一维对流扩散反应方程的高精度数值方法

通过将原方程变换为对流扩散方程,将所得方程的对流项采用四阶组合紧致迎风格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的空间半离散格式采用四阶龙格库塔方法进行时间推进,得到了一种求解非定常对流扩散反应问题的高精度方法,其收敛阶为O(h4+τ4).经数值实验并与文献结果进行对比,表明该格式适用于对流占优问题的数值模拟,验证了格式的良好性能.     

对流占优扩散问题的并行计算

１．引言 在刻画流体运动的某些物理现象,以及研究热的传导、粒子的扩散等问题时,都会归结到求解对流扩散方程．用有限差分方法求解该方程,若采用显式方法,计算格式简单,但它们都是条件稳定的,时间步长必须取得非常小;若采用隐式方法,方法是无条件稳定的,但要解代数方程组,求解比较困难．Ｄ．Ｊ．ＥＶＡＮＳ和Ａ．Ｒ．ＡＨＭＡＤ在文［２］中提出了用显式交替方向法求解定态椭圆型方程,对Ｌａｐｌａｃｅ方程做了数值实验．本文将这个方法推广到了时间依赖的问题,而且适用于对流占优扩散问题的求解．基于二阶迎风格式［１］;本…     

TVD格式求解对流扩散方程的最优限制器研究

为探究不同通量限制器应用于TVD(Total Variation Diminishing)格式求解对流扩散方程时的适用性，基于3种典型的TVD格式与10种常用的通量限制器，分别求解了线性对流扩散方程、非线性对流扩散方程、拟线性对流扩散方程。数值结果表明，相比于MUSCL(Monotonic Upstream-centered Scheme for Conservation Laws)和MTVDLF(Modified TVDLF)格式，采用TVDLF(TVD Lax-Friedrichs)格式时，计算结果出现了较为严重的数值耗散；对MUSCL和MTVDLF格式进行具体分析发现，关于阶跃型纯对流问题，Superbee限制器的误差最小，Minmod误差最大。关于高斯型对流扩散问题，Minmod误差最大，Woodward误差最小。而关于阶跃型对流扩散问题及Burgers方程，限制器的类型对实验结果影响并不明显。     

一维非定常对流扩散方程的高阶组合紧致迎风格式

通过将对流项采用四五阶组合迎风紧致格式离散,扩散项采用四阶对称紧致格式离散之后,对得到的半离散格式在时间方向采用四阶龙格库塔方法求解,从而得到了一种求解非定常对流扩散方程问题的高精度组合紧致有限差分格式,其收敛阶为O(h~4+τ~4).经Fourier精度分析和数值验证,证实了格式的良好性能.三个数值算例包括线性常系数问题,矩形波问题和非线性问题,数值结果表明:该格式具有很高的分辨率,且适用于对高雷诺数问题的数值模拟.     

基于非等距网格高阶紧致差分格式的多重网格算法研究

本文结合非等距网格高精度紧致差分格式的优越性与多重网格方法的快速收敛性,求解二维对流扩散方程。研究结果表明,对于处理物理量在不同的空间方向呈现不同的性态特征或不同变化规律的物理问题时,用非等距网格离散的四阶紧致格式的多重网格算法和二阶中心差分格式的多重网格算法都比等距网格离散得高效。同时,在非等距网格下下,部分半粗化多重网格算法比完全粗化多重网格算法具有更高的计算效率。针对不同的松弛算子对误差残量的磨光效果比较研究表明,线松弛算子是最高效的。而且,非等距网格离散的高精度紧致格式的多重网格算法对于对流扩散问题中大网格雷诺数情形也是收敛的。     

三维对流扩散方程的多重网格算法研究

研究了三维对流扩散方程基于有限差分法的多重网格算法。差分格式采用一般网格步长下的二阶中心差分格式和四阶紧致差分格式,建立了与两种格式相适应的部分半粗化的多重网格算法,构造了相应的限制算子和插值算子,并与传统的等距网格下的完全粗化的多重网格算法进行了比较。数值研究结果表明,对于各向异性问题,一般网格步长下的部分半粗化多重网格算法比等距网格下的完全粗化多重网格算法具有个更高的精度和更好的收敛效率。     

机车车轮对流传热系数计算

对流传热系数的准确计算对研究机车车轮温度场和应力场及其疲劳寿命预测有重要意义.针对HXD2机车车轮踏面制动过程,建立车轮及其绕流流场计算模型,应用CFD方法通过仿真得到机车车轮在不同运行速度下的对流传热系数.结果表明:由于车轮自身旋转,车轮表面不同位置处的对流传热系数不同;车轮上半迎风面的对流传热系数较大,下半迎风面较小,且都大于背风面数值.计算结果为研究机车车轮对流传热、蠕滑和制动等传热过程提供参考.     

可压缩SIMPLE算法中密度的处理方法

在可压缩流的计算中,针对准确有效地得到激波间断解一直是研究中的难点.由于密度对激波的分辨率具有重要影响,当用有限体积法在同位网格上采用离散Navier-Stokes方程时,为了保证稳定性,对流质量通量的计算中密度运用一阶迎风(FUD)格式进行捅值,结果难以得到高分辨率的激波间断解.为了提高激波分辨率,改善计算精度,提出了对流性项中界面密度的计算方法,采用流质量通量与压力修正值方程源项里质量通量计算中界面密度的插值方法统一起来,都采用具有高分辨率格式.通过跨音速和超音速团弧凸包两个算例的仿真,结果表明,方法有效地克服了FUD格式过扩散性的缺陷,又保持了FUD格式良好的稳定性,显著提高了激波分辨率,因而是一种能够改善计算精度的有效方法.     

各向异性扩散方程的高精度算法

针对多介质各向异性扩散方程,本文设计了一种非结构多边形网格高精度有限体积计算格式.为了能适应网格大变形,在构造格式框架时除了用到单元中心量外还引入了节点量作为中间变量,并通过推广孪生逼近算法于各向异性扩散系数情形消除节点量,使算法回归于单元中心量计算流程.数值算例表明,该方法能较好适应大变形网格及间断系数各向异性扩散方程计算.     

广义平均值差分格式在对流—扩散方程中的应用

§1.引言 从逼近的角度看,微分方程的各种数值方法均可认为是对解函数的某种方式的逼近。当解具有大梯度时,线性逼近的效果往往不好。一般的克服办法是细分网格或采用高阶多项式插值。本文考虑从非线性逼近的角度处理微分方程大梯度问题。前几年孙家昶导出广义平均值以及一类半线性数值微分公式,并且运用这种工具解常微分方程的初边值问题,取得良好效果。本文在此基础上对于对流—扩散方程用广义平均值构造了一种自适应的差分格式,使之具有根据解的局部性态选择格式的特点,并分析了格式的截断误差和所引入参数的选取,以及格式的稳定性和保单调性条件。对于一维及二维问题的一     

False diffusion in three-dimensional flow calculations

This paper presents the derivation of the three principle components of the false or numerical diffusivities which result from the use of first-order upwind difference schemes in the calculation of steady three-dimensional flows. The effects of the false diffusion are illustrated by comparison of the first-order calculations with those with a higher-order scheme and experimental data, for the flow of a row of turbulent jets issuing at right angles to a nearly uniform cross-stream. The adverse effect of the false diffusivities in suppressing the physics of the flow is shown.     

An analysis and comparison of conservative upwind difference schemes for ideal and non-ideal gases

In a recent paper [P. Glaister, Conservative upwind difference schemes for the Euler equations, Comput. Math. Appl. 45 (2003) 1673–1682] a number of numerical schemes were presented for the Euler equations governing compressible flows of an ideal gas, the principal one of which is based on a conservative linearisation approach. This scheme was subsequently extended to encompass compressible flows of real gases where the equation of state allows for non-ideal gases [P. Glaister, Conservative upwind difference schemes for compressible flows of a real gas, Comput. Math. Appl. 48 (2004) 469–480]. These schemes use different parameter vectors in their construction and, consequently, the scheme in [P. Glaister, Conservative upwind difference schemes for compressible flows of a real gas, Comput. Math. Appl. 48 (2004) 469–480] when applied to the special case of an ideal gas is not identical to the principal ideal gas scheme in [P. Glaister, Conservative upwind difference schemes for the Euler equations, Comput. Math. Appl. 45 (2003) 1673–1682]. In this paper it is shown how these schemes are related, followed by a numerical comparison when each is applied to two standard test problems.     

A finite difference scheme for the incompressible advection-diffusion equation

A representation of continuum space by a second order grid system is proposed. A new finite difference scheme for the two-dimensional incompressible advection-diffusion equation is derived for the model. The difference scheme is flux conserving and contains no spatial truncation error with respect to the model except through approximation of the local velocity. It contains no false diffusion and preserves the sign of positive definite quantities. It is simple to program and is subject only to the usual diffusion and Courant-Friedrichs-Lewy stability conditions. It is stable at all grid mesh sizes or cell Reynolds numbers.A sample one-dimensional problem is presented and comparison made with the standard central difference and “upwind” difference schemes.     

Simulation of a simple reacting gas flow with an upwind scheme

Good results have been obtained using the Random Choice Method (RCM) in the computation of reacting gas flow problems. The RCM is an unfamiliar method and difficult to program. The question arises as to whether a simpler difference approximation can obtain as effective results with less computational difficulty. Among all difference schemes upwind methods have been proven to have excellent properties. Thus, such methods serve as models for the effectiveness of all difference schemes.A standard upwind scheme modified to include a fractional heat conduction step is used to compute solutions of one dimensional compressible fluid flow equations with a finite heat conduction coefficient. The gas is assumed to be chemically reacting and thus to deposit energy in the field. Comparison is made to the known qualitative behavior of the solutions for different ratios of the reaction rate and the heat conduction coefficient. This difference scheme is seen to compare unfavorably with the RCM.     

一阶非定常双曲问题的间断-差分流线扩散法及其误差估计

本文提出了求解一阶非定常双曲问题的一种新型有限元方法．间断-差分流线扩散法(DFDSD方法),建立了Euler型DFDSD格式,并对格式解的稳定性和收敛性进行了理论分析,最后给出了数值算例说明算法的有效性．     

A discontinuous finite difference streamline diffusion method for time-dependent hyperbolic problems

In this article, a new finite element method, discontinuous finite difference streamline diffusion method (DFDSD), is constructed and studied for first-order linear hyperbolic problems. This method combines the benefit of the discontinuous Galerkin method and the streamline diffusion finite element method. Two fully discrete DFDSD schemes (Euler DFDSD and Crank–Nicolson (CN) DFDSD) are constructed by making use of the difference discrete method for time variables and the discontinuous streamline diffusion method for space variables. The stability and optimal L² norm error estimates are established for the constructed schemes. This method makes contributions to the discontinuous methods. Finally, a numerical example is provided to show the benefit of high efficiency and simple implementation of the schemes.     

A Stability Criterion for Semi-Discrete Difference Schemes of Hyperbolic Conservation Laws on Uniform Grids

In the present study, the stability condition for semi-discrete difference schemes of hyperbolic conservation laws obtained from Fourier analysis is simplified. This stability condition can be applied only to linear difference schemes with constant coefficients implemented with periodic boundary treatment. It could often give useful results for other cases, such as schemes with variable coefficients, schemes for nonperiodic problem and nonlinear problem. However, this condition usually leads to a trigonometric inequality, which makes it not convenient to use. For explicit difference schemes on uniform grids, this trigonometric inequality can be converted to polynomial form. Furthermore, if the scheme is a high-order one, the polynomial can be factorized into a simple form. Thus, it is much easier to solve than the inequality obtained directly from Fourier analysis. For compact difference schemes and conservative schemes, similar results are obtained. Some applications of this new stability criterion are shown, including judging the stability of two schemes, proving the upstream central schemes to be stable, constructing a stable upwind dissipation relation preserving (DRP) scheme and constructing an optimized weighted essentially non-oscillatory (WENO) scheme. Since WENO schemes are nonlinear schemes, the stability analysis in the present study is performed on their underlying linear schemes. According to the numerical tests, the underlying linear scheme should be stable, otherwise the corresponding WENO scheme may display instability. These applications demonstrate that this criterion is convenient and efficient for judging the linear stability of semi-discrete difference schemes and constructing stable upwind difference schemes.     

非线性扩散方程的一种高精度差分格式

§1.引言 在计算流体力学中,Lagrange方法因其具有计算公式简单、物质界面清晰等优点被广泛采用,在Lagrange方法中,网格随流体而运动,初始网格即便具有很好的正交性,也会随着流体的不断运动而发生扭曲乃至相交,从而导致许多计算格式的精度下降,甚至使运算