with patched grids. A comparison between the numerical results obtained by the authors and the experimental results given by Enayet[3] shows that the numerical results are reasonable.     

PENALTY UPWIND FINITE ELEMENT METHOD FOR DIFFUSIVE CONVECTION PROBLEM IN DOUBLE－DIFFUSIVE SYSTEM

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A FINITE ELEMENT SIMULATION OF NEW YORK BIGHT TIDES FROM A WAVE EQUATION

A finite element model is used to simulate the tidal circulations in the New YorkBight.In this simulation a generalized wave continuity equation coupled with the primitive mo-mentum equations is used to produce a stable and accurate algorithm.The simulation is carriedout for 30 days to allow for a direct comparison with field measurement.The computed resultsagree well with the observed data.     

WAVE EQUATION MODEL FOR NUMERICAL SOLUTIONS OF ADVECTION－DOMINANT HEAT TRANSFER

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FINITE VOLUME METHOD FOR SIMULATION OF VISCOELASTIC FLOW THROUGH A EXPANSION CHANNEL

A finite volume method for the numerical solution of viscoelastic flows is given. The flow of a differential Upper-Convected Maxwell (UCM) fluid through an abrupt expansion has been chosen as a prototype example. The conservation and constitutive equations are solved using the Finite Volume Method (FVM) in a staggered grid with an upwind scheme for the viscoelastic stresses and a hybrid scheme for the velocities. An enhanced-in-speed pressure-correction algorithm is used and a method for handling the source term in the momentum equations is employed. Improved accuracy is achieved by a special discretization of the boundary conditions. Stable solutions are obtained for higher Weissenberg number (We), further extending the range of simulations with the FVM. Numerical results show the viscoelasticity of polymer solutions is the main factor influencing the sweep efficiency.     

NUMERICAL SOLUTION FOR TIDAL FLOW IN OPENING CHANNEL USING COMBINED MACCORMACK-FINITE ANALYSIS METHOD

A new numerical scheme for solving the tidal flow in an opening channel using the advective-diffusion shallow-water equations as the governing equations is proposed based on the combination of the MacCormack and the finite analysis methods. In the present scheme, the finite analysis method is used to discretize the momentum equation and the MacCormack technique is used to discretize the continuity equation in a single grid system. The matrix of the discretized momentum equation is characterized by predominantly main diagonal elements, which ensures favorable convergence and stability for the numerical simulation by the combined method. To verify the present method, hydraulics simulation is carried out for a section down mainstream of the Huangpu River. The computational results agree with the measured data. By use of orthogonal curvilinear coordinate system, the methods can be easily extended to the numerical simulation of the tidal flow in a tortuous channel.