Reduced finite difference scheme and error estimates based on POD method for non-stationary Stokes equation

The proper orthogonal decomposition (POD) is a model reduction technique for the simulation of physical processes governed by partial differential equations (e.g.,fluid flows). It has been successfully used in the reduced-order modeling of complex systems. In this paper, the applications of the POD method are extended, i.e., the POD method is applied to a classical finite difference (FD) scheme for the non-stationary Stokes equation with a real practical applied background. A reduced FD scheme is established with lower dimensions and sufficiently high accuracy, and the error estimates are provided between the reduced and the classical FD solutions. Some numerical examples illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD scheme based on the POD method is feasible and efficient in solving the FD scheme for the non-stationary Stokes equation.     

A new alternating group explicit-implicit algorithm with high accuracy for dispersive equation

In this paper, a new alternating group explicit-implicit （nAGEI） scheme for dispersive equations with a periodic boundary condition is derived. This new unconditionally stable scheme has a fourth-order truncation error in space and a convergence ratio faster than some known alternating methods such as ASEI and AGE. Comparison in accuracy with the AGEI and AGE methods is presented in the numerical experiment.     

非线性双曲型守恒律的高精度MmB差分格式

构造了一维非线性双曲型守恒律方程的一个高精度、高分辨率的广义G odunov型差分格式。其构造思想是:首先将计算区间划分为若干个互不相交的小区间,再根据精度要求等分小区间,通过各细小区间上的单元平均状态变量,重构各等分小区间交界面上的状态变量,并加以校正;其次,利用近似R iem ann解算子求解细小区间交界面上的数值通量,并结合高阶R unge-K u tta TVD方法进行时间离散,得到了高精度的全离散方法。证明了该格式的Mm B特性。然后,将格式推广到一、二维双曲型守恒方程组情形。最后给出了一、二维Eu ler方程组的几个典型的数值算例,验证了格式的高效性。     

高阶边界元奇异积分的一种通用高效计算方法

针对三维边界元法中曲面单元上的(弱、强、超)奇异积分提出了一种通用高效的计算方法。经极坐标变换,将奇异积分转化为常规积分;采用数值方法计算Cauchy主值积分和Hadamard有限项积分系数;引入保角变换和反曲变换消除因单元畸形或因积分点靠近单元边界而引起的周向积分奇异性。该方法可以统一处理(弱、强、超)奇异积分,并且只需要知道核函数的奇异阶数和少数几个点上的被积函数值,不依赖于积分和函数的具体选取;所需的积分点少,精度高,并且受单元畸形程度影响较小,稳定性好。采用该方法计算了声学和弹性力学中的典型奇异积分,并结合二阶Nystrm方法求解了弹性力学的边界积分方程,验证了方法的高精度和高效性。本文数值积分程序可向作者索取。     

A design of residual error estimates for a high order BDF‐DGFE method applied to compressible flows

We deal with the numerical solution of the non‐stationary compressible Navier–Stokes equations with the aid of the backward difference formula – discontinuous Galerkin finite element method. This scheme is sufficiently stable, efficient and accurate with respect to the space as well as time coordinates. The nonlinear algebraic systems arising from the backward difference formula – discontinuous Galerkin finite element discretization are solved by an iterative Newton‐like method. The main benefit of this paper are residual error estimates that are able to identify the computational errors following from the space and time discretizations and from the inexact solution of the nonlinear algebraic systems. Thus, we propose an efficient algorithm where the algebraic, spatial and temporal errors are balanced. The computational performance of the proposed method is demonstrated by a list of numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Improved error and work estimates for high‐order elements

Work estimates for high‐order elements are derived. The comparison of error and work estimates shows that even for relative accuracy in the 0.1% range, which is one order below the typical accuracy of engineering interest (1% range), linear elements may outperform all higher‐order elements. As expected, the estimates also show that the optimal order of element in terms of work and storage demands depends on the desired relative accuracy. The comparison of work estimates for high‐order elements and their finite difference counterparts reveals a work‐ratio of several orders of magnitude. It thus becomes questionable if general geometric flexibility via micro‐unstructured grids is worth such a high cost. Copyright © 2013 John Wiley & Sons, Ltd.     

Characteristic finite difference method and application for moving boundary value problem of coupled system

The coupled system of multilayer dynamics of fluids in porous media is to describe the history of oil-gas transport and accumulation in basin evolution.It is of great value in rational evaluation of prospecting and exploiting oil-gas resources.The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values.A kind of characteristic finite difference schemes is put forward,from which optimal order estimates in l^2 norm are derived for the error in the approximate solutions.The research is important both theoretically and practically for the model analysis in the field,the model numerical method and software development.     

时滞抛物型方程的高精度精细积分法

对一类时滞抛物型方程初边值问题,提出了关于空间步长是四阶精度的高精度无条件稳定的精细积分法.数值算例表明,本文提出的精细积分法具有很高的精度,因而是一种有效的数值方法.     

Upwind finite difference method for miscible oil and water displacement problem with moving boundary values

The research of the miscible oil and water displacement problem with moving boundary values is of great value to the history of oil-gas transport and accumulation in the basin evolution as well as to the rational evaluation in prospecting and exploiting oil-gas resources. The mathematical model can be described as a coupled system of nonlinear partial differential equations with moving boundary values. For the twodimensional bounded region, the upwind finite difference schemes are proposed. Some techniques, such as the calculus of variations, the change of variables, and the theory of a priori estimates, are used. The optimal orderl2-norm estimates are derived for the errors in the approximate solutions. The research is important both theoretically and practically for the model analysis in the field, the model numerical method, and the software development.     

二维对流扩散方程的高精度全隐式多重网格方法

提出了数值求解二维非定常变系数对流扩散方程的一种时间二阶、空间四阶精度的三层全隐紧致差分格式。为了加快迭代求解隐格式时在每一个时间步上的收敛速度,采用多重网格加速技术,建立了适用于本文高精度金隐紧致格式的多重网格算法。数值实验结果验证了本文方法的精确性、稳定性和对高网格雷诺数问题的强适应性。     

High order schemes for the scalar transport equation

A finite volume hybrid scheme for the spatial discretization that combines a fixed stencil and a stencil determined by the classical essentially non‐oscillatory (ENO) scheme is presented. Evolution equations are obtained for the mean values of each cell by means of piecewise interpolation. Time discretization is accomplished by a classical fourth‐order Runge–Kutta. Interpolation polynomials are determined using information of adjacent cells. While smooth regions are interpolated by means of a fixed molecule, discontinuous or sharp regions are interpolated by the classical ENO algorithm. The algorithm estimates the interpolation error at each time step by means of two interpolants of order q and q+1. The main computational load of the resultant scheme is in the interpolation, which is performed by the divided differences table. This table involves O(qN) operations, where q is the interpolation order and N is the number of cells. Finally, linear test cases of continuous and discontinuous initial conditions are integrated to see the goodness of the hybrid scheme. It is well known that, for some particular initial conditions, the classical ENO scheme does not perform properly, not attaining the truncation error of the scheme. It is shown that, for the smooth initial condition, sin⁴(x), the classical ENO scheme does not preserve the character of stability of the initial value problem, giving rise to unstable eigenvalues. The proposed hybrid scheme solves this problem, choosing a fixed stencil over the whole computational domain. The resultant schemes are equivalent to the classical finite difference schemes, which preserve the character of stability. It is also known that the same degeneracy of the error can be encountered for discontinuous solutions. It is shown for the initial discontinuous solution, e^−x, that the classical ENO algorithm does not perform properly due to the conflict between the selection of the stencil to smoother regions (downwind region) and the hyperbolic character of the problem, which obliges us to take information from downwind. The proposed hybrid scheme solves this problem by choosing a fixed stencil over the whole computational domain except at the discontinuity. Copyright © 2001 John Wiley & Sons, Ltd.     

Diagonal form fast multipole boundary element method for 2D acoustic problems based on Burton-Miller boundary integral equation formulation and its applications

This paper describes formulation and implementation of the fast multipole boundary element method (FMBEM) for 2D acoustic problems. The kernel function expansion theory is summarized, and four building blocks of the FMBEM are described in details. They are moment calculation, moment to moment translation, moment to local translation, and local to local translation. A data structure for the quad-tree construction is proposed which can facilitate implementation. An analytical moment expression is derived, which is more accurate, stable, and efficient than direct numerical computation. Numerical examples are presented to demonstrate the accuracy and efficiency of the FMBEM, and radiation of a 2D vibration rail mode is simulated using the FMBEM.     

裂纹问题的一致性高阶无网格法

一致性高阶无网格法能高效精确地求解连续体问题,尤其是能得到高精度的应力场。本文将该方法拓展到应力解析精度至关重要的裂纹问题（即非连续体问题）的数值分析。采用背景积分网格描述裂纹几何,基于无需增加节点额外自由度的虚拟节点法描述裂纹处位移场的间断,提出了虚拟节点的引入算法和断裂单元的数值积分方法。为进一步模拟裂纹扩展,采用相互作用积分方法计算应力强度因子,裂纹的扩展方向由最大周向应力准则确定。数值结果表明,本文发展方法能够精确地通过间断分片试验;相较于标准的高阶无网格法和低阶一致性无网格法,本文的一致性高阶无网格法显著改善了应力强度因子的计算精度,能够准确预测裂纹扩展路径。     

Least-squares mixed finite element method for a class of stokes equation

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A－HIGH－ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THE EQUATION OF TWO－DIMENSIONAL PARABOLIC TYPE

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用差分法与超松弛迭代法求高维平稳FPK方程的解

用不同精度的差分格式将高维平稳FPK方程离散化为线性代数方程组,然后用超松弛迭代法求解该线性代数方程组得到平稳FPK方程的近似解。讨论了不同的差分格式、网格密度及超松弛因子对解精度及收敛速度的影响,并与其他方法的计算精度进行比较,提出用多重网格算法提高计算效率。研究了典型的二维及四维随机系统的稳态响应,算例表明,该算法具有简洁、节省存储量且精度高的特点,是求解高维平稳FPK方程解的有效算法。     

Modified characteristic finite difference fractional step method for moving boundary value problem of percolation coupled system

For the coupled system with moving boundary values of multilayer dynamicsof fluids in porous media,a characteristic finite difference fractional step scheme appli-cable to the parallel arithmetic is put forward.Some techniques,such as the change ofregions,the calculus of variations,the piecewise threefold quadratic interpolation,themultiplicative commutation rule of difference operators,the decomposition of high orderdifference operators,and the prior estimates,are adopted.The optimal order estimatesin the l2norm are derived to determine the error in the approximate solution.This nu-merical method has been successfully used to simulate the flow of migration-accumulationof the multilayer percolation coupled system.Some numerical results are well illustratedin this paper.     

基于最小二乘的工程结构边界载荷反求误差消减方法

机械装备工程结构强度计算和疲劳寿命预测等都需要准确获知结构的工作或边界载荷。然而,由于土壤和砂石等工作介质的切削及铲运等精确阻力模型难以建立,以及机械装备动力传动过程中的摩擦等损耗难以精确计算,导致准确获得装备工作结构的边界载荷非常困难。依据圣维南原理,采用有限元法建立复杂结构中局部结构应变与其边界载荷的关系,对边界载荷反求过程中误差来源和范围进行了分析和估计,提出了基于最小二乘法的复杂结构局部边界载荷误差消减方法。对推耙机横梁结构的边界载荷进行分析,验证了该误差消减方法的有效性。