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1.
For dissipative differential equations y' = f (y) it is knownthat contractivity of the exact solution is reproduced by algebraicallystable RungeKutta methods. In this paper we investigatewhether a different property of the exact solution also holdsfor RungeKutta solutions. This property, called equilibriumattractivity, means that the norm of the righthand side f neverincreases. It is a property dual to algebraic stability sinceneither is sufficient for the other, in general. We derive sufficientalgebraic conditions for RungeKutta methods and proveequilibrium attractivity of the high-order algebraically stableRadau-IIA and Lobatto-IIIC methods and the Lobatto-IIIA collocationmethods (which are not algebraically stable). No smoothnessassumptions on f and no stepsize restrictions are required but,except for some simple cases, f has to satisfy certain additionalproperties which are generalizations of the simple one-sidedLipschitz condition using more than two argument points. Thesemultipoint conditions are discussed in detail. 相似文献
2.
An implicit RungeKutta method, applied to an initial-valueproblem, gives systems of algebraic equations. It is shown that,under natural assumptions concerning the differential system,these equations have unique solutions if the method satisfiesa condition related to algebraic stability. In particular, thiscondition is satisfied if the method is irreducible and (k,l)-algebraically stable for some l 0. 相似文献
3.
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
4.
An analysis of global error estimation for RungeKuttasolutions of ordinary differential equations is presented. Thebasic technique is that of Zadunaisky in which the global erroris computed from a numerical solution of a neighbouring problemrelated to the main problem by some method of interpolation.It is shown that RungeKutta formulae which permit validglobal error estimation using low-degree interpolation can bedeveloped, thus leading to more accurate and computationallyconvenient algorithms than was hitherto expected. Some specialRungeKutta processes up to order 4 are presented togetherwith numerical results. 相似文献
5.
van der HOUWEN P. J.; WOLKENFELT P. H. M.; BAKER C. T. H. 《IMA Journal of Numerical Analysis》1981,1(3):303-328
In this paper modified and conventional RungeKutta methodsfor second kind Volterra integral equations are discussed ina uniform way. The modification presented takes into accountthe residual of the previous step with the aim of improvingthe stability behaviour. A general convergence theorem is givenwhich establishes that the modified methods may lose one orderof accuracy. Furthermore, the stability behaviour of the methodsis analysed and explicit stability results are derived. It transpiresthat every A-stable RungeKutta method for ordinary differentialequations generates mixed methods which can be made A-stableby a suitable modification. 相似文献
6.
An analysis of global error estimation using the Zadunaiskytechnique with RungeKutta methods is presented. Threeforms of interpolant which can lead to valid asymptotic estimationare considered. Test results indicate that the Hermite formcoupled with special RungeKutta formulae is to be preferred,particularly when two-term global error estimation can be obtained.Very reliable estimation can be achieved and it is suggestedthat the technique could form the basis of a production code. 相似文献
7.
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
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.
J.C. Butcher 《Numerical Algorithms》1998,17(3-4):193-221
Almost Runge–Kutta methods (or “ARK methods”) have many of the advantages of Runge–Kutta methods but, for many problems, are
capable of greater accuracy. In this paper a complete classification of fourth order ARK methods with 4 stages is presented.
The paper also analyzes fifth order methods with 5 or with 6 stages. Some limited numerical experiments show that the new
methods are capable of excellent performance, comparable to that of known highly efficient Runge–Kutta methods.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
10.
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
11.
Diagonally Implicit RungeKutta (DIRK) methods are developedand applied to differentialalgebraic systems arisingfrom dynamic process simulation. In particular, an embeddedfamily of DIRK methods is developed for implementation as avariable-step variable-order algorithm. The methods developedallow easy assessment of local solution error as well as theability to change the order of approximation. The stabilityproperties of the methods are chosen to make them suitable foruse on stiff systems. Some important aspects of implementation of DIRK methods arediscussed within the context of the solution of differentialalgebraicsystems. The performance of this algorithm is compared withan alternative variable-order approach based on "triples" whichallows the patching together of several fixed-order formulae.The results indicate that the fully embedded DIRK algorithmis generally more efficient than the algorithm based on "triples".Areas of further investigation in the context of differentialalgebraicsystems are outlined. 相似文献
12.
An error analysis is given for convolution quadratures based on strongly A-stable Runge–Kutta methods, for the non-sectorial
case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation
depends on the classical order and stage order of the Runge–Kutta method and on the growth exponent of the Laplace transform.
Numerical experiments with convolution quadratures based on the Radau IIA methods are given on an example of a time-domain
boundary integral operator. 相似文献
13.
In this article, we study positivity properties of exponential Runge–Kutta methods for abstract evolution equations. Our problem
class includes linear ordinary differential equations with a time-dependent inhomogeneity. We show that the order of a positive
exponential Runge–Kutta method cannot exceed two. On the other hand there exist second-order methods that preserve positivity
for linear problems. We give some examples for the latter. 相似文献
14.
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
15.
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
16.
An error analysis of Runge–Kutta convolution quadrature is presented for a class of non-sectorial operators whose Laplace
transform satisfies, besides the standard assumptions of analyticity in a half-plane Re s > σ
0 and a polynomial bound
\operatornameO(|s|m1){\operatorname{O}(|s|^{\mu_1})} there, the stronger polynomial bound
\operatornameO(sm2){\operatorname{O}(s^{\mu_2})} in convex sectors of the form
|\operatorname*arg s| £ p/2-q{|\operatorname*{arg} s| \leq \pi/2-\theta} for θ > 0. The order of convergence of the Runge–Kutta convolution quadrature is determined by μ
2 and the underlying Runge–Kutta method, but is independent of μ
1. Time domain boundary integral operators for wave propagation problems have Laplace transforms that satisfy bounds of the
above type. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour
of Runge–Kutta convolution quadrature for this class of applications. Our results show in particular that the full classical
order of the Runge–Kutta method is attained away from the scattering boundary. 相似文献
17.
K. J. In 't Hout 《BIT Numerical Mathematics》2001,41(2):322-344
This paper deals with the adaptation of Runge—Kutta methods to the numerical solution of nonstiff initial value problems for delay differential equations. We consider the interpolation procedure that was proposed in In 't Hout [8], and prove the new and positive result that for any given Runge—Kutta method its adaptation to delay differential equations by means of this interpolation procedure has an order of convergence equal to min {p,q}, where p denotes the order of consistency of the Runge—Kutta method and q is the number of support points of the interpolation procedure.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
18.
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
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.
The De Donder–Weyl (DW) Hamilton–Jacobi equation is investigated in this paper, and the connection between the DW Hamilton–Jacobi
equation and multi-symplectic Hamiltonian system is established. Based on the DW Hamilton–Jacobi theory, generating functions
for multi-symplectic Runge–Kutta (RK) methods and partitioned Runge–Kutta (PRK) methods are presented.
The work is supported by the Foundation of ICMSEC, LSEC, AMSS and CAS, the NNSFC (No.10501050, 19971089 and 10371128) and
the Special Funds for Major State Basic Research Projects of China (2005CB321701). 相似文献