CONJUGATE-SYMPLECTICITY OF LINEAR MULTISTEP METHODS

For the numerical treatment of Hamiltonian differential equations, symplectic integrators are the most suitable choice, and methods that are conjugate to a symplectic integrator share the same good long-time behavior. This note characterizes linear multistep methods whose underlying one-step method is conjugate to a symplectic integrator. The bounded- hess of parasitic solution components is not addressed.     

Runge—Kutta方法的G—正交性

１Ｇ－正交矩阵微分方程考虑ＲＮ×Ｎ上常微分矩阵方程初值问题这里Ｗ：［０,＋∞）×ＲＮ×Ｎ→ＲＮ×Ｎ为一光滑的映射,Ｙ（０）ＲＮ×Ｎ为给定的初值,Ｇ为实常正定矩阵．定义１．１如果问题（１．１）的真解ｙ（ｔ）满足ＹＴ（ｔ）ＧＹ（ｔ）＝Ｇ,ｔ≥０,则称该问题真解Ｙ（ｔ）是Ｇ－正交的,以下简称该问题是Ｇ－正交的．特别地,当Ｇ＝ＩＮ时,称该问题是正交的,这里ＩＮ为Ｎ×Ｎ单位降．引理１．１［２］问题（１．１）是正交的当且仅当Ｗ（ｔ,Ｙ）＝Ｆ（ｔ,Ｙ）Ｙ,这里Ｆ;［０．＋∞）×ＲＮ×Ｎ→Ｎ×Ｎ为一反对称矩阵函…     

含分布时滞的时滞微分系统多步龙格-库塔方法的时滞相关稳定性

本文研究了一类含分布时滞的时滞微分系统的多步龙格-库塔方法的稳定性.基于辐角原理，本文给出了多步龙格-库塔方法弱时滞相关稳定性的充分条件，并通过数值算例验证了理论结果的有效性.     

多步Runge-Kutta方法的保单调性

一类重要的常微分方程源自用线方法求解非线性双曲型 偏微分方程,这类常微分方程的解具有单调性, 因此要求数值方法能保持原系统的这种性质.本文研究多步Runge-Kutta方法求解常微分方程初值问题的保单调性.分别获得了多步Runge-Kutta方法是条件单调和无条件单调的充分条件.       

CONVERGENCE OF PARALLEL DIAGONAL ITERATION OF RUNGE-KUTTA METHODS FOR DELAY DIFFERENTIAL EQUATIONS

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.     

Schur积多步方法

1.引 言考虑常微方程组其中,y,f∈Rm,Rm表示m维实空间, y0∈Rm为初值. 本文将显式线性多步方法与隐式Euler方法结合起来,构造了如下一类Schur积多步方法.     

THE GPL-STABILITY OF RUNGE-KUTTA METHODS FORDELAY DIFFERENTIAL SYSTEMS

1.IntroductionBeforedealingwitlltilellumcricalstabililtyallalysisoftileIRKInethodsforsystelllsofDDEs,weconsidertilefOllowingillitialvalueproblemIy(t)~f(t,y(t)),t>to,(l)y(to)=yo,(2)wherefisagivenfunctionalldy(t)isunknowllf'Ort>to'FOrtheinitialproblenl(l)--(2),consideranImplicitRunge--Kuttamethod,Km,i=hf(t. cih,aam EailK.,j),i~1,2,...?v,(3)j=1UVhiK.,i,rs~0,1,2,...5(4)yn 1~ac ZbiK.,i,rs~0,1,2,...5(4)i~1whereE:=,hi~1,ci~E;1ail?15i517,yn~y(t.),t.~to ahalldh>0isastepsize.Whenwewallttoanaly…     

ORDER PROPERTIES AND CONSTRUCTION OF SYMPLECTIC RUNGE-KUTTA METHODS

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, …     

D-CONVERGENCE OF RUNGE-KUTTA METHODS FOR STIFF DELAY DIFFERENTIAL EQUATIONS

1. IntroductionWhen considering the applicability of numerical methods for the solution of the delay differential equation (DDE) y'(t) = f(t, y(t), y(t - T)), it is necessary to analyze the error behaviourof the methods. In fact, many papers have investigated the local and global error behaviour ofDDE solvers (cL[1,2,14]). These error analyses are based on the assumption that the fUnctionf(t,y,z) satisfies Lipschitz conditions in both the last two variables. They are suitable fornonstiff …     

D-CONVERGENCE AND STABILITY OF A CLASS OF LINEAR MULTISTEP METHODS FOR NONLINEAR DDES

1. IntroductionConsider the following nonlinear delay problem{:;\f>>:v{t(tf,?,,<,>3,<'~">>,:: 5:,3Ti:,,,,, [l:::;where y: R - C",T > 0 is a delay term, f: [t.,T] x CN x CN - CN and W(t):[to -- T, tol - CN denotes a given initial function. Thoroughout this paper 9 the problem(1.1) is supposed to have a unique solution y(t), which satisfies11 y(')(t) 115 Mi, t e [to ~ T,T]here norm 11. 11 is defined by 11 x II'=< xgx > (Vx E C"), and Mi > 0 are someconstants.Definition 1.1.[1] The clas…     

Runge-Kutta方法关于时滞奇异摄动问题的误差分析

１．引言 用（,）表示Ｅｕｃｌｉｄｅａｎ空间的内积,｜｜·｜｜为相应范数,考虑时滞奇异摄动问题（ＳＰＰＤｓ）这里。∈,ｒ（ｒ＞０）是常数,　和　是给定的函数,ｆ：　　　　　　　　　　　和　　　　　　　　　　　　　　是给定的充分光滑的映射,它们满足下面的条件这里ｗ１和－ｗ２是具有适度大小的常数且　　　　　　　　　分别关于其它变量满足 Ｌｉｐｓｃｈｉｔｚ 条件．不失一般性,假设ｗ２＝１（参见［１］） 与经典 Ｌｉｐｓｃｈｉｔｚ条件相比,条件（１．２ａ）更弱．事实上,当（１．３）中的 Ｌ具有适度大小时,就能…     

SINGLY DIAGONALLY IMPLICIT RUNGE-KUTTA METHODS COMBINING LINE SEARCH TECHNIQUES FOR UNCONSTRAINED OPTIMIZATION

There exists a strong connection between numerical methods for the integration of ordinary differential equations and optimization problems. In this paper, we try to discover further their links. And we transform unconstrained problems to the equivalent ordinary differential equations and construct the LRKOPT method to solve them by combining the second order singly diagonally implicit Runge-Kutta formulas and line search techniques.Moreover we analyze the global convergence and the local convergence of the LRKOPT method. Promising numerical results are also reported.     

一类多步方法求解Banach空间中试验问题的非线性稳定性

本文讨论了一类多步方法求解Banach空间中试验问题类K(μ,λ*,ε)的非线性稳定性, 这一试验问题类的基础是李寿佛[1]引进的试验问题类K(μ,λ*)。我们将证明在Hibert空间中类K(μ,λ*,ε)等价于类K(μ,λ*)。我们给出了试验问题类K(μ,λ*,ε)中微分方程的任何二解之差所满足的不等式,这一结果可看作是李寿佛[1]对试验问题类K(μ,λ*)所获结果的推广。并得到了一类线性多步方法关于K(μ,λ*,ε)(μ为任意实数)类问题的一些稳定性结果．     

Hamilton系统的对称辛广义加性Runge-Kutta 方法

Sandu和Günther[SIAM J.Numer.Anal.53(2015)17--42]对形如$\dot{y}=\sum\limits_{k=1}^{N}f^{[k]}(y)$的微分方程提出广义加性Runge-Kutta (GARK)方法.本文利用双色有根树导出GARK方法的阶条件,给出辛条件和对称性条件,并构造了三个二阶对称辛GARK (SSGARK)方法和两个四阶SSGARK方法.对三个经典测试问题的数值实验结果显示,与文献中几个非对称或非辛的ARK/GARK方法相比,新的SSGARK方法能更有效地保持Hamilton量.     

求解非线性中立型延迟微分方程一类线性多步方法的收敛性

本文致力于带有Lagrang插值的一类线性多步法求解非线性中立型延迟微分方程的误差分析.证明了一个p′阶的线性多步方法配上一个q阶的Lagrang插值导致一个minf[p′,q 1]阶的E-(或EB-)收敛的非线性中立型延迟微分方程数值方法.     

ORDER RESULTS FOR ALGEBRAICALLY STABLE MONO-IMPLICIT RUNGE-KUTTA METHODS

1.IntroductionConsidertheinitialvalueproblemwhichisassumedtohaveauniquesolutiony(t)ontheinterval[0, co).Forsolving(1.1),considerthes--stageimplicitRunge-Kutta(IRK)methodandthes-stagemono-implicitRunge-Kutta(MIRK)method{2,51swhereh)0isthestepsize,hi,c...     

一类求解奇异延迟微分方程的两步连续Runge-Kutta方法的收敛性

本文给出了一类求解延迟落在当前积分步内延迟微分方程的两步连续Runge-Kutta方法。在一定条件下我们证明了方法收敛性,数值试验表明方法是有效的。     

STABILITY ANALYSIS OF RUNGE-KUTTA METHODS FOR NONLINEAR SYSTEMS OF PANTOGRAPH EQUATIONS

This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on （k, l）-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stability conditions for the presented methods are derived.     

CONVERGENCE RESULTS OF RUNGE-KUTTA METHODS FOR MULTIPLY-STIFF SINGULAR PERTURBATION PROBLEMS

AbstractThe main purpose of this paper is to present some convergence results for algebraically stable Runge-Kutta methods applied to some classes of one- and two-parameter multiply-stiff singular perturbation problems whose stiffness is caused by small parameters and some other factors. A numerical example confirms our results.