浅水方程组合型超紧致差分格式

提出一族组合型超紧致差分格式(CSCD),对CSCD的数值特性作了分析,并同其他中心型差分格式进行比较。从定性角度,得出同阶中心差分格式中,CSCD格式的截断误差系数最小的结论。从定量角度,利用Fou-rier分析方法分析了CSCD格式的分辨率,并同其他中心型差分格式比较,得出CSCD格式有较高的分辨率的结论。把10阶CSCD格式应用于KdV-Burgers方程和浅水方程的数值模拟,给出两个应用算例。数值实验表明CSCD格式不仅有理论上的高精度,而且有良好的稳定性和收敛性。     

数值求解Navier—Stokes方程的迎风紧致差分格式

从迎风紧致逼近^[1]出发,提出数值求解可压Navier-Stokes方程的一种高精度的数值方法。利用Steger-Warming的通量分裂技术^[2]将守恒型方程中的流通向量分裂成两部分,在此基础上据风向构造逼近于无粘项的三阶迎风紧致有限差分格式。对方程中的粘性部分采用通常的二阶差分逼近。所建立的差分格式被用来数值求解了三维粘性绕流问题。     

高阶紧致格式求解二维粘性不可压缩复杂流场

提出了一种求解二维不可压缩复杂流场的高精度算法．控制方程为原始变量、压力Ｐｏｉｓｓｏｎ方程提法．在任意曲线坐标下，采用四阶紧致格式求解Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程组，时间推进采用交替方向隐式（ＡＤＩ）格式，在非交错网格上用松弛法求解压力Ｐｏｉｓｓｏｎ方程．对于复杂的流场，采用了区域分解方法，并在每一时间步对各子域实施松弛迭代使之能精确地反映非定常流场．利用该算法计算了二维受驱空腔流动，弯管流动和垂直平板的突然起动问题．计算结果与实验结果和其他研究者的计算结果相比较吻合良好．对于平板起动流动，成功地模拟了流场中旋涡的生成以及Ｋａｒｍａｎ涡街的形成     

污染扩散方程高精度紧致格式及其稳定性分析

针对污染扩散方程提出了时间任意阶精度的显式格式,并对该格式的稳定性和精度进行了分析,理论结果表明:一阶精度的计算格式是传统的显格式,其稳定条件为:s≤1/2(s=D.Δt/Δx2,D为扩散系数,Δt为时间步长,Δx为空间步长),随着保留精度阶数的增加,稳定性范围也会随之增大;当保留无穷阶精度时,格式是无条件稳定的。这也就从一个侧面揭示了稳定性与时间精度之间的关系,为高性能数值计算格式的构思提供了可以借鉴的原则。数值算例的结果表明,本文格式具有一定的实用性。     

可压缩混合层流动近十年研究进展

从稳定性分析、实验研究和数值模拟3个方面回顾了过去近10年间关于可压缩混合层流动的研究.这些工作研究了可压缩性对混合层其它性质的影响,这些性质包括混合层增长率、不稳定模式、流场结构、平均速度和脉动速度、以及标志物的混合等.3种研究方法相互补充, 共同提供了认识可压缩混合层流动机制的大量信息.这些研究是湍流机理研究的重要组成部分,也是混合层流动在应用技术领域成功应用的前提.      

可压缩燃烧反应转捩混合层直接数值模拟

针对三维时间发展可压缩氢/氧非预混燃烧反应平面自由剪切混合层,采用5阶迎风/6阶对称紧致混合差分格式以及3阶显式Runge-Kutta时间推进方法,直接数值模拟了伴随燃烧产物生成和反应能量释放, 流动受扰动激发失稳并转捩的演化过程. 在转捩初期, 获得了${\it\Lambda}$涡、马蹄涡等典型的大尺度拟序结构,观察到了流动失稳后发生双马蹄涡三维对并的现象, 大尺度结构呈较好的对称性.在流动演化后期, 大尺度结构逐次破碎形成小尺度结构, 混合层进入转捩末期,呈明显的不对称性.      

基于有限差分强度折减法的略阳电厂边坡稳定性分析

将强度折减理论应用于边坡稳定性分析中,借助FLAC/SLOPE有限差分分析程序,选择弹塑性Mohr-Coulomb模型及其破坏准则,以大唐略阳电厂边坡作为工程实例,分析了该边坡的稳定性,并与传统的Bishop法、Janbu法等方法计算所得边坡稳定系数进行了对比分析。结果表明,有限差分强度折减法能更加真实地反映边坡的实际情况,求得的边坡稳定系数更接近边坡的实际稳定状态,显示出其在边坡稳定性分析中的一定优势。     

高阶Schroedinger型方程的两层高精度恒稳差分格式

众所周知，高阶Schroedinger方程在量子力学、非线性光学及流体力学中都有广泛的应用。本文对高阶Schroedinger型方程δu/δt=i(-1)^mδ2m/δx^2m(其中i=√-1，m为正整数)，利用待定系数法，构造出一个两层高精度的隐式差分格式。其截断误差阶为O((△t)^2 (Δx)^6)，比同类格式精度高2～4阶，并用Fourier分析法证明了它是绝对稳定的。最后，数值例子表明本文格式比著名的Crank-Nicolson格式精度高10^-2～10^-7，这说明我们的格式是有效的，理论分析与实际计算相吻合。     

一种求解双曲方程新的有限差分算法

本文提出一种适于求解一阶双向系统的新的差分格式。它的建立方法是：将所要求解的方程与解的空间导数所满足的微分方程同时离散化，然后再通过插值函数构成封闭的离散变量代数方程。在线性情况下的误差分析表明：该格式的幅值与位相误差均小于常用的一、二阶差分格式；当其应用于非线性气动方程求解时，基本上可以消除数值扩散与振荡这两种非正常现象。     

求解二维污染扩散方程的时空任意阶的三点紧致显格式

通过在泰勒级数展开中运用逐阶迭代的方法,推导出了空间二阶导数任意精度的三点紧致的表达式,并在半高散方程中通过二维扩散方程本身把时间导数转换为空间导数,从而推导出了时空任意阶的三点紧致显格式.数值实验表明,本文格式的精度很高,而且具有使用简单,易于编程的优点,对求解二维污染扩散方程具有很好的应用前景.     

Numerical simulations of compressible mixing layers with a discontinuous Galerkin method

Discontinuous Galerkin(DG) method is known to have several advantages for flow simulations,in particular,in fiexible accuracy management and adaptability to mesh refinement. In the present work,the DG method is developed for numerical simulations of both temporally and spatially developing mixing layers. For the temporally developing mixing layer,both the instantaneous fiow field and time evolution of momentum thickness agree very well with the previous results. Shocklets are observed at higher convective Mach numbers and the vortex paring manner is changed for high compressibility. For the spatially developing mixing layer,large-scale coherent structures and self-similar behavior for mean profiles are investigated. The instantaneous fiow field for a three-dimensional compressible mixing layer is also reported,which shows the development of largescale coherent structures in the streamwise direction. All numerical results suggest that the DG method is effective in performing accurate numerical simulations for compressible shear fiows.     

The stability of difference schemes of a higher dimensional parabolic equation

This paper proposes a new method to improve the stability condition of differencescheme of a parabolic equation.Necessary and sufficient conditions of the stability of thisnew method are given and proved.Some numerical examples show that this method hassome calculation advantages.     

Numerical solution of the incompressible Navier–Stokes equations with an upwind compact difference scheme

A new finite difference method for the discretization of the incompressible Navier–Stokes equations is presented. The scheme is constructed on a staggered‐mesh grid system. The convection terms are discretized with a fifth‐order‐accurate upwind compact difference approximation, the viscous terms are discretized with a sixth‐order symmetrical compact difference approximation, the continuity equation and the pressure gradient in the momentum equations are discretized with a fourth‐order difference approximation on a cell‐centered mesh. Time advancement uses a three‐stage Runge–Kutta method. The Poisson equation for computing the pressure is solved with preconditioning. Accuracy analysis shows that the new method has high resolving efficiency. Validation of the method by computation of Taylor's vortex array is presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Four-point explicit difference schemes for the dispersive equation

A class of three-level explicit difference schemes for the dispersive equationu_1=au_(xxx)are established These schemes have higher stability and involve four meshpoints at the middle level.Their local truncation errors are O(τ+h)and stabilityconditions are from|R|≤0.25 to|R|≤10,where|R|=|a|τ/h~3,which is muchbetter than|R|≤0.25.     

色散方程的高稳定性两层四点显格式的单点精细积分法

基于单点精细积分的思想,对色散方程Ｕｔ＝ａＵｘｘｘ构造了一类高稳定性的两层四点显式差分格式,其局部截断误差为Ｏ（τ＋ｈ）稳定性条件为│Ｒ│＝│ａτ／ｈ＾３│≤ｆ（β）,对任意正实数β为单调递增函数,它们不仅显著地改善了同类格式的稳定性条件│Ｒ│≤０．２５而且也优于众多三层多点（５点或５点以上）显格式的稳定性条件。     

超音速混合层稳定性分析及增强混合的研究

利用流动稳定性提高超音速混合层的混合效率,对于提高超音速流的高效混合是一个有效途径。研究结果表明,有展向曲率的三维混合层中,三维扰动的增长率很大,且法向的掺混能力也较强,可以有效地增强混合。对于高马赫数来流的超音速混合层,这一特性依然存在,这将有利于提高高超音速混合层的混合能力。     

色散方程的高稳定性三层五点显格式的广义单点精细积分法

基于文（１）中的单点精细积分方法,对色散方程Ｕｔ＝ａＵｘｘｘ提出了一种构造高稳定性三层五点（蛙跳）显格式的广义单点精细积分法,文中格式的局部截断误差为Ｏ（ｘ＾２＋ｈ＾２）,而稳定性条件为｜Ｒ｜≤ｇ（β）（其中ｇ对任意正实数是单调递增函数）,同时类格式中最好的。     

Kinetic theory based multi-level adaptive finite difference WENO schemes for compressible Euler equations

In this paper we proposed the kinetic framework based fifth-order adaptive finite difference WENO schemes abbreviated as WENO-AO-K schemes to solve the compressible Euler equations, which are quasi-linear hyperbolic equations that can admit discontinuous solutions like shock and contact waves. The formulation of the proposed schemes is based on the kinetic theory where one can recover the Euler equations by applying a suitable moment method strategy to the Boltzmann equation. The kinetic flux vector splitting strategy is used in WENO-AO framework, which produces the computationally expensive error and exponential functions. Thus, to reduce the computational cost, a physically more relevant peculiar velocity based splitting strategy is used, which is more efficient than the kinetic flux vector splitting. High order of accuracy in time is achieved using the third-order total variation diminishing Runge–Kutta (TVD-RK) scheme. Several one- and two-dimensional test cases are solved for the compressible Euler equations using the proposed fifth-order WENO-AO-K schemes and the results are compared with conventional WENO-AO scheme. Proposed schemes capture the complex flow features in a smooth region accurately, and discontinuity is also well resolved. Error analysis of the proposed schemes shows optimal convergence rates in various norms.     

A family of high-order accuracy explicit difference schemes with branching stability for solving 3-D parabolic partial differential equation

ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｏｆｔｅｎｍｅｅｔｔｈｅｐｒｏｂｌｅｍｏｆｓｏｌｖｉｎｇｅｑｕａｔｉｏｎｏｆｐａｒａｂｏｌｉｃｔｙｐｅｉｎｍａｎｙｆｉｅｌｄｓｓｕｃｈａｓｓｅｅｐａｇｅ ,ｄｉｆｆｕｓｉｏｎ ,ｈｅａｔｃｏｎｄｕｃｔｉｏｎａｎｄｓｏｏｎ .Ｉｎｔｈｅｃａｓｅｏｆ3＿ｄｉｍｅｎｓｉｏｎ ,ｔｈｅｍｏｄｅｌｉｓａｎｉｎｉｔｉａｌａｎｄｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍａｓｆｏｌｌｏｗｓ: ｕ ｔ = 2 ｕ ｘ2 2 ｕ ｙ2 2 ｕ ｚ2 　　　　 (0 <ｘ,ｙ,ｚ<1 ;ｔ>0 ) ,ｕ(ｘ ,ｙ,ｚ,0 ) =φ(ｘ ,ｙ ,ｚ)…     

Low-dimensional model for vortex merging in the two-dimensional temporal mixing layer

The two-dimensional temporal mixing layer shows spiraling and merging vortices and is an example of a flow problem in which, despite the complexity, the vortices as individual coherent structures can be clearly visualized. In this paper we present a method for the analysis of the data that describes the spiraling and merging of vortices. To that end we define a parameterized set of structures, the ‘phenomenological model manifold’, which approximates the apparent spatial structures. Then we let the parameters of the manifold vary in such a way that the succession of states resembles the evolving flow as well as possible. Two different model manifolds were designed, one model for which the vortices are described with Gaussian profiles, and another in which a more optimal spatial structure is used. Projection of the numerical data on these manifolds results in information about the strength, ellipticity and trajectories of the vortices. The method is also used to study the successive merging of vortices; differing from scaling arguments for an inviscid flow, the results show that the first pairwise merging evolves approximately 2.11 times faster than the second merging. Efficient procedures are described for the required extensive optimisation problems.