Numerical methods for Hamilton Jacobi functional differential equations

Initial and initial boundary value problems for first order partial functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are construted.     

Expanding stability regions of explicit advective-diffusive finite difference methods by Jacobi preconditioning

Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel. The disadvantage of explicit methods is the severe restrictions that are placed on stable time-step intervals. Stability bounds for explicit time differencing methods on advective–diffusive problems are generally determined by the diffusive part of the problem. These bounds are very small and implicit methods are used instead. The linear systems arising from these implicit methods are generally solved by iterative methods. In this article we develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods. A Jacobi preconditioned time differencing method is defined and analyzed for both diffusion and advection–diffusion equations. Several computational examples of both linear and nonlinear advective-diffusive problems are solved to demonstrate the accuracy and improved stability limits. © 1995 John Wiley & Sons, Inc.     

The global Hessenberg and CMRH methods for linear systems with multiple right-hand sides

In this paper, we introduce two new methods for solving large sparse nonsymmetric linear systems with several right-hand sides. These methods are the global Hessenberg and global CMRH methods. Using the global Hessenberg process, these methods are less expensive than the global FOM and global GMRES methods [9]. Theoretical results about the new methods are given, and experimental results that show good performances of these new methods are presented.     

几类改进的新的两步六阶Chebyshev-Halley方法

利用权函数法,给出非线性方程求根的Chebyshev-Halley方法的几类改进方法,证明方法六阶收敛到单根.Chebyshev-Halley方法的效率指数为1.442,改进后的两步方法的效率指数为1.565.最后给出数值试验,且与牛顿法,Chebyshev-Halley 方法及其它已知的方程求根方法做了比较.结果表明方法具有一定的优越性.     

Mehar’s methods for fuzzy assignment problems with restrictions

In this paper, limitations of existing methods [5, 11] for solving fuzzy assignment problems (FAPs) are pointed out. In order to overcome the limitations of existing methods, two new methods named Mehar’s methods are proposed. To show the advantages of Mehar’s methods over existing methods, some FAPs are solved. The Mehar’s methods can solve the problems solved by existing methods as well as those which cannot be solved by existing methods.     

求解界约束优化问题的有效集算法综述

主要介绍了求解界约束优化问题的有效集方法,包括投影共轭梯度法和有效集识别函数法,讨论了各自的优点和不足.最后,指出了有效集法的研究趋势及应用前景.     

多组变量典型相关分析SSQCOR准则的数值求解

本文研究多组变量相关分析SSQCOR.准则的数值方法.从KKT条件出发引入了Gauss-Seidel型方法,从SSQCOR出发引入了交替变量法.证明了前者是后者的非精确形式,都具有单调上升性.为了提高得到全局解的可能性,引入了初始点策略.用实际数据和模拟数据进行了数值试验以说明算法的有效性.     

New Multivalue Methods for Differential Algebraic Equations

Multivalue methods are slightly different from the general linear methods John Butcher proposed over 30 years ago. Multivalue methods capable of solving differential algebraic equations have not been developed. In this paper, we have constructed three new multivalue methods for solving DAEs of index 1, 2 or 3, which include multistep methods and multistage methods as special cases. The concept of stiff accuracy will be introduced and convergence results will be given based on the stage order of the methods. These new methods have the diagonal implicit property and thus are cheap to implement and will have order 2 or more for both the differential and algebraic components. We have implemented these methods with fixed step size and they are shown to be very successful on a variety of problems. Some numerical experiments with these methods are presented.     

一般线性方法的线性与非线性稳定性间的关系

１．引言１９６３年,Ｄａｈｌｑｕｉｓｔ以一类线性问题为模型提出了Ａ－稳定性概念,此后有关如何判断方法是否Ａ－稳定或确定其稳定域的研究十分活跃,袁兆鼎等~［１］中对此有详细的讨论．Ｂｕｒｒａｇｅ与Ｂｕｔｃｈｅｒ［２］以一类非线性问题为模型,就一般线性方法引入了代数稳定性概念．Ｂｕｔｃｈｅｒ［４］探讨了代数稳定性与Ａ－稳定性间的内在联系．为确保代数稳定性蕴涵Ａ－稳定性,Ｂｕｔｃｈｅｒ［５］进一步要求代数稳定性定义中涉及的矩阵Ｇ是正定的。然而这样一来,正如李寿佛［６］中指出的那样,许多ＡＮ稳定且按［４…     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

A REGULARIZED CONJUGATE GRADIENT METHOD FOR SYMMETRIC POSITIVE DEFINITE SYSTEM OF LINEAR EQUATIONS

AbstractA class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.     

时滞积分微分方程的Rosenbrock方法

本文研究时滞积分微分方程的数值方法.通过改造现有常及离散型延迟微分方程的数值方法,并匹配以适当数值求积公式,构造了求解时滞积分微分方程的Rosenbrock方法,导出了其稳定性准则.数值例子阐明了所获方法的计算有效性.     

Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.     

Applications of doubly companion matrices

Doubly companion matrices are used as a tool to analyze the ESIRK and DESIRE methods, and the general linear methods satisfying the IRKS property. These methods can be considered as extensions of the DIRK and SIRK Runge–Kutta methods, and were introduced to overcome some of the undesirable properties of these methods. A connection between the ESIRK and DESIRE methods, and IRKS methods is also established. Several low order methods are given.     

Delay-dependent stability analysis of multistep methods for delay differential equations

This paper deals with the delay-dependent stability of numerical methods for delay differential equations. First, a stability criterion of Runge-Kutta methods is extended to the case of general linear methods. Then, linear multistep methods are considered and a class of r（0）-stable methods are found. Later, some examples of r（0）-stable multistep multistage methods are given. Finally, numerical experiments are presented to confirm the theoretical results.     

A note on some improvements of the simultaneous methods for determination of polynomial zeros

Applying Gauss-Seidel approach to the improvements of two simultaneous methods for finding polynomial zeros, presented in [9], two iterative methods with faster convergence are obtained. The lower bounds of the R-order of convergence for the accelerated methods are given. The improved methods and their accelerated modifications are discussed in view of the convergence order and the number of numerical operations. The considered methods are illustrated numerically in the example of an algebraic equation.     

Direct prediction methods in Hilbert space with applications to control problems

In this paper, the convergence of variable-metric methods without line searches (direct prediction methods) applied to quadratic functionals on a Hilbert space is established. The methods are then applied to certain control problems with both free endpoints and fixed endpoints. Computational results are reported and compared with earlier results. The methods discussed here are found to compare favorably with earlier methods involving line searches and with other direct prediction quasi-Newton methods.     

New eighth-order iterative methods for solving nonlinear equations

In this paper, three new families of eighth-order iterative methods for solving simple roots of nonlinear equations are developed by using weight function methods. Per iteration these iterative methods require three evaluations of the function and one evaluation of the first derivative. This implies that the efficiency index of the developed methods is 1.682, which is optimal according to Kung and Traub’s conjecture [7] for four function evaluations per iteration. Notice that Bi et al.’s method in [2] and [3] are special cases of the developed families of methods. In this study, several new examples of eighth-order methods with efficiency index 1.682 are provided after the development of each family of methods. Numerical comparisons are made with several other existing methods to show the performance of the presented methods.     

无约束优化的一类共轭梯度法(英)

本文给出了一类具有4个参数的共轭梯度法，并且分析了其中两个子类的方法．证明了在步长满足更一般的Wolfe条件时，这两个子类的方法是下降算法．同时还证明了这两个子类算法的全局收敛性．     

A simple technique for constructing two-step Runge-Kutta methods

A technique is proposed for constructing two-step Runge-Kutta methods on the basis of one-step methods. Explicit and diagonally implicit two-step methods with the second or third stage order are examined. Test problems are presented showing that the proposed methods are superior to conventional one-step techniques.