一种新的求解双曲型守恒律的三阶中心差分格式

本文提出了一种求解双曲型守恒律新的三阶中心差分格式，主要是引入了一种推广的三阶重构，并证明了这种重构在网格边界无振荡．所提的格式保持了中心差分格式简单的优点，不需用Riemann解算器，避免了进行特征解耦．数值试验结果表明本文格式是高精度、高分辨率的。     

设计多层差分格式的一种有效途径

本文推广了传统的余项裣方法并将其应用于多层差分格式的设计中。由此不仅可以设计出高精度的多层差分格式,并且能有效的控制格式的耗散与色散效应以满足不同数值模拟的需要。     

基于无振荡中心格式求解非均匀道路上的多车种LWR交通流模型

将无振荡中心格式推广于多车种LWR交通流模型,给出了一种求解非均匀道路上模型的高分辨率数值方法.为保证格式无振荡,采用非线性限制器近似离散斜率.通量的离散微分可以按分量来近似,使得雅可比矩阵的计算都可以避免.方法具有形式简单、计算量小的优点.应用该方法对信号灯控制等问题进行数值模拟,验证了方法的稳定性和有效性.     

以中心- 迎风数值格式研究具有外力项的p 系统

本文以半离散中心- 迎风数值格式研究具有外力项的p 系统. 中心型数值格式用来处理双曲型守恒律或系统的优势是快速且简单, 因为不需要使用近似Riemann 解, 也不需要做特征分解. 我们的数值模拟验证了理论研究结果: 具有外力项的p 系统的解的收敛及爆破行为, 同时也指出一些尚待理论研究的问题.     

Cahn-Hilliard方程的高阶保能量散逸性方法

能量散逸性是物理和力学中某些微分方程一项重要的物理特性.构造精确地保持微分方程能量散逸性的数值格式对模拟具有能量散逸性的微分方程具有重要的意义.本文利用四阶平均向量场方法和傅里叶谱方法构造了Cahn-Hilliard方程高阶保能量散逸性格式.数值结果表明高阶保能量散逸性格式能很好地模拟Cahn-Hilliard方程在不同初始条件下解的行为,并且很好地保持了Cahn-Hilliard方程的能量散逸特性.     

对流扩散方程的数值计算

本文研究了对流扩散方程的一种并行格式.利用一组saul'yev型非对称格式进行二次构造,分别得到了一类并行GE格式和GEL、GER格式;进一步推广,得到绝对稳定的交替分组显式AGE格式,并用数值例子检验AGE格式的数值计算效果.     

解含转向点问题的完全指数型拟合差分方法

本文对含转向点的微分方程边值问题建立了完全指数型拟合差分格式,证明了此格式具有一阶一致收敛性.推广了Miller[1]的方法,简化了证明过程.数值结果表明本格式比Il'in[2]格式要好.     

高精度隐式参差光滑格式的构造

参照Lax-Wendroff格式的构造方法,就双曲型方程、抛物型方程和双曲-抛物型方程,构造了一种新的IRS(implicit residual smoothing)格式。该IRS格式有二阶或三阶时间精度且大大地拓宽了解的稳定区域和CFL数。这种新的IRS格式有中心加权型和迎风偏向型两种,并用von-Neumann分析方法分析了格式的稳定范围。讨论了在透平机械中广泛应用的Dawes方法的局限性,发现该方法对稳态问题得出的解与时间步长的选取有关,对粘性问题求解时,时间步长受严格限制。最后,结合TVD(total variation diminishing)格式和四阶Runge-Kutta技术,用IRS格式和Dawes方法对二维反射激波场进行了数值模拟,数值结果支持本文的分析结论。     

非饱和水流问题的迎风差分法及其数值模拟

本文考察了非饱和水流问题模型方程的守恒型迎风差分法.我们基于有限体积方法建立的非饱和流动的守恒形式,分别提出了一阶和二阶迎风差分格式,并对差分格式进行了误差估计,给出了收敛性定理.最后,数值模拟验证了计算格式的有效性.     

弱有限元方法在线弹性问题中的应用

本文考虑弱有限元（简称WG）方法在线弹性问题中的应用.WG方法是传统有限元方法的推广,用于偏微分方程的数值求解.和传统有限元一样,它的基本思想源于变分原理.WG方法的特点是使用在剖分单元内部和剖分单元边界上分别有定义的分片多项式函数（即弱函数）作为近似函数来逼近真解,并针对弱函数定义相应的弱微分算子代入数值格式进行计算.除此之外,WG方法允许在数值格式中引进稳定子以实现近似函数的弱连续性.WG方法具有允许使用任意多边形或多面体剖分,数值格式与逼近函数构造简单,易于满足相应的稳定性条件等优点.本文考虑WG方法在求解线弹性问题中的应用.围绕线弹性问题数值求解中常见的三个问题,即：数值格式的强制性,闭锁性,应力张量的对称性介绍WG方法在线弹性问题求解中的应用.     

Carleman estimates for the regularization of ill-posed Cauchy problems

This work is a survey of results for ill-posed Cauchy problems for PDEs of the author with co-authors starting from 1991. A universal method of the regularization of these problems is presented here. Even though the idea of this method was previously discussed for specific problems, a universal approach of this paper was not discussed, at least in detail. This approach consists in constructing of such Tikhonov functionals which are generated by unbounded linear operators of those PDEs. The approach is quite general one, since it is applicable to all PDE operators for which Carleman estimates are valid. Three main types of operators of the second order are among them: elliptic, parabolic and hyperbolic ones. The key idea is that convergence rates of minimizers are established using Carleman estimates. Generalizations to nonlinear inverse problems, such as problems of reconstructions of obstacles and coefficient inverse problems are also feasible.     

Estimating the statistical error in calculating velocity and temperature components by the direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is widely used to solve problems in rarefied gas dynamics. In solving steady state problems, a special feature of the method is in using dependent sample values of random variables to calculate the macroparameters of gas flow. In this paper, the possibilities of methods of statistical physics to estimate the statistical error of the DSMC method are theoretically analyzed. A simple approach to approximate estimation of the statistical error in calculating temperature and velocity components is proposed. The approach is tested on a number of problems.     

A global optimization approach for the linear two-level program

Linear two-level programming deals with optimization problems in which the constraint region is implicity determined by another optimization problem. Mathematical programs of this type arise in connection with policy problems to which the Stackelberg leader-follower game is applicable. In this paper, the linear two-level programming problem is restated as a global optimization problem and a new solution method based on this approach is developed. The most important feature of this new method is that it attempts to take full advantage of the structure in the constraints using some recent global optimization techniques. A small example is solved in order to illustrate the approach.The paper was completed while this author was visiting the Department of Mathematics of Linköping University.     

Continuous-Time Generalized Fractional Programming Problems. Part I: Basic Theory

This study, that will be presented as two parts, develops a computational approach to a class of continuous-time generalized fractional programming problems. The parametric method for finite-dimensional generalized fractional programming is extended to problems posed in function spaces. The developed method is a hybrid of the parametric method and discretization approach. In this paper (Part I), some properties of continuous-time optimization problems in parametric form pertaining to continuous-time generalized fractional programming problems are derived. These properties make it possible to develop a computational procedure for continuous-time generalized fractional programming problems. However, it is notoriously difficult to find the exact solutions of continuous-time optimization problems. In the accompanying paper (Part II), a further computational procedure with approximation will be proposed. This procedure will yield bounds on errors introduced by the numerical approximation. In addition, both the size of discretization and the precision of an approximation approach depend on predefined parameters.     

A new approach based on the surrogating method in the project time compression problems

This paper develops a mathematical model for project time compression problems in CPM/PERT type networks. It is noted this formulation of the problem will be an adequate approximation for solving the time compression problem with any continuous and non-increasing time-cost curve. The kind of this model is Mixed Integer Linear Program (MILP) with zero-one variables, and the Benders' decomposition procedure for analyzing this model has been developed. Then this paper proposes a new approach based on the surrogating method for solving these problems. In addition, the required computer programs have been prepared by the author to execute the algorithm. An illustrative example solved by the new algorithm, and two methods are compared by several numerical examples. Computational experience with these data shows the superiority of the new approach.     

Solving the ordered one-median problem in the plane

In this paper, we propose a general approach solution method for the single facility ordered median problem in the plane. All types of weights (non-negative, non-positive, and mixed) are considered. The big triangle small triangle approach is used for the solution. Rigorous and heuristic algorithms are proposed and extensively tested on eight different problems with excellent results.     

Exponentially-fitted Runge-Kutta-Nyström method for the numerical solution of initial-value problems with oscillating solutions

A new approach for constructing efficient Runge-Kutta-Nyström methods is introduced in this paper. Based on this new approach a new exponentially-fitted Runge-Kutta-Nyström fourth-algebraic-order method is obtained for the numerical solution of initial-value problems with oscillating solutions. The new method has an extended interval of periodicity. Numerical illustrations on well-known initial-value problems with oscillating solutions indicate that the new method is more efficient than other ones.     

A novel algorithm based on parameterization method for calculation of curvature of the free surface flows

In this paper, a new approach based on parameterization method is presented for calculation of curvature on the free surface flows. In some phenomena such as droplet and bubble, surface tension is prominent. Therefore in these cases, accurate estimation of the curvature is vital. Volume of fluid (VOF) is a surface capturing method for free surface modeling. In this method, free surface curvature is calculated based on gradient of scalar transport parameter which is regarded as original method in this paper. However, calculation of curvature for a circle and other known geometries based on this method is not accurate. For instance, in practice curvature of a circle in interface cells is constant, while this method predicts different curvatures for it. In this research a novel algorithm based on parameterization method for improvement of the curvature calculation is presented. To show the application of parameterization method, two methods are employed. In the first approach denoted by, three line method, a curve is fitted to the free surface so that the distance between curve and linear interface approximation is minimized. In the second approach namely four point method, a curve is fitted to intersect points with grid lines for central and two neighboring cells. These approaches are treated as calculus of variation problems. Then, using the parameterization method, these cases are converted into the sequences of time-varying nonlinear programming problems. With some treatments a conventional equivalent model is obtained. It is finally proved that the solution of these sequences in the models tends to the solution of the calculus of variation problems. For verification of the presented methods, curvature of some geometrical shapes such as circle, elliptic and sinusoidal profile is calculated and compared with original method used in VOF process and analytical solutions. Finally, as a more practical problem, spurious currents are studied. The results showed that more accurate curve prediction is obtained by these approaches than the original method in VOF approach.     

New Alternating Direction Method for a Class of Nonlinear Variational Inequality Problems

The alternating direction method is an attractive method for a class of variational inequality problems if the subproblems can be solved efficiently. However, solving the subproblems exactly is expensive even when the subproblem is strongly monotone or linear. To overcome this disadvantage, this paper develops a new alternating direction method for cocoercive nonlinear variational inequality problems. To illustrate the performance of this approach, we implement it for traffic assignment problems with fixed demand and for large-scale spatial price equilibrium problems.     

Numerical evaluation of the two-dimensional partition of unity boundary integrals for Helmholtz problems

There has been considerable attention given in recent years to the problem of extending finite and boundary element-based analysis of Helmholtz problems to higher frequencies. One approach is the Partition of Unity Method, which has been applied successfully to boundary integral solutions of Helmholtz problems, providing significant accuracy benefits while simultaneously reducing the required number of degrees of freedom for a given accuracy. These benefits accrue at the cost of the requirement to perform some numerically intensive calculations in order to evaluate boundary integrals of highly oscillatory functions. In this paper we adapt the numerical steepest descent method to evaluate these integrals for two-dimensional problems. The approach is successful in reducing the computational effort for most integrals encountered. The paper includes some numerical features that are important for successful practical implementation of the algorithm.