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。     

求解半线性椭圆问题的牛顿-瀑布型两层网格法

选取一对合适的步长使用中心差分格式离散半线性椭圆问题形成粗网格和细网格,使用三次样条插值算子将粗网格上高精度近似解插值到细网格为其提供初始值,结合牛顿法提出了牛顿-瀑布型两层网格法．数值实验表明该算法具有稳健性强、计算效率高的优点．     

非结构网格的并行多重网格解算器

多重网格方法作为非结构网格的高效解算器,其串行与并行实现在时空上都具有优良特性.以控制方程离散过程为切入点,说明非结构网格在并行数值模拟的流程,指出多重网格方法主要用于求解时间推进格式产生的大规模代数系统方程,简述了算法实现的基本结构,分析了其高效性原理;其次,综述性地概括了几何多重网格与代数多种网格研究动态,并对其并行化的热点问题进行重点论述.同时,针对非结构网格的实际应用,总结了多重网格解算器采用的光滑算子;随后列举了非结构网格应用的部分开源项目软件,并简要说明了其应用功能;最后,指出并行多重网格解算器在非结构网格应用中的若干关键问题和未来的研究方向.     

用于图像重构的代数多重网格算法

通过分析代数多重网格(algebraic multi-grid,AMG)算法中粗网格提取过程,提出了一种基于代数多重网格算法的图像重构算法.在代数多重网格算法的粗网格序列中,下一层粗网格保留上一层网格的强连接部分.将这种机制运用到图像,提取的粗网格可以较好的保留图像的有效信息部分,在图像变化剧烈的细节区域网格点分布不均匀,平滑模糊部分网格点分布均匀一致.以粗网格像素点进行插值,可以得到较好的重建结果.以均方误差为评价参数,与小波算法进行了比较,比较结果表明该算法在一定程度上优于传统的小波算法,且有一个图像融合应用实例,优于小波融合方法.     

Cahn-Hilliard方程多重网格求解器收敛性分析

Cahn-Hilliard(CH)方程是相场模型中的一个基本的非线性方程，通常使用数值方法进行分析。在对CH方程进行数值离散后会得到一个非线性的方程组，全逼近格式(Full Approximation Storage, FAS)是求解这类非线性方程组的一个高效多重网格迭代格式。目前众多的求解CH方程主要关注数值格式的收敛性，而没有论证求解器的可靠性。文中给出了求解CH方程离散得到的非线性方程组的多重网格算法的收敛性证明，从理论上保证了计算过程的可靠性。针对CH方程的时间二阶全离散差分数值格式，利用快速子空间下降(Fast Subspace Descent, FASD)框架给出其FAS格式多重网格求解器的收敛常数估计。为了完成这一目标，首先将原本的差分问题转化为完全等价的有限元问题，再论证有限元问题来自一个凸泛函能量形式的极小化，然后验证能量形式及空间分解满足FASD框架假设，最终得到原多重网格算法的收敛系数估计。结果显示，在非线性情形下，CH方程中的参数ε对网格尺度添加了限制，太小的参数会导致数值计算过程不收敛。最后通过数值实验验证了收敛系数与方程参数及网格尺度的依赖关系。     

TTI介质井间地震波场紧致交错网格高阶有限差分模拟及边界条件

研究井间地震波场的形成过程以及波场的传播机理、规律,对于指导实际井间地震勘探有着重要的意义．从具有倾斜对称轴的横向各向同性介质（TTI）的二维三分量一阶速度-应力弹性波方程出发,采用高阶紧致交错网格差分算子对方程进行差分离散,得到了TTI介质中井间地震波场正演的高阶有限差分格式．并推导了TTI介质完全匹配层吸收边界条件公式和相应的紧致交错网格高阶差分格式,在此基础上实现了二维三分量TTI介质中井间地震波场模拟．数值算例表明：紧致交错网格高阶有限差分方法模拟的记录精度高,数值频散小,该方法能够精确的模拟复杂各向异性介质中的地震波传播过程,可以得到高精度的正演记录．完全匹配层吸收边界能有效地解决人工边界问题,是一种高效的边界吸收算法．     

不等距网格上求解ODE特征值问题若干高精度格式的计算与分析

本文针对不等距网格,从Raylei曲商(Raylei曲quotient)角度出发,构造了若干求解ODE特征值问题的高阶格式,并进行误差分析.文中高阶格式的构造是基于线性有限元及其对应的差分格式进行的.单纯的线性有限元及其对应的差分格式求解PDE特征值问题都只有二阶精度,我们利用质量集中和加权组合的思想通过将二者结合得到四阶精度的算法.本文从理论和实验的角度构造高阶格式并进行了相应的误差分析.通过在五种网格上计算四阶精度格式的误差阶系数,将四阶格式加权组合的新格式甚至可以达到六阶精度.最后用数值实验验证了构造的高阶格式的误差阶.同时,本文构造的两种四阶格式相对于传统的线性有限元方法,在同等量级误差的要求下,需要的网格数有量级的减少.     

多重网格方法求解两类Helmholtz方程

详细给出了多重网格方法的实现过程,借助正定Helmholtz方程及不定Helmholtz方程的求解来探讨多重网格方法的特性。对多重网格V环、W环以及F环三种不同迭代格式的收敛效果进行了对比。通过正定Helmholtz方程的求解,发现多重网格的确有很高的计算效率。对于不定Helmholtz方程,随着波数的增加,利用多重网格方法得到结果不收敛,原因出在细网格光滑和粗网格矫正过程。如何针对此问题对多重网格进行有效改进还有待进一步研究。     

多重网格技术对SIMPLER算法的加速性能研究

将多重网格技术引入SIMPLER算法以加快其收敛速度，从而节约计算时间。通过计算不同雷诺数下的二维方腔顶盖驱动流，研究了多重网格方法中的V循环、W循环对SIMPLER算法的加速效果，并讨论了网格层数对加速性能的影响。研究结果表明，在不同雷诺数下，多重网格方法均可以起到良好的加速效果；在相同雷诺数和精度要求下，W循环方式的外迭代次数少于V循环方式的外迭代次数，而且网格层数对多重网格加速性能的影响并不显著。     

层适应网格上求解奇异摄动问题的粒子群算法

针对一类奇异摄动对流扩散问题,将粒子群算法与差分格式相结合,在Bakhvalov-Shishkin网格上进行求解。对于Bakhvalov-Shishkin网格中的网格参数,采用粒子群算法进行优化,构造了求误差范数最小值的目标函数。对两个算例进行了数值计算,实验结果表明,与选择固定的网格参数相比,采用粒子群算法计算能得到更好的数值结果,并且数值结果具有收敛性,验证了该方法的有效性和优越性。     

The non-integer difference of the discrete-time function and its application to the control system synthesis

A definition of a non-integer nth order backward difference of the discrete-time function is given. It is a generalization of the well-known nth order difference, where n is a positive integer. Some simple examples of the non-integer nth order differences of several fundamental functions are given. The application of the nth order difference in the closed-loop discrete-time systems control algorithms are illustrated by numerical examples.     

完全状态差表生成法则

根据完全封闭升序n端输出时序机的各种定义，建立了位状态差、序状态差、表状态差运算规律，构成了完全状态差表及自状态差表，导出了完全状态差表的一系列重要性质。完全状态差表在海量数据块检测、模式识别、自然语言理解等方面有广泛应用。自状态差表可成功直观简明精确定位时序机隐式软故障。     

Extended Kalman filters for fractional‐order nonlinear continuous‐time systems containing unknown parameters with correlated colored noises

By using the Grünwald‐Letnikov (G‐L) difference method and the Tustin generating function method, this study presents extended Kalman filters to achieve satisfactory state estimation for fractional‐order nonlinear continuous‐time systems that containing some unknown parameters with the correlated fractional‐order colored noises. Based on the G‐L difference method and the Tustin generating function method, the difference equations corresponding to fractional‐order nonlinear continuous‐time systems are constructed respectively. The first‐order Taylor expansion is used to linearize the nonlinear functions in the estimated system, which provides the system model for extended Kalman filters. Using the augmented vector method, the unknown parameters are regarded as new state vectors, and the augmented difference equation is constructed. Based on the augmented difference equation, extended Kalman filters are designed to estimate the state of fractional‐order nonlinear systems with process noise as fractional‐order colored noise or measurement noise as fractional‐order colored noise. Meanwhile, the extended Kalman filters proposed in this paper can also estimate the unknown parameters effectively. Finally, the effectiveness of the proposed extended Kalman filters is validated in simulation with two examples.     

重节点埃尔米特插值算法与程序实现

研究了具有任意阶导数信息Hermite插值问题,使用广义差商的一种新的表示方法和构造广义差商表的一种新方法,给出具有任意阶导数信息Hermite插值算法和程序实现,拓展了牛顿差商插值公式和余项公式。     

Stability analysis of six-point finite difference schemes for the constant coefficient convective-diffusion equation

A comprehensive and systematic study is presented to derive stability properties of various two-level, six-point finite difference schemes (in particular, difference schemes of Padé type) for the approximation to the constant coefficient convective-diffusion equation. First, the modified equivalent partial differential equation (MEPDE) for a general six-point difference scheme is derive. The MEPDE provides direct information on the order of accuracy of a difference scheme. The von Neumann and matrix methods are then employed to deduce the necessary and sufficient conditions for the numerical stability for the six-point difference schemes. An unified technique is developed to find the stability regions for the difference schemes. Some new second and third order six-point difference schemes for the approximation of the constant coefficient convective-diffusion equation are presented.     

迎风紧致格式与热流计算

１．引 言 众所周知,ＴＶＤ格式是能够高质量地捕捉激波的方法,但在计算粘性绕流时许多ＴＶＤ格式数值耗散太大,不能正确模拟粘性流动,因而无法正确计算热流值．文献［３］指出,采用高精度格式可适当放松对网格雷诺数的要求,因此发展三阶或三阶以上的格式是需要的．文献［４］研究了迎风紧致群速度控制格式（ＵＣＧＶＣ格式）在 Ｅｕｌｅｒ方程中的应用,提高了对激波的分辨率,优于通常二阶精度ＴＶＤ格式．本文在文献［４］的基础上给出了利用迎风紧致格式求解ＮＳ方程．它是ＵＣＧＶＣ格式在粘性流计算中的推广．对于方程中的无粘…     

The principle of the difference of difference quotients as a key to the self-adaptive solution of nonlinear partial differential equations

A unified method for the numerical solution of nonlinear ordinary and partial differential equations (elliptic and parabolic) is developed. The discretization error is estimated from the difference of difference quotients of a family of difference formulae. The difference grid and the difference star of arbitrary order are self-adapted and selected so that an optimum method results in which all error terms are well balanced to a prescribed relative tolerance.     

Algorithms based on difference equations of infinite order and the computation of Laplace-type integrals

Certain processes in analysis, e. g., the computation of Laplace-Stieltjes moment integrals, lead to difference equations of infinite order. In this paper, we seek to apply an analog of the famous Miller algorithm for difference equations of finite order to these systems. In certain situations we are able to demonstrate the convergence of the algorithm.     

屏蔽差集偶与伪随机屏蔽二进序列偶的研究

为了进一步研究伪随机屏蔽二元序列偶,提出了一类新的区组设计——屏蔽差集偶,证明了一类特殊的屏蔽差集偶和伪随机屏蔽二元序列偶是等价的,并研究了伪随机屏蔽二元序列偶存在的充分必要和必要条件;最后利用差集构造出一类伪随机屏蔽二元序列偶,丰富了伪随机屏蔽二元序列偶的研究内容。     

Multidomain WENO Finite Difference Method with Interpolation at Subdomain Interfaces

High order finite difference WENO methods have the advantage of simpler coding and smaller computational cost for multi-dimensional problems, compared with finite volume WENO methods of the same order of accuracy. However a main restriction is that conservative finite difference methods of third and higher order of accuracy can only be used on uniform rectangular or smooth curvilinear meshes. In order to overcome this difficulty, in this paper we develop a multidomain high order WENO finite difference method which uses an interpolation procedure at the subdomain interfaces. A simple Lagrange interpolation procedure is implemented and compared to a WENO interpolation procedure. Extensive numerical examples are shown to indicate the effectiveness of each procedure, including the measurement of conservation errors, orders of accuracy, essentially non-oscillatory properties at the domain interfaces, and robustness for problems containing strong shocks and complex geometry. Our numerical experiments have shown that the simple and efficient Lagrange interpolation suffices for the subdomain interface treatment in the multidomain WENO finite difference method, to retain essential conservation, full high order of accuracy, essentially non-oscillatory properties at the domain interfaces even for strong shocks, and robustness for problems containing strong shocks and complex geometry. The method developed in this paper can be used to solve problems in relatively complex geometry at a much smaller CPU cost than the finite volume version of the same method for the same accuracy. The method can also be used for high order finite difference ENO schemes and an example is given to demonstrate a similar result as that for the WENO schemes.