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1.
A One-step Method of Order 10 for y' = f(x, y) 总被引:1,自引:0,他引:1
In some situations, especially if one demands the solution ofthe differential equation with a great precision, it is preferableto use high-order methods. The methods considered here are similarto RungeKutta methods, but for the second-order equationy'= f(x, y). As for RungeKutta methods, the complexityof the order conditions grows rapidly with the order, so thatwe have to solve a nonlinear system of 440 algebraicequations to obtain a tenthorder method. We demonstratehow this system can be solved. Finally we give the coefficients(20 decimals) of two methods with small local truncation errors. 相似文献
2.
Summary. This paper studies the convergence properties of general Runge–Kutta methods when applied to the numerical solution of a
special class of stiff non linear initial value problems. It is proved that under weaker assumptions on the coefficients of
a Runge–Kutta method than in the standard theory of B-convergence, it is possible to ensure the convergence of the method
for stiff non linear systems belonging to the above mentioned class. Thus, it is shown that some methods which are not algebraically
stable, like the Lobatto IIIA or A-stable SIRK methods, are convergent for the class of stiff problems under consideration.
Finally, some results on the existence and uniqueness of the Runge–Kutta solution are also presented.
Received November 18, 1996 / Revised version received October 6, 1997 相似文献
3.
Order stars and stability for delay differential equations 总被引:3,自引:0,他引:3
Summary. We consider Runge–Kutta methods applied to delay differential equations with real a and b. If the numerical solution tends to zero whenever the exact solution does, the method is called -stable. Using the theory of order stars we characterize high-order symmetric methods with this property. In particular, we
prove that all Gauss methods are -stable. Furthermore, we present sufficient conditions and we give evidence that also the Radau methods are -stable. We conclude this article with some comments on the case where a andb are complex numbers.
Received June 3, 1998 / Published online: July 7, 1999 相似文献
4.
Andreas Rößler 《BIT Numerical Mathematics》2007,47(3):657-680
The weak approximation of the solution of a system of Stratonovich stochastic differential equations with a m–dimensional Wiener process is studied. Therefore, a new class of stochastic Runge–Kutta methods is introduced. As the main
novelty, the number of stages does not depend on the dimension m of the driving Wiener process which reduces the computational effort significantly. The colored rooted tree analysis due
to the author is applied to determine order conditions for the new stochastic Runge–Kutta methods assuring convergence with
order two in the weak sense. Further, some coefficients for second order stochastic Runge–Kutta schemes are calculated explicitly.
AMS subject classification (2000) 65C30, 65L06, 60H35, 60H10 相似文献
5.
In this paper, a new class of Runge–Kutta methods is introduced. Some basic properties of this subgroup of algebraically stable methods are presented and a complete parametric representation is given. Necessary and sufficient order conditions for lower order methods as well as sufficient order conditions for higher order methods are derived yielding a significantly reduced number of conditions when compared with general Runge–Kutta methods. Design examples conclude this paper. 相似文献
6.
Georgios E. Zouraris 《Numerische Mathematik》1997,77(1):123-142
Summary. We analyze a class of algebraically stable Runge–Kutta/standard Galerkin methods for inhomogeneous linear parabolic equations,
with time–dependent coefficients, under Neumann boundary conditions, and derive an error bound of provided is bounded.
Received June 25, 1994 / Revised version received February 26, 1996 相似文献
7.
J. G. Verwer 《Numerische Mathematik》2009,112(3):485-507
We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous
wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated
in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer
efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special
attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.
相似文献
8.
Chengming Huang 《Numerische Mathematik》2009,111(3):377-387
This paper is concerned with the study of the delay-dependent stability of Runge–Kutta methods for delay differential equations.
First, a new sufficient and necessary condition is given for the asymptotic stability of analytical solution. Then, based
on this condition, we establish a relationship between τ(0)-stability and the boundary locus of the stability region of numerical methods for ordinary differential equations. Consequently,
a class of high order Runge–Kutta methods are proved to be τ(0)-stable. In particular, the τ(0)-stability of the Radau IIA methods is proved. 相似文献
9.
Tree and forest spaces, which are at the heart of the theory of Runge–Kutta methods, are formulated recursively, and it is
shown that the forest space is an algebra. To obtain order conditions in a systematic manner, Banach algebras are introduced
to generate both the elementary weights for a general Runge–Kutta method and the corresponding quantities based on the Picard
integral. To connect these two concepts, the Picard integral is written as the limiting case of an s-stage Runge–Kutta method, equivalent to s steps of the Euler method, as s tends to infinity. This approach makes it possible to make direct use of the tree space without going over to the dual space.
By choosing linear combinations of trees, appropriate to a particular application, it is shown how to obtain alternative ways
of writing the order conditions. This leads to a simpler and more direct derivation of particular methods. 相似文献
10.
** Email: kzhang{at}math.cuhk.edu.hk The non-linear wave equation is taken as a model problem forthe investigation. Different multi-symplectic reformulationsof the equation are discussed. Multi-symplectic RungeKuttamethods and multi-symplectic partitioned RungeKutta methodsare explored based on these different reformulations. Some popularand efficient multi-symplectic schemes are collected and constructed.Stability analyses are performed for these schemes. 相似文献
11.
In our previous paper [3], the performance of a variable step‐size implementation of Parallel Iterated Methods based on Multistep
Runge–Kutta methods (PIMRK) is far from satisfactory. This is due to the fact that the underlying parameters of the Multistep
Runge–Kutta (MRK) method, and the splitting matrices W that are needed to solve the nonlinear system are designed on a fixed step‐size basis. Similar unsatisfactory results based
on this method were also noted by Schneider [12], who showed that the method is only suitable when the step‐size does not
vary too often. In this paper, we design the Variable step‐size Multistep Runge–Kutta (VMRK) method as the underlying formula
for Parallel Iterated methods. The numerical results show that Parallel Iterated Variable step‐size MRK (PIVMRK) methods improve
substantially on the PIMRK methods and are usually competitive with Parallel Iterated Runge–Kutta methods (PIRKs).
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
12.
Using the property of inherent Runge—Kutta stability, it is possible to construct diagonally implicit general linear methods with stability regions exactly the same as for Runge—Kutta methods. In addition to A-stable methods found in this way, it is also possible to construct explicit methods with stability regions identical to those of explicit Runge—Kutta methods. The use of doubly companion matrices makes it possible to find all explicit and diagonally-implicit methods possessing the inherent Runge—Kutta stability property.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
13.
W.M. Wright 《Numerical Algorithms》2002,31(1-4):381-399
A class of general linear methods is derived for application to non-stiff ordinary differential equations. A property known as inherent Runge–Kutta stability guarantees the stability regions of these methods are the same as for Runge–Kutta methods. Methods with this property have high stage order which enables asymptotically correct error estimates and high order interpolants to be computed conveniently. Some preliminary numerical experiments are given comparing these methods with some well known Runge–Kutta methods. 相似文献
14.
The successful use of mono-implicit Runge—Kutta methods has been demonstrated by several researchers who have employed these methods in software packages for the numerical solution of boundary value ordinary differential equations. However, these methods are only applicable to first order systems of equations while many boundary value systems involve higher order equations. While it is straightforward to convert such systems to first order, several advantages, including substantial gains in efficiency, higher continuity of the approximate solution, and lower storage requirements, are realized when the equations can be treated in their original higher order form. In this paper, we consider generalizations of mono-implicit Runge—Kutta methods, called mono-implicit Runge—Kutta—Nyström methods, suitable for systems of second order ordinary differential equations having the general form, y(t) = f(t,y(t),y(t)), and derive optimal symmetric methods of orders two, four, and six. We also introduce continuous mono-implicit Runge—Kutta—Nyström methods which allow us to provide continuous solution and derivative approximations. Numerical results are included to demonstrate the effectiveness of these methods; savings of 65% are attained in some instances.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
15.
Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non
necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous
one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward
Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity,
of the exact solution. Numerical illustrations are provided. 相似文献
16.
We construct A‐stable and L‐stable diagonally implicit Runge–Kutta methods of which the diagonal vector in the Butcher matrix
has a minimal maximum norm. If the implicit Runge–Kutta relations are iteratively solved by means of the approximately factorized
Newton process, then such iterated Runge–Kutta methods are suitable methods for integrating shallow water problems in the
sense that the stability boundary is relatively large and that the usually quite fine vertical resolution of the discretized
spatial domain is not involved in the stability condition.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
17.
This paper is concerned with the numerical dissipativity of a class of nonlinear neutral delay integro-differential equations. The dissipativity results are obtained for algebraically stable Runge–Kutta methods when they are applied to above problems. 相似文献
18.
When dealing with implicit RungeKutta methods, the equationsdefining the stages are usually solved by iterative methods.The closer the first iterate is to the solution, the fewer iterationsare required. In this paper the author presents and analysesnew high order algorithms to compute such initial iterates.Numerical experiments are given to illustrate the performanceof the new procedures when combined with a variable-step symplecticintegrator. 相似文献
19.
General Linear Methods (GLMs) were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep
methods. An extension of GLMs, so-called SGLMs (GLM with second derivative), was introduced to the case in which second derivatives,
as well as first derivatives, can be calculated. In this paper, we introduce the definitions of consistency, stability and
convergence for an SGLM. It will be shown that in SGLMs, stability and consistency together are equivalent to convergence.
Also, by introducing a subclass of SGLMs, we construct methods of this subclass up to the maximal order which possess Runge–Kutta
stability property and A-stability for implicit ones. 相似文献
20.
We investigate conservative properties of Runge–Kutta methods for Hamiltonian partial differential equations. It is shown
that multi-symplecitic Runge–Kutta methods preserve precisely the norm square conservation law. Based on the study of accuracy
of Runge–Kutta methods applied to ordinary and partial differential equations, we present some results on the numerical accuracy
of conservation laws of energy and momentum for Hamiltonian PDEs under Runge–Kutta discretizations.
J. Hong, S. Jiang and C. Li are supported by the Director Innovation Foundation of ICMSEC and AMSS, the Foundation of CAS,
the NNSFC (No. 19971089, No. 10371128, No. 60771054) and the Special Funds for Major State Basic Research Projects of China
2005CB321701. 相似文献