数值方程

Based on the Volume of Fluid Method, the wave damping absorber of the numerical wave flume for irregular waves has been investigated. Setting the Turbulence motion equation and the Renault equation as the governing equations and using the wall function technique in the near-wall region, an absorbing wave-maker is set at the left side of the flume.

本文对基于VOF方法的不规则波数值波浪水槽进行了阻尼消波研究，以湍流运动方程一雷诺方程为控制方程，在近壁区应用壁函数技术，在数值波浪水槽的左端设置了主动吸收式造波机。

An extended hyperbolic mild-slope equation, which can take such terms as wind input, bottom friction and nonlinearity of wave into account, here has been deduced into a parabolic one in the form of a series of governing equations described with complex amplitude. It became a complete wave model when integrated with corresponding physical and imaginary boundaries, and can be solved through the improved Crank-Nicholson or the well known Alternating Direction Implicit difference method, both of which can accelerate the convergent speed , expand limitation of the mathematical method and improve the quality of the solution.

本文基于综合考虑底摩阻、风能输入及非线性影响的推广双曲型缓坡方程，将其进行转换，产生一个演变方程，其复振幅控制方程为抛物线性，并采用ADI差分格式、改进的Crank-Nicholson格式及相应的物理和虚拟边界条件进行求解，以提高数值模拟的适用性、数值计算的稳定性、收敛速度及精度。

Chapter 9: We report a large quantity of numerical experiments of 13 different algebraic multigrid algorithms for solving the Poisson equation, anisotropic equation, equation with cross-derivative terms, general matrix problems with large off-diagonal positive entries, biharmonic equation, Toeplity matrix, elasticity systems, finite element discretization of the Laplacian and even 3D problems. Particular attention is focused on asymptotic convergence factors and CPU-time consumed. Numerical results for many different types of practical problems demonstrate the efficiency and robustness of the proposed algebraic multigrid methods.

第九章：在各种代数多重网格算法的基础上，进行了大量的数值试验，具体给出了十三种不同的代数多重网格方法求解泊松方程，各向异性方程，带混合导数项的方程，带有大的非对角正元素的一般矩阵问题，重调和方程，托普利兹矩阵，弹性力学方程组，拉普拉斯算子的有限元离散，甚至三维问题的较为丰富的数值结果，重点关注它们的渐近收敛因子和所需的CPU时间，来源于不同类型问题的计算结果既为代数多重网格理论分析和算法的改进提供了很实用的资料，同时也证实了本文给出的代数多重网格算法的效绩和稳健性。

This paper presents the boundary value problem solved as the biharmonic equation of upper plane and strip domain with Green functional method and it deals with the numerical solution of bihormonic equation, using the Matlab program calculation to realize the visualization of numeric solution.

用Green函数法求解了区域为上半平面和带形区域的双调和泊松方程的边值问题；以及探讨了双调和方程的数值解，并用Matlab编程计算实现了双调和方程数值解的可视化。

In Chapter 2, starting from the basic fractional ordinary differential equations,weapply a high order approximation of fractional derivative advanced by Lubich to frac-tional differential equation, construct a high numerical difference scheme to solve thefractional differential equation, present error analysis of the algorithms theoretically,and prove the consistency ,convergency and stability.

接下来的第二章中，首先从基本的分数阶常微分方程出发，对Lubich提出的一个关于分数阶导数的高阶近似，将其应用于分数阶微分方程，构造高阶数值差分格式来进行分数阶微分方程的数值求解，并在理论上给出这一算法的误差分析，证明了它的相容性，收敛性和稳定性。

The main works in this paper are as follows:(1). In this thesis, a new method, continuous linearization model, is constructed for the non-linear dynamic systems. The precision Duhamel integration solution for the linearization equation is also given from the theory of linear differential equation. The calculating results show that the new method can improve the calculation precision greatly. In fact, the truncation error is not limited by this method, it can arrive the precision according to the requirement.

本文的主要工作有以下几个方面：(1)由于非线性项的存在，使得多数情况下难以得到非线性动力方程的解析解表达式，目前常使用数值积分方法来求解非线性动力学方程的解，但常规计算方法的计算精度不很理想，本文提出了含Taylor展开项中高阶余项的非线性动力方程的线性化数值计算方法，构造了瞬态线性化方程，给出了Duhamel积分的表达式。

The main contents of this course include: introduction, extraction of root from an equation, numerical solution of a system of linear equations, interpolation method, the method of fitting of a curve, numerical integration and differentiation, numerical solution of a differential equation, one-dimensional search method of extreme value problem, etc.

本课程包括以下内容：绪论、方程求根、线性方程组数值解法、插值法、曲线拟合法、数值积分与数值微分、微分方程数值解法、极值问题的一维搜索法。

An inhomogeneous Benjamin-Davis-Ono-Burgers equation including topographic forcing and turbulent dissipation is derived in terms of the quasi-geostrophic vorticity equation, an approximate analytic solution of the BDO-Burgers equation with a small dissipation is obtained, the time variations of mass and energy of the algebraic solitary waves are discussed, and finally, the forced BDO-Burgers equation is integrated numerically and the numerical solutions are given for a given basic flow with a weak shear and a localized topographic forcing.

利用包括地形外源和湍流耗散的准地转涡度方程导出了包含强迫项和湍流耗散项的非齐次强迫BDO-Burgers方程，求出了小耗散存在情况下BDO-Burgers方程的近似解析解，讨论了代数孤立波的质量和能量的守恒关系，最后，对给定的弱切变基本气流和局地地形强迫，数值积分强迫BDO-Burgers方程，求得了不同参数下的数值解。

This book reviews the many areas of numerical analysis, including the configuration polynomial, finite difference, factorial polynomials, summation, Newton formula, operator and configuration polynomial, Cheung section, close polynomials, TaylM more item type, interpolation, numerical differentiation, numerical integration, and with the series, differential equations, differential equations, least squares polynomial approximation, minimax polynomial approximation, rational function approximation, triangular approximation, non-linear algebra, linear equations, linear programming, boundary value problems, MonteCarIo methods and so on.

本书综述了数值分析领域的诸多内容，包括配置多项式、有限差分、阶乘多项式、求和法、Newton公式、算子与配置多项式、祥条、密切多项式、TaylM多项式、插值、数值微分、数值积分、和与级数、差分方程、微分方程、最小二乘多项式逼近、极小化极大多项式逼近、有理函数逼近、三角逼近、非线性代数、线性方程组、线性规划、边值问题、MonteCarIo方法等内容。本书的特色主要表现在利用例题及大量详细的题解来透彻地阐明所述内容的内涵，同时附有大量的补充题以便读者进一步巩固和深化从书中获得的数值分析知识。

This thesis is divided into six parts. The first chapter is preface, the current status of research in the inverse problems for parabolic partial differential equations is reported; the second chapter is "regularization methods for numerical differentiation and their applications ", in this chapter we investigate many regularization methods from a viewpoint of regularization theory and algorithm, some applications in the inverse problems for parabolic partial differential equations are given; the third chapter is "spectral regularization methods". Based on Fourier analysis, within the framework of regularization theory, we apply the spectral methods to some ill-posed problems. Many numerical experiments are done in order to show the validity of the methods; the fourth chapter is devoted to wavelet dual least squares method and a revised wavelet method; in the fifth chapter,we combine finite difference method with method of lines and apply it to the backward heat conduction problem in time; in the sixth chapter "identification problems for unknown source ", the essence and the degree of two problems related to source identification are pointed out, at the same time, some numerical methods are reported.

本文分为六个部分，第一章前言简要分析了国内外抛物型偏微分方程反问题的研究现状；第二章数值微分的正则化及其应用从正则化理论和算法的角度出发，考察了许多正则化方法，还给出了数值微分在抛物型偏微分方程反问题的一些应用；第三章谱正则化方法是在Fourier分析的基础上，在一般正则化理论的框架下，给出了这种方法在各种不适定问题中的应用，数值实验表明谱方法是有效的；第四章研究了小波对偶最小二乘方法和改进的小波方法；第五章主要研究了有限差分方法结合线方法在时间反向热传导问题中的应用；第六章是未知源识别问题，主要指出了两类未知源问题的不适定程度和不适定本质，同时报告了一些数值方法。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。