求解二维不可压缩Navier-Stokes方程的混合型微分求积法

微分求积法（ＤＱＭ）能以较少的网格点求得微分方程的高精度数值解,但采用单纯的微分求积法求解二维不可压缩Ｎａｖｉｅｒ＿Ｓｔｏｋｅｓ 方程时,只能对低雷诺数流动获得较好的数值解,当雷诺数较高时会导致数值解不收敛·　为此,提出了一种微分求积法与迎风差分法混合求解二维不可压缩Ｎａｖｉｅｒ＿Ｓｔｏｋｅｓ 方程的预估＿校正数值格式,用伪时间相关算法以较少的网格点获得了较高雷诺数流动的数值解·　作为算例,对１∶１ 和１∶２ 驱动方腔内的流动进行了计算,得到了较好的数值结果·     

一种抑制激波计算中数值振荡现象的双重小波收缩方法

在激波数值计算中,容易出现数值振荡的问题,振荡激烈时会掩盖真实解,为此提出了许多高精度复杂计算格式或采用人工粘性抑制数值振荡.从信号处理的角度,提出双重小波收缩方法,它能自适应提取激波数值振荡解中的真实物理解.先用局部微分求积法求解浅水波方程和理想流体Euler运动方程中的激波问题,发现其数值振荡现象严重,然后采用双重小波收缩方法对其处理,获得了无数值振荡解,它能准确捕捉激波的位置并且保持激波结构.相比于复杂的Riemann(黎曼)求解格式,借助小波收缩方法,可以采用相对简单的计算格式如微分求积法求解激波问题.     

基于第二粘性用局部微分求积法计算激波

考虑第二粘性效应,采用局部微分求积法数值求解激波问题．首先解释了在激波计算时,有必要考虑第二粘性,然后基于粘性模型,对一维和二维激波进行了数值模拟,还分别考察了剪切粘性力和第二粘性力对数值结果的影响．结果表明,采用粘性模型加上局部微分求积法能够模拟出激波特征,具有客观、简单的优点．     

基于径向基函数的微分求积区域分裂法及其应用

1 引言本文提出的基于径向基函数的微分求积区域分裂法是以径向基函数(RBFs)作为微分求积法(DQM)的基函数,并结合区域分裂法(DDM)提出的,结合了上述三种方法的优点,对解决不规则区域上的问题有很高的实用价值．     

服从OldroydB型微分模型的粘弹性流体问题的数值解法

本文对服从OldroydB型微分模型的粘弹性流体问题给出了一种数值逼近算法．该算法对压力方程采用标准混合有限元方法,对速度方程采用并行非重叠区域分解方法和特征线法．这种并行算法在子区域上用Galerkin方法,通过积分平均方法显式地给出内边界的数值流．在本文最后还给出了该算法的最优L^2。一误差估计．     

多项式基函数法

提出一种新型的数值计算方法--基函数法.此方法直接在非结构网格上离散微分算子,采用基函数展开逼近真实函数,构造出了导数的中心格式和迎风格式,取二阶多项式为基函数,并采用通量分裂法及中心格式和迎风格式相结合的技术以消除激波附近的非物理波动,构造出数值求解无粘可压缩流动二阶多项式的基函数格式,通过多个二维无粘超音速和跨音速可压缩流动典型算例的数值计算表明,该方法是一种高精度的、对激波具有高分辨率的无波动新型数值计算方法,与网格自适应技术相结合可得到十分满意的结果.     

求解三维第一类Fredholm积分方程的GMRES法

利用数值求积公式,将三维第一类Fredholm积分方程进行离散,通过引入正则化方法,将离散后的积分方程转化为一离散适定问题,通过广义极小残余算法得到了其数值解．数值模拟结果表明该方法的可行有效性．     

Navier—Stokes方程区域分解法的收敛性

０引言区域分解方法是近年来迅速发展的偏微分方程数值方法．区域分解方法及其收敛性的研究大多是在线性偏微分方程下得到的,对于非线性问题,经典的技巧在收敛性证明时遇到了困难．流体计算是一个较为复杂的非线性问题,数值模拟过程中因节点多．网格复杂,所以计算量很大．由于区域分解方法不但可以缩小求解规模,进行并行计算,而且可以在不同区域选取不同离散方法和模型,因此对Ｎ－Ｓ方程区域分解方法的研究会有较高的实用价值,也可以对其它非线性问题数值方法研究提供新的途径．本文首先给出了Ｎ－Ｓ方程的最优控制方法以及一些重要…     

一种基于积分方程的数值微分方法

本文研究了目前一些求解数值微分的方法无法求出端点导数或是求出的端点附近导数不可用的问题.利用构造一类积分方程的方法,将数值微分问题转化为这类积分方程的求解,并用一种加速的迭代正则化方法来求解积分方程. 数值实验结果表明该算法可以有效求出端点的导数,且具有数值稳定、计算简单等优点.     

高分辨KFVS有限体积方法及其CFD应用

1．引言文中研究三维Euler方程组的数值求解·儿1）中p,（。。,。。,。z）,p和E分别表示流体密度,流体速度矢量,压力和总能．方程组（1．1）是不封闭的,除非增加一个额外的方程一状态方程p一pk句,e表示单位质量内能．本文仅限于理想气体,此时状态方程为p一（、－1加e．队2）近H十年来,涌现了许多求解方程组（1．1）的无振荡、高分辨格式,例如TVD格式问,＊＊O格式问等,它们在一定程度上促进了航空航天和造船事业的发展．其中有一类根据双曲方程组（1．l）特征值的符号建立的迎风格式尤为突出,与中心格式相比,迎风格式的耗…     

Block‐centered upwind multistep difference method and convergence analysis for numerical simulation of oil reservoir

The three‐dimensional displacement of two‐phase flow in porous media is a preliminary problem of numerical simulation of energy science and mathematics. The mathematical model is formulated by a nonlinear system of partial differential equations to describe incompressible miscible case. The pressure is defined by an elliptic equation, and the concentration is defined by a convection‐dominated diffusion equation. The pressure generates Darcy velocity and controls the dynamic change of concentration. We adopt a conservative block‐centered scheme to approximate the pressure and Darcy velocity, and the accuracy of Darcy velocity is improved one order. We use a block‐centered upwind multistep method to solve the concentration, where the time derivative is approximated by multistep method, and the diffusion term and convection term are treated by a block‐centered scheme and an upwind scheme, respectively. The composite algorithm is effective to solve such a convection‐dominated problem, since numerical oscillation and dispersion are avoided and computational accuracy is improved. Block‐centered method is conservative, and the concentration and the adjoint function are computed simultaneously. This physical nature is important in numerical simulation of seepage fluid. Using the convergence theory and techniques of priori estimates, we derive optimal estimate error. Numerical experiments and data show the support and consistency of theoretical result. The argument in the present paper shows a powerful tool to solve the well‐known model problem.     

Local discontinuous Galerkin approximations to fractional Bagley-Torvik equation

In this research, numerical approximation to fractional Bagley-Torvik equation as an important model arising in fluid mechanics is investigated. Our discretization algorithm is based on the local discontinuous Galerkin (LDG) schemes along with using the natural upwind fluxes, which enables us to solve the model problem element by element. This means that we require to solve a low-order system of equations in each subinterval, hence avoiding the need for a full global solution. The proposed schemes are tested on a range of initial- and boundary-value problems including a variable coefficient example, a nonsmooth problem, and some oscillatory test cases with exact solutions. Also, the validation of the proposed methods was compared with those obtained available existing computational procedures. Overall, it was found that LDG methods indicated highly satisfactory performance with comparatively lower degree of polynomials and number of elements compared with other numerical models.     

Numerical solution of the Orr–Sommerfeld equation using the viscous Green function and split-Gaussian quadrature

We continue our study of the construction of numerical methods for solving two-point boundary value problems using Green functions, building on the successful use of split-Gauss-type quadrature schemes. Here we adapt the method for eigenvalue problems, in particular the Orr–Sommerfeld equation of hydrodynamic stability theory. Use of the Green function for the viscous part of the problem reduces the fourth-order ordinary differential equation to an integro-differential equation which we then discretize using the split-Gaussian quadrature and product integration approach of our earlier work along with pseudospectral differentiation matrices for the remaining differential operators. As the latter are only second-order the resulting discrete equations are much more stable than those obtained from the original differential equation. This permits us to obtain results for the standard test problem (plane Poiseuille flow at unit streamwise wavenumber and Reynolds number 10 000) that we believe are the most accurate to date.     

Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid

In this paper, Adomian’s decomposition method is used to solve non-linear differential equations which arise in fluid dynamics. We study basic flow problems of a third grade non-Newtonian fluid between two parallel plates separated by a finite distance. The technique of Adomian decomposition is successfully applied to study the problem of a non-Newtonian plane Couette flow, fully developed plane Poiseuille flow and plane Couette–Poiseuille flow. The results obtained show the reliability and efficiency of this analytical method. Numerical solutions are also obtained by solving non-linear ordinary differential equations using Chebyshev spectral method. We present a comparative study between the analytical solutions and numerical solutions. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the Adomian decomposition method.     

Application of differential quadrature to transport processes

The methodology and numerical solution of problems concerning transport processes via the method of differential quadrature are presented. Application of the method is demonstrated by solving a simple one-dimensional, time-dependent (transient) diffusion process involving an irreversible reaction without any flux across the end boundary. In addition, the same technique is used (for the first time to the authors' knowledge) to solve a steady-state problem. For this purpose, a convection-diffusion problem involving an irreversible reaction is considered. The demonstration is carried out in two ways, (1) using the Bellman et al. technique which employs approximation formulas for higher order partial derivatives derived by iterating the linear quadrature approximation for the first order partial derivative, and (2) using individual quadratures to approximate the partial derivatives of first, as well as higher orders, as suggested by Mingle. Both approaches give the same results; however, the latter saves an appreciable amount of iterative computing effort despite the fact that it requires separate weighting coefficients for each individual quadrature. Since the technique of differential quadrature can produce solutions with sufficient accuracy even when using as few as three discrete points, both the programming task and computational effort are alleviated considerably. For these reasons the differential quadrature approach appears to be very practical in solving a variety of problems related to transport phenomena.     

Numerical simulation of three-dimensional incompressible fluid in a box flow passage considering fluid–structure interaction by differential quadrature method

A numerical simulation scheme of 3D incompressible viscous fluid in a box flow passage is developed to solve Navier–Stokes (N–S) equations by firstly taking fluid–structure interaction (FSI) into account. This numerical scheme with FSI is based on the polynomial differential quadrature (PDQ) approximation technique, in which motions of both the fluid and the solid boundary structures are well described. The flow passage investigated consists of four rectangular plates, of which two are rigid, while another two are elastic. In the simulation the elastic plates are allowed to vibrate subjected to excitation of the time-dependent dynamical pressure induced by the unsteady flow in the passage. Meanwhile, the vibrating plates change the flow pattern by producing many transient sources and sinks on the plates. The effects of FSI on the flow are evaluated by running numerical examples with the incoming flow’s Reynolds numbers of 3000, 7000 and 10,000, respectively. Numerical computations show that FSI has significant influence on both the velocity and pressure fields, and the DQ method developed here is effective for modelling 3D incompressible viscous fluid with FSI.     

An optimal asymptotic-numerical method for convection dominated systems having exponential boundary layers

The convection dominated diffusion problems are studied. Higher order accurate numerical methods are presented for problems in one and two dimensions. The underlying technique utilizes a superposition of given problem into two independent problems. The first one is the reduced problem that refers to the outer or smooth solution. Stretching transformation is used to obtain the second problem for inner layer solution. The method considered for outer or degenerate problems are based on higher order Runge–Kutta methods and upwind finite differences. However, inner problem is solved analytically or asymptotically. The schemes presented are proved to be consistent and stable. Possible extensions to delay differential equations and to nonlinear problems are outlined. Numerical results for several test examples are illustrated and a comparative analysis is presented. It is observed that the method presented is highly accurate and easy to implement. Moreover, the numerical results obtained are not only comparable with the exact solution but also in agreement with the theoretical estimates.     

Application of the differential transformation method to vibration analysis of pipes conveying fluid

In this paper, a relatively new semi-analytical method, called differential transformation method (DTM), is generalized to analyze the free vibration problem of pipes conveying fluid with several typical boundary conditions. The natural frequencies and critical flow velocities are obtained using DTM. The results are compared with those predicted by the differential quadrature method (DQM) and with other results reported in the literature. It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of pipes conveying fluid.     

Analysis of a new BEM‐FEM coupling for two‐dimensional fluid‐solid interaction

In this work we present a new numerical method, based on a coupling of finite and boundary elements, to solve a fluid‐solid interaction problem in the plane. The discrete method uses classical Lagrange finite elements adapted to curved boundaries for the field variable and spectral approximation of the unknowns on the artificial boundary. We provide error estimates for this Galerkin scheme and propose a full discretization based on elementary quadrature formulae, showing that the perturbation due to numerical integration preserves the optimal rate of convergence. We also suggest an iterative method to solve the complicated linear systems arising from this type of schemes. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Rational Chebyshev collocation method for the similarity solution of two dimensional stagnation point flow

In this study, we propose an efficient and accurate numerical technique that is called the rational Chebyshev collocation (RCC) method to solve the two dimensional flow of a viscous fluid in the vicinity of a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow, are reduced to a third-order ordinary differential equation of a boundary value problem with a semi-infinite domain by using similarity transformation. The rational Chebyshev method reduces this nonlinear ordinary differential equation to a system of algebraic equations. This technique is a powerful type of the collocation methods for solving the boundary value problems over a semi-infinite interval without truncating it to a finite domain. We also present the comparison of this work with others and show that the present method is more accurate and efficient.