Symplectic Partitioned Runge-Kutta Methods

For partitioned Runge-Kutta methods, in the integration of Hamiltonian systems, a condition for symplecticness and its characterization which is based on the W-transformation of Hairer and Wanner are presented. Examples for partitioned Runge-Kutta methods which satisfy the symplecticness condition are given. A special class of symplectic partitioned Runge-Kutta methods is constructed.     

A Generalized W-Transformation for Constructing Symplectic Partitioned Runge-Kutta Methods

This paper deals with the construction of implicit symplectic partitioned Runge–Kutta methods (PRKM) of high order for separable and general partitioned Hamiltonian systems. The main tool is a generalized W-transformation for PRKM based on different quadrature formulas. Methods of high order and special properties can be determined using the transformed coefficient matrices. Examples are given.     

Symplectic Runge-Kutta methods with real eigenvalues

Iserles [1] constructed symplectic Runge-Kutta methods with real eigenvalues with the help of perturbed collocation. This note shows that such methods can comfortably be obtained using theW-transformation of [2].     

关于高阶Runge-Kutta方法的对称性及藕对性

借助于Haier及Wanner的W变换,研究高阶Runge－Kutta方法的表征,简化了SunGeng定理的条件．     

Symplectic Runge-Kutta methods of high order based on W-transformation

In this paper , characterizations of symmetric and symplectic Runge-Kutta methods based on the W-transformation of Hairer and Wanner are presented. Using these characterizations, we construct two families symplectic (symmetric and algebraically stable or algebraically stable) Runge-Kutta methods of high order. Methods constructed in this way and presented in this paper include and extend the known classes of high order implicit Runge-Kutta methods.     

Non-Minimal Runge-Kutta Methods

Results in Mathematics -     

Symplectic RK Methods and Symplectic PRK Methods with Real Eigenvalues

Properties of symplectic Runge-Kutta (RK) methods and symplectic partitioned Runge-Kutta (PRK) methods with real eigenvalues are discussed in this paper. It is shown that an s stage such method can‘t reach order more than s 1. Particularly, we prove that no symplectic RK method with real eigenvalues exists in stage s of order s 1 when s is even. But an example constructed by using the W-transformation shows that PRK method of this type does not necessarily meet this order barrier. Another useful way other than W-transformation to construct symplectic PRK method with real eigenvalues is then presented. Finally, a class of efficient symplectic methods is recommended.     

Order conditions for partitioned Runge-Kutta methods

We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations.     

改进的两步高阶Euler-Chebyshev方法

给出非线性方程求根的Euler-Chebyshev方法的改进方法,证明了方法的收敛性,它们七次和九次收敛到单根.给出数值试验,且与牛顿法及其它较高阶的方程求根方法做了比较.结果表明方法具有很好的优越性,它丰富了非线性方程求根的方法,在理论上和应用上都有一定的价值.     

Multirate Partitioned Runge-Kutta Methods

The coupling of subsystems in a hierarchical modelling approach leads to different time constants in the dynamical simulation of technical systems. Multirate schemes exploit the different time scales by using different time steps for the subsystems. The stiffness of the system or at least of some subsystems in chemical reaction kinetics or network analysis, for example, forbids the use of explicit integration schemes. To cope with stiff problems, we introduce multirate schemes based on partitioned Runge—Kutta methods which avoid the coupling between active and latent components based on interpolating and extrapolating state variables. Order conditions and test results for such a lower order MPRK method are presented.This revised version was published online in October 2005 with corrections to the Cover Date.     

传统隐式Runge-Kutta方法的转换方法

多级隐式Runge-Kutta(RK)方法簇中,除Gauss类方法是s级2s阶的辛方法以外,Radau类方法和Lobatto类方法既不是s级2s阶的方法也不是辛方法.基于隐式RK方法是一类转换RK方法这一特征,利用V-变换和Pade对角逼近,提出了构造高阶RK方法的转换定理.依据转换定理,导出了s级2s阶的Radau方法和s级2s阶的Lobatto方法.利用V-变换和待定系数法,导出了辛Radau方法和辛Lobatto方法.在此基础上,发现并证明了辛Radau方法是s级2s阶的方法.     

16阶二次幻方的膨胀构造法

2017年詹森构造了 6个异基因的8阶二次幻方兼完美幻方,根据它们的特殊性质,创立用一个4阶矩阵代替原有元素的膨胀法,构造出16阶二次幻方兼完美幻方;并对另外2个具有相似性质的8阶二次幻方,也通过膨胀法构造出了 16阶二次幻方.     

High Order Transformation Methods for Curved Finite Elements

The problem of producing basis functions for curved elementsin the finite element method is discussed. An alternative methodto the Rational Basis Functions method and the IsoparametricTransformation method is given. This method, called a High OrderTransformation method, allows any degree of approximation, likethe Rational Basis Function method, while retaining the simplicityof the Isoparametric method. A discussion of the calculationof typical integrals for the case of the three-sided elementwith two straight sides and one conic arc is given.     

高次Galerkin有限元法的超收敛性

通过局部误差估计,对具有光滑边界的二阶常系数椭圆型方程,给出了高次Ｇａｌｅｒｋｉｎ 有限元法的超收敛性． 运用对称技巧和积分恒等式技巧,在局部对称矩形网格或三角形网格上,我们得到了改进的超收敛性（提高１－ ３ 阶）．     

高阶偏微分方程的算子方法

本文建立了偏微分方程[δ2/δt^2-a^2P(δX)]^mu=f(x,t),其中m≥1初值问题显式解,并且用算子sh/(tP(δx)1/2)/p(δx)1^1/2表示出来,得到了能把表示成给定函数的积分形式的一解公式。     

高阶偏微分方程与概率方法

二阶偏微分方程与扩散过程的是概率界众所财知的。前者为后者提供了分析依据，后者为前者的解给出了概率表示；如何把这种联系推广到高阶偏微分方程的情形，是很多概率学家近十几年来一直关心的问题。     

Landau-Ginzburg-Higgs方程的多辛Runge-Kutta方法

非线性波动方程作为一类重要的数学物理方程吸引着众多的研究者,基于Hamilton空间体系的多辛理论研究了Landau-Ginzburg-Higgs方程的多辛算法,讨论了利用Runge-Kutta方法构造离散多辛格式的途径,并构造了一种典型的半隐式的多辛格式,该格式满足多辛守恒律、局部能量守恒律和局部动量守恒律.数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性.     

Runge-Kutta Methods with Global Error Estimates

In the numerical solution of ordinary differential initial valueproblems, it is desirable to have a global error estimate. Inthis paper the practical use of a global error estimate introducedby H. J. Stetter is discussed, and specific procedures involvingexplicit Runge-Kutta methods, which effect a certain savingin function evaluations, are presented. Some numerical resultsshow the accuracy of the estimate.     

高阶双参量Runge-Kutta方法

构造了仅由两个参量确定的方法类RK,（μ,δ）,一切节点属于区间［0,1］且至少2s－1阶相容的s级RK的方法,如Radau|A,Radau||A,Gauss方法等,均是其特例．此类方法的代数稳定性与A-稳定性均等价于参量的μ的非负性,这一准则改进了Burrage的如下结论：一个满足简化条件B（s）和C（s）的s级RK的方法代数稳定的必要条件是它至少2s—1阶相容．基于此类方法构造了高阶指数拟合的RK公式,且公式是代数稳定的,因而适于求解非线性stiff问题．特别,当用k（k＞1）步方法求解stiff问题时,用拟会得当的RK公式确定k－1个附加初值是行之有效的．     

High order embedded Runge-Kutta formulae

The criteria to be satisfied by embedded Runge-Kutta pairs of formulae are reviewed. Two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use.