Comparisons of CIP,compact and CIP‐CSL2 schemes for reproducing internal solitary waves

Six different models were evaluated for reproducing internal solitary waves which occur and propagate in a stratified flow field with a sharp interface. Three stages were used to compute internal solitary waves in a stratified field: (1) first‐phase computation of momentum equations, (2) second‐phase computation of momentum equations, which corresponds to computing the Poisson's equation, and (3) density computation. The six models discussed in this paper consisted of combinations of four different schemes, a three‐point combined compact difference scheme (CCD), a normal central difference scheme (CDS), a cubic‐polynomial interpolation (CIP), and an exactly conservative semi‐Lagrangian scheme (CIP‐CSL2). The residual cutting method was used to solve the Poisson's equation. Three tests were used to confirm the validity of the computations using KdV theory; i.e. the incremental wave speed and amplitude of internal solitary waves, the maximum horizontal velocity and amplitude, and the wave form. In terms of the shape of an internal solitary wave, using CIP for momentum equations was found to provide better performance than CCD. These results suggest one of the most appropriate scheme for reproducing internal solitary waves may be one in which CIP is used for momentum equations and CCD to solve the Poisson's equation. Copyright © 2005 John Wiley & Sons, Ltd.     

A study of upstream-weighted high-order differencing for approximation to flow convection

This paper is concerned with a number of upstream-weighted second- and third-order difference schemes. Also considered are the conventional upwind and central difference schemes for comparison. It commences with a general difference equation which unifies all the given first-, second- and third-order schemes. The various schemes are evaluated through the use of the general equation. The unboundedness and accuracy of the solutions by the difference schemes are assessed via various analyses: examination of the coefficients of the difference equation, Taylor series truncation error analysis, study of the upstream connection to numerical diffusion, single-cell analysis. Finally, the difference schemes are tested on one- and two-dimensional model problems. It is shown that the high-order schemes suffer less from the problem of numerical diffusion than the first-order upwind difference scheme. However, unboundedness cannot be avoided in the solutions by these schemes. Among them the linear upwind difference scheme presents the best compromise between numerical diffusion and solution unboundedness.     

Augmented upwind numerical schemes for the groundwater transport advection‐dispersion equation with local operators

When solute transport is advection‐dominated, the advection‐dispersion equation approximates to a hyperbolic‐type partial differential equation, and finite difference and finite element numerical approximation methods become prone to artificial oscillations. The upwind scheme serves to correct these responses to produce a more realistic solution. The upwind scheme is reviewed and then applied to the advection‐dispersion equation with local operators for the first‐order upwinding numerical approximation scheme. The traditional explicit and implicit schemes, as well as the Crank‐Nicolson scheme, are developed and analyzed for numerical stability to form a comparison base. Two new numerical approximation schemes are then proposed, namely, upwind–Crank‐Nicolson scheme, where only for the advection term is applied, and weighted upwind‐downwind scheme. These newly developed schemes are analyzed for numerical stability and compared to the traditional schemes. It was found that an upwind–Crank‐Nicolson scheme is appropriate if the Crank‐Nicolson scheme is only applied to the advection term of the advection‐dispersion equation. Furthermore, the proposed explicit weighted upwind‐downwind finite difference numerical scheme is an improvement on the traditional explicit first‐order upwind scheme, whereas the implicit weighted first‐order upwind‐downwind finite difference numerical scheme is stable under all assumptions when the appropriate weighting factor (θ) is assigned.     

GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of high-order accuracy. To attain high accuracy in time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Her-mite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation (nonlinear convection- diffusion problem) and KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.     

A numerical method for KdV equation using collocation and radial basis functions

Recently, there has been an increasing interest in the study of initial boundary value problems for Korteweg–de Vries (KdV) equations. In this paper, we propose a numerical scheme to solve the third-order nonlinear KdV equation using collocation points and approximating the solution using multiquadric (MQ) radial basis function (RBF). The scheme works in a similar fashion as finite-difference methods. Numerical examples are given to confirm the good accuracy of the presented scheme.     

摄动有限差分方法研究进展

振动有限差分（ＰＦＤ）方法,既离散徽商项也离散非微商项（包括微商系数）,在微商用直接差分近似的前提下提高差分格式的精度和分辨率．ＰＦＤ方法包括局部线化微分方程的摄动精确数值解（ＰＥＮＳ）方法和摄动数值解（ＰＮＳ）方法以及考虑非线性近似的摄动高精度差分（ＰＨＤ）方法。论述了这些方法的基本思想、具体技巧、若干方程（对流扩散方程、对流扩散反应方程、双曲方程、抛物方程和ＫｄＶ方程）的ＰＥＮＳ、ＰＮＳ和ＰＨＤ格式,它们的性质及数值实验．并与有关的数值方法作了必要的比较．最后提出值得进一步研究的一些课题．      

Chebyshev finite spectral method with extended moving grids

A Chebyshev finite spectral method on non-uniform meshes is proposed. An equidistribution scheme for two types of extended moving grids is used to generate grids. One type is designed to provide better resolution for the wave surface, and the other type is for highly variable gradients. The method has high-order accuracy because of the use of the Chebyshev polynomial as the basis function. The polynomial is used to interpolate the values between the two non-uniform meshes from a previous time step to the current time step. To attain high accuracy in the time discretization, the fourth-order Adams-Bashforth-Moulton predictor and corrector scheme is used. To avoid numerical oscillations caused by the dispersion term in the Korteweg-de Vries (KdV) equation, a numerical technique on non-uniform meshes is introduced. The proposed numerical scheme is validated by the applications to the Burgers equation (nonlinear convectiondiffusion problems) and the KdV equation (single solitary and 2-solitary wave problems), where analytical solutions are available for comparisons. Numerical results agree very well with the corresponding analytical solutions in all cases.     

Solitary wave solution to Aw-Rascle viscous model of traffic flow

A traveling wave solution to the Aw-Rascle traffic flow model that includes the relaxation and diffusion terms is investigated. The model can be approximated by the well-known Kortweg-de Vries (KdV) equation. A numerical simulation is conducted by the first-order accurate Lax-Friedrichs scheme, which is known for its ability to capture the entropy solution to hyperbolic conservation laws. Periodic boundary conditions are applied to simulate a lengthy propagation, where the profile of the derived KdV solution is taken as the initial condition to observe the change of the profile. The simulation shows good agreement between the approximated KdV solution and the numerical solution.     

Phase transitions in the two-lane density difference lattice hydrodynamic model of traffic flow

In this paper, we derive the KdV equation from the two-lane lattice hydrodynamic traffic model considering density difference effect. The soliton solution is obtained from the KdV equation. Under periodical boundary, the KdV soliton of traffic flow is demonstrated by numerical simulation. The numerical simulation result is consistent with the nonlinear analytical result. Under open system, the density fluctuation of the downstream last one lattice is designed to explore the empirical congested traffic states. A phase diagram is presented which includes free traffic, moving localized cluster, triggered stop-and-go traffic, oscillating congested traffic, and homogeneous congested traffic. Finally, the spatiotemporal evolution of all the traffic states described in phase diagram are reproduced. Results suggest that the two-lane density difference hydrodynamic traffic model is suitable to describe the actual traffic.     

对流扩散方程的摄动有限体积(PFV)方法及讨论

在有限体积(FV)方法的重构近似中,引入数值摄动处理,即把界面数值通量摄动展开成网格间距的幂级数,并利用积分方程自身的性质求出幂级数的系数,同时获得高精度迎风和中心型摄动有限体积(PFV)格式.对标量输运方程给出积分近似为二阶、重构近似为二、三和四阶迎风和中心型PFV格式,这些PFV格式的结构形式及使用基点数与一阶迎风格式完全一致,迎风PFV格式满足对流有界准则;二阶和四阶中心PFV格式对网格Peclet数的任意值均为正型格式,比常用的二阶中心格式优越.用一维标量输运和方腔流动算例说明PFV格式的优良性能,并把PFV方法与性质相近的摄动有限差分(PFD)方法及相关的高精度方法作了对比分析.     

A survey of some second-order difference schemes for the steady-state convection–diffusion equation

This paper presents a survey of several finite difference schemes for the steady-state convection–diffusion equation in one and two dimensions. Most difference schemes have O(h2) truncation error. The behaviour of these schemes on a one-dimensional model problem is analysed in detail, especially for the case when convection dominates diffusion. It is concluded that none of these schemes is universally second order. One recently proposed scheme is found to yield highly inaccurate solutions for the case of practical interest, i.e. when convection dominates diffusion. Extensions to two and threedimensions are also discussed.     

随机结构静力反应概率密度演化方程的差分方法

随机结构分析的概率密度演化方法是分析随机结构静力反应的一种具有良好前景的方法。本文研究了求解随机结构静力反应概率密度演化方程的差分方法，分别探讨了单边差分格式和Lax-Wendroff格式的计算性态。二者均能满足概率相容性条件并且能够保证均值线性增长。以八层框架结构的静力随机反应为例，对两种差分格式的结果及精确解答进行了具体的比较分析。研究表明，两种差分格式均是收敛和稳定的，在不连续点处存在角点效应．单边差分格式能够保证概率非负性，而Lax-Wendroff格式具有往往更快的收敛速度。就变异系数而言，通常单边差分格式的变异系数随着区间离散数的增长而趋于稳定值，Lax-Wendroff格式则一开始就可得到恒定的值。     

Development of a class of multiple time‐stepping schemes for convection–diffusion equations in two dimensions

In this paper we present a class of semi‐discretization finite difference schemes for solving the transient convection–diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection–diffusion (CD) equation to the inhomogeneous steady convection–diffusion‐reaction (CDR) equation after using different time‐stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one‐dimensional framework. For the sake of increasing accuracy, the exact solution for the one‐dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one‐dimensional problem. Development of the proposed time‐stepping schemes is rooted in the Taylor series expansion. All higher‐order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection–diffusion‐reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

The uniformly convergent difference schemes for a singular perturbation problem of a self-adjoint ordinary differential equation

In this paper, we construct a class of difference schemes with fitted factors for a singular perturbation problem of a self-adjoint ordinary differential equation. Using a different method from [1], by analyzing the truncation errors of schemes, we give the sufficient conditions under which the solution of the difference scheme converges uniformly to the solution of the differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence, and applying them to example, obtain the numerical results.     

Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows

This study examines the effect of discretization schemes for the convection term in the constitutive equation on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a fully developed turbulent channel flow are selected as test cases, and eight different discretization schemes are considered. Among them, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much by these schemes and the corresponding flow fields are quite different from those obtained by higher-order upwind difference schemes. Among higher-order upwind difference schemes investigated in this study, a third-order compact upwind difference scheme (CUD3) with locally added AD shows stable and most accurate solutions for highly extensional flows even at relatively high Weissenberg numbers.     

New numerical schemes based on a criterion for constructing essentially stable and accurate numerical schemes for convection-dominated equations

In order to obtain stable and accurate numerical solutions for the convection-dominated steady transport equations, we propose a criterion for constructing numerical schemes for the convection term that the roots of the characteristic equation of the resulting difference equation have poles. By imposing this criterion on the difference coefficients of the convection term, we construct two numerical schemes for the convection-dominated equations. One is based on polynomial differencing and the other on locally exact differencing. The former scheme coincides with the QUICK scheme when the mesh Reynolds number (Rm) is $\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $, which is the critical value for its stability, while it approaches the second-order upwind scheme as Rm goes to infinity. Hence the former scheme interpolates a stable scheme between the QUICK scheme at Rm = $\mathop \[{\textstyle{{\rm 8} \over {\rm 3}}}\] $ and the second-order upwind scheme at Rm = ∞. Numerical solutions with the present new schemes for the one-dimensional, linear, steady convection-diffusion equations showed good results.     

Diffraction of a solitary wave by a thin wedge

The diffraction of a solitary wave by a thin wedge with vertical walls is studied when the incident solitary wave is directed along the wedge axis. The method of multiple scales is extended to this problem and reduces the task to that of solving the two-dimensional KdV equation with proper boundary and initial conditions. The finite-difference numerical procedure is carried out with the fractional step algorithm in which difference schemes are all implicit. Except the maximum run-up at the wall, the results in this paper are found to corroborate the Melville's experiments not only qualitatively but also quantitatively. The maximum run-up of our results agrees well with Funakoshi's numerical one but it is considerably larger than that in Melville's experiment. An important reason for this discrepancy is believed to be the effect of viscous boundary layer on the vertical side wall.     

Time-fractional KdV equation: formulation and?solution using variational methods

In this work, the semi-inverse method has been used to derive the Lagrangian of the Korteweg?Cde Vries (KdV) equation. Then the time operator of the Lagrangian of the KdV equation has been transformed into fractional domain in terms of the left-Riemann?CLiouville fractional differential operator. The variational of the functional of this Lagrangian leads neatly to Euler?CLagrange equation. Via Agrawal??s method, one can easily derive the time-fractional KdV equation from this Euler?CLagrange equation. Remarkably, the time-fractional term in the resulting KdV equation is obtained in Riesz fractional derivative in a direct manner. As a second step, the derived time-fractional KdV equation is solved using He??s variational-iteration method. The calculations are carried out using initial condition depends on the nonlinear and dispersion coefficients of the KdV equation. We remark that more pronounced effects and deeper insight into the formation and properties of the resulting solitary wave by additionally considering the fractional order derivative beside the nonlinearity and dispersion terms.     

Uniform convergence for the difference scheme in the conservation form of ordinary differential equation with a small parameter

In this paper, we consider a singular perturbation boundary problem for a self-adjoint ordinary differential equaiton. We construct a class of difference schemes with fitted factors, and give the sufficient conditions under which the solution of difference scheme converges uniformly to the solution of differential equation. From this we propose several specific schemes under weaker conditions, and give much higher order of uniform convergence.