Adjoint and Selfadjoint Lie-group Methods

In the past few years, a number of Lie-group methods based on Runge—Kutta schemes have been proposed. One might extrapolate that using a selfadjoint Runge—Kutta scheme yields a Lie-group selfadjoint scheme, but this is generally not the case: Lie-group methods depend on the choice of a coordinate chart which might fail to comply to selfadjointness.In this paper we discuss Lie-group methods and their dependence on centering coordinate charts. The definition of the adjoint of a numerical method is thus subordinate to the method itself and the choice of the chart. We study Lie-group numerical methods and their adjoints, and define selfadjoint numerical methods. The latter are defined in terms of classical selfadjoint Runge—Kutta schemes and symmetric coordinates, based on geodesic or on flow midpoint. As result, the proposed selfadjoint Lie-group numerical schemes obey time-symmetry both for linear and nonlinear problems.This revised version was published online in October 2005 with corrections to the Cover Date.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Construction of Runge–Kutta methods of Crouch–Grossman type of high order

An approach is described to the numerical solution of order conditions for Runge–Kutta methods whose solutions evolve on a given manifold. This approach is based on least squares minimization using the Levenberg–Marquardt algorithm. Methods of order four and five are constructed and numerical experiments are presented which confirm that the derived methods have the expected order of accuracy.     

Convergence of Runge-Kutta Methods for Delay Differential Equations

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.     

Cost Efficient Lie Group Integrators in the RKMK Class

In this work a systematic procedure is implemented in order to minimise the computational cost of the Runge—Kutta—Munthe-Kaas (RKMK) class of Lie-group solvers. The process consists of the application of a linear transformation to the stages of the method and the analysis of a graded free Lie algebra to reduce the number of commutators involved. We consider here RKMK integration methods up to order seven based on some of the most popular Runge—Kutta schemes.This revised version was published online in October 2005 with corrections to the Cover Date.     

Mono-Implicit Runge-Kutta-Nyström Methods with Application to Boundary Value Ordinary Differential Equations

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.     

The Construction of Practical General Linear Methods

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.     

Explicit General Linear Methods with Inherent Runge–Kutta Stability

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.     

Experiments with a New Fifth Order Method

Almost Runge–Kutta methods were introduced to obtain many of the advantages of Runge–Kutta methods without their disadvantages. We consider the construction of fourth order methods of this type with a special choice of the free parameters to ensure that, at least for constant stepsize, order 5 behaviour is achieved. It is shown how this can be extended to variable stepsize.     

Characterizations and construction of Poisson/symplectic and symmetric multi-revolution implicit Runge–Kutta methods of high order

This paper deals with the characterizations and construction of Poisson/symplectic and (φ⁻¹)-symmetric implicit high-order multi-revolution Runge–Kutta methods (MRRKMs). The basic tool is a modified W-transformation based on quadrature formulas and orthogonal polynomials. Two sufficient conditions can be obtained under which MRRKMs are Poisson/symplectic or (φ⁻¹)-symmetric. We construct two classes of high order implicit MRRKMs by using these sufficient conditions. Our results can be considered as an extension of related results of the standard Runge–Kutta methods in some references.     

Passive Runge–Kutta Methods—Properties, Parametric Representation, and Order Conditions

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.     

An improvement of the numerical stability results for nonlinear neutral delay-integro-differential equations

This paper deals with stability of the extended Runge–Kutta methods for nonlinear neutral delay-integro-differential equations. The stability results in the reference [Y. Yu, L. Wen, S. Li, Nonlinear stability of Runge–Kutta methods for neutral delay integro-differential equations, Appl. Math. Comput. 191 (2007) 543–549] are improved. With this improvement, several new numerical stability criteria are obtained, it is proven that the extended Runge–Kutta methods are globally and asymptotically stable under the suitable conditions.     

Positivity of exponential Runge–Kutta methods

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.     

Equilibrium attractivity of Runge-Kutta methods

For dissipative differential equations y' = f (y) it is knownthat contractivity of the exact solution is reproduced by algebraicallystable Runge–Kutta methods. In this paper we investigatewhether a different property of the exact solution also holdsfor Runge–Kutta 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 Runge–Kutta 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.     

A One-step Method of Order 10 for y' = f(x, y)

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 Runge—Kutta methods, but for the second-order equationy'= f(x, y). As for Runge—Kutta methods, the complexityof the order conditions grows rapidly with the order, so thatwe have to solve a non—linear system of 440 algebraicequations to obtain a tenth—order method. We demonstratehow this system can be solved. Finally we give the coefficients(20 decimals) of two methods with small local truncation errors.     

Two-Step High Order Starting Values for Implicit Runge–Kutta Methods

In this article a simple form of expressing and studying the order conditions to be satisfied by starting algorithms for Runge–Kutta methods, which use information from the two previous steps is presented. In particular, starting algorithms of highest order for Runge–Kutta–Gauss methods up to seven stages are derived. Some numerical experiments with Hamiltonian systems to compare the behaviour of the new starting algorithms with other existing ones are presented.     

Second order Runge–Kutta methods for Stratonovich stochastic differential equations

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     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

Accuracy of classical conservation laws for Hamiltonian PDEs under Runge–Kutta discretizations

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.     

Variable-Order ESIRK Methods for Stiff Differential Equations

ESIRK methods (Effective order Singly-Implicit Runge–Kutta methods) have been shown to be efficient for the numerical solution of stiff differential equations. In this paper, we consider a new implementation of these methods with a variable order strategy. We show that the efficiency of the ESIRK method for stiff problems is improved by using the proposed variable order schemes.