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1.
1. IntroductionFOr a given s stage Runge-Kutta methodwith A = [ail], p = [pl, PZt... 5 P.]T and ac = [afl, ry23... ) %]T / 0, we introduce thefollowing simplifying conditions as in Butcher [1]and make the notational convensionwhere 1 5 m? pi(x), i ~ 1, 2, 3,'' ? are arbitrarily given i--th polynomials with the property that pi(0) = 0,Note that B(P), C(P) and D(P) are equivalent to BI,. = 0, CI,P = 0 and DI,. = 0respectively. We shall always denote BI,., CI,., DI,. and VI,. by B, …  相似文献   

2.
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.  相似文献   

3.
The paper presents a sixth-order numerical algorithm for studying the completely integrable Camassa-Holm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a sixth-order symplectic Runge-Kutta method and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The sixth-order symplectic Runge-Kutta time integrator is preferable for an equation that possesses a Hamiltonian structure. We illustrate the ability of the proposed scheme by examining examples involving peakon or peakon-like solutions. We compare the computed solutions with exact solutions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accuracy, elapsed computing time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method.  相似文献   

4.
1.IntroductionandPreliminariesLetfibeademainintheorientedEuclideanspaceRZdofpoint(p,q)~((PI,...5pd)",(ql,'5qd)").IfH(P,q)isasufficientlysmoothrealfunctiondefinedinfi,thentheHamiltoniansystemofdifferentialequationswithHamiltonianH(P,q)isgivenbydpiOHdqiOH~~~~~:fi(p,q),}qi~OH~.dtoqidtOPtTheintegerdiscalledthenumberofdegreesoffreedomandfiisthephasespace.HereweassumethatallHamiltoniansconsideredareautonomous,i.e.,time--independent.Definition1.1.Aone-stepmethodiscalledsymplecticif,asappl…  相似文献   

5.
将摄动配置方法应用到含时薛定谔方程,在计算实现的基础上结合摄动配置的特征提出了一类新的数值积分方法,并给出了一个2级2阶和一个3级4阶的辛摄动配置方法对含时薛定谔方程的数值算例.为了检验新的数值积分方法,我们还给出了与两个辛摄动配置格式在理论上等价的辛龙格-库塔方法以及同阶的非辛方法的数值模拟.展示了一些数值结果,并给出了一些分析.  相似文献   

6.
本文旨在提出一种求解带电粒子系统的任意高阶能量守恒格式.在使用能量不变二次化方法将原始哈密顿能量泛函转化为一个二次形式后,辛龙格-库塔方法被用来构造了一种新的能量守恒格式来求解洛伦兹力系统.新格式不仅能保持能量守恒,而且可以达到任意高阶.通过与经典的Boris方法和另一个二阶能量守恒方法对比,数值实验验证了所提算法的显著优越性.  相似文献   

7.
8.
Symplecticness, stability, and asymptotic properties of Runge-Kutta, partitioned Runge-Kutta, and Runge-Kutta-Nystrom methods applied to the simple Hamiltonian system p = -vg, q = kp are studied. Some new results in connection with P-stability are presented. The main part is focused on backward error analysis. The numerical solution produced by a symplectic method with an appropriate stepsize is the exact solution of a perturbed Hamiltonian system at discrete points. This system is studied in detail and new results are derived. Numerical examples are presented.  相似文献   

9.
This paper deals with some relevant properties of Runge–Kutta (RK) methods and symplectic partitioned Runge–Kutta (PRK) methods. First, it is shown that the arithmetic mean of a RK method and its adjoint counterpart is symmetric. Second, the symplectic adjoint method is introduced and a simple way to construct symplectic PRK methods via the symplectic adjoint method is provided. Some relevant properties of the adjoint method and the symplectic adjoint method are discussed. Third, a class of symplectic PRK methods are proposed based on Radau IA, Radau IIA and their adjoint methods. The structure of the PRK methods is similar to that of Lobatto IIIA–IIIB pairs and is of block forms. Finally, some examples of symplectic partitioned Runge–Kutta methods are presented.  相似文献   

10.
Implicit Runge-Kutta method is highly accurate and stable for stiff initial value prob-lem.But the iteration technique used to solve implicit Runge-Kutta method requires lotsof computational efforts.In this paper,we extend the Parallel Diagonal Iterated Runge-Kutta(PDIRK)methods to delay differential equations(DDEs).We give the convergenceregion of PDIRK methods,and analyze the speed of convergence in three parts for theP-stability region of the Runge-Kutta corrector method.Finally,we analysis the speed-upfactor through a numerical experiment.The results show that the PDIRK methods toDDEs are efficient.  相似文献   

11.
当Poisson系统中的Poisson矩阵是非常数时,经典的辛方法如辛Runge_Kutta方法,生成函数法一般不能保持Poisson系统的Poisson结构,利用非线性变换可把非常数Poisson结构转化成辛结构,然后任意阶的辛方法可以长时间计算Poisson系统的辛结构.自由刚体问题中Euler方程被转换成辛结构并用辛中点格式进行数值求解,数值结果给出了这种非线性变换的有效性.  相似文献   

12.
A symplectic reduction method for symplectic G-spaces is given in this paper without usingthe existence of momentum mappings.By a method similar to the above one,the arthorsgive a symplectic reduction method for the Poisson action of Poisson Lie groups on symplecticmanifolds,also without using the existence of momentum mappings.The symplectic reductionmethod for momentum mappings is thus a special case of the above results.  相似文献   

13.
The paper is concerned with construction of symmetric and symplectic Runge-Kutta methods for Hamiltonian systems. Based on the symplectic and symmetrical properties, a sixth-order diagonally implicit symmetric and symplectic Runge-Kutta method with seven stages is presented, the proposed method proved to be P-stable. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing Runge-Kutta methods in scientic literature.  相似文献   

14.
50. IntroductionThe construction and the factorization of harmonic maps from R2 (or its simPlyconnecteddomain) into the uIiltary group U(N) were firstly solved by K.Ulilenbeck in [11, wherethe conception of unitons was iniroduced. Since then various developmenis have beencoatributed[2--5]. Recently, by introducing (singular) Darboux transformations, a purelya1gebraic method to construct harmonic maPs and unitons illto U(N) has been shownin t6'7]. This method can be aIso aPplied to the ca…  相似文献   

15.
A symplectic reduction method for symplectic G-spaces is given in this paper without using the existence of momentum mappings. By a method similar to the above one, the arthors give a symplectic reduction method for the Poisson action of Poisson Lie groups on symplectic manifolds, also without using the existence of momentum mappings. The symplectic reduction method for momentum mappings is thus a special case of the above results.  相似文献   

16.
A modified Runge-Kutta method with minimal phase-lag is developed for the numerical solution of Ordinary Differential Equations with oscillating solutions. The method is based on the accurate Runge-Kutta method of Sharp and Smart RK4SS(5) (see [1]) of order five. Numerical and theoretical results show that this new approach is more efficient, compared with the fifth order Runge-Kutta Sharp and Smart method.  相似文献   

17.
用龙格-库塔法求解非线性方程组   总被引:2,自引:0,他引:2  
本文介绍了一种求解非线性方程组的新方法龙格-库塔法。  相似文献   

18.
The aim of this paper is to design a new family of numerical methods of arbitrarily high order for systems of first-order differential equations which are to be termed pseudo two-step Runge-Kutta methods. By using collocation techniques, we can obtain an arbitrarily high-order stable pseudo two-step Runge-Kutta method with any desired number of implicit stages in retaining the two-step nature. In very first investigations, the pseudo two-step Runge-Kutta methods are shown to be promising numerical integration methods.AMS(MOS) subject classifications (1991) 65M12 65M20CR subject classifications G.1.7This work was partly supported by DAAD, N.R.P.F.S. and QG-96-02  相似文献   

19.
Total variation diminishing Runge-Kutta schemes   总被引:14,自引:0,他引:14  
In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

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20.
精细辛几何算法的误差估计   总被引:1,自引:0,他引:1       下载免费PDF全文
该文讨论了精细辛几何算法的计算误差,先展开二阶和四阶精细辛几何算法的表达式得到误差同精细剖分数目的关系,然后分析了任意阶精细辛几何算法的误差,得到了一致简洁的结果,总的误差可近似表示为单个精细步长的误差乘以剖分数目,最后讨论了在要求控制精度下剖分数目的选取,该方法克服了算法精度对积分时间步长的依赖性.  相似文献   

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