A General Family of Pseudo Two-step Runge-Kutta Methods

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     

Two-Step Runge-Kutta: Theory and Practice

Local and global error for Two-Step Runge-Kutta (TSRK) methods are analyzed using the theory of B-series. Global error bounds are derived in both constant and variable stepsize environments. An embedded TSRK pair is constructed and compared with the RK5(4)6M pair of Dormand and Prince on the DETEST set of problems. Numerical results show that the TSRK performs competitively with the RK method.     

On fifth order Runge-Kutta methods

A 6 stage Runge-Kutta method is derived with the property that its order is 5 when used to solve a scalar differential equation but only 4 when used to solve a general system of differential equations. The existence of such a method underlines the necessity of carrying out theoretical analyses in a vector valued setting rather than in a one-dimensional setting as in the work of Kutta and some more recent authors.This research was supported by the New Zealand Foundation for Research, Science and Technology     

Parallel linear system solvers for Runge-Kutta methods

If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form I-A hJ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form I - B hJ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LU-decompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor s ³ . It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode.     

The orders of embedded continuous explicit Runge-Kutta methods

Previously, the authors [9] classified various types of continuous explicit Runge-Kutta methods of order 5. Here, new lower bounds on the numbers of stages required for a sequence of continuous methods of increasing orders which are embedded in a continuouss-stage method of orderp are obtained. Carnicer [2] showed for each continuous explicit Runge-Kutta method of orderp in a mildly restricted family that at least 2p – 2 stages are required. Here, the same bound is established for all such methods of orderp.This research was supported by the Natural Sciences and Engineering Research Council of Canada, and the Information Technology Research Centre of Ontario. In addition, the second author was supported by the Ministero dell'Università e della Ricerca Scientifica e Tecnologica of Italy.     

An improvement on the positivity results for 2-stage explicit Runge-Kutta methods

In this paper, we investigate the positivity property for a class of 2-stage explicit Runge-Kutta (RK2) methods of order two when applied to the numerical solution of special nonlinear initial value problems (IVPs) for ordinary differential equations (ODEs). We also pay particular attention to monotonicity property. We obtain new results for positivity which are important in practical applications. We provide some numerical examples to illustrate our results.     

A modified Runge-Kutta method for the numerical solution of ODE''s with oscillation solutions

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.     

A composition law for Runge-Kutta methods applied to index-2 differential-algebraic equations

This paper presents a new composition law for Runge-Kutta methods when applied to index-2 differential-algebraic systems. Applications of this result to the study of the order of composite methods and of symmetric methods are given.     

积分微分方程Runge-Kutta方法的散逸性

本文针对一类积分微分方程讨论Runge-Kutta方法的散逸性,当积分项用PQ公式逼近时,证明了(k,l)-代数稳定的Runge-Kutta方法是D(l)-散逸的.     

On the construction of stiffly accurate and B-stable Runge-Kutta methods

In the solution of stiff ODEs and especially DAEs it is desirable that the method used is stiffly accurate and B-stable. In this paper guidelines for the construction of Runge-Kutta methods with these properties are presented.     

Two-derivative Runge-Kutta methods with optimal phase properties

In this work, we consider two-derivative Runge-Kutta methods for the numerical integration of first-order differential equations with oscillatory solution. We construct methods with constant coefficients and special properties as minimum phase-lag and amplification errors with three and four stages. All methods constructed have fifth algebraic order. We also present methods with variable coefficients with zero phase-lag and amplification errors. In order to examine the efficiency of the new methods, we use four well-known oscillatory test problems.     

求解多延迟微分方程的Runge-Kutta方法的收缩性

该文涉及多延迟微分方程MDDEs系统的理论解与数值解的收缩性．为此,一些新的稳定性概念诸如：BN_f^(m)-稳定性及GRN_m-稳定性稳定性被引入．该探讨得出：Ｒｕｎｇｅ Ｋｕｔｔａ（ＲＫ）方法及相应的连续插值的BN^(m)-稳定性导致求解MDDEs的方法的收缩性（GRN_m-稳定性）．     

Dissipativity of Runge-Kutta methods for dynamical systems with delays

This paper is concerned with the numerical solution of dissipativeinitial value problems with delays by Runge-Kutta methods. Asufficient condition for the dissipativity of the systems isgiven. The concepts of D(l)-dissipativity and GD(l)-dissipativityare introduced. We investigate the dissipativity propertiesof (k,l)-algebraically stable Runge-Kutta methods with piecewiseconstant or linear interpolation procedures for finite-dimensionaland infinite-dimensional dynamical systems with delays.     

A quasi-randomized Runge-Kutta method

We analyze a quasi-Monte Carlo method to solve the initial-value problem for a system of differential equations . The function is smooth in and we suppose that and are of bounded variation in and that is bounded in a neighborhood of the graph of the solution. The method is akin to the second order Heun method of the Runge-Kutta family. It uses a quasi-Monte Carlo estimate of integrals. The error bound involves the square of the step size as well as the discrepancy of the point set used for quasi-Monte Carlo approximation. Numerical experiments show that the quasi-randomized method outperforms a recently proposed randomized numerical method.

     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

A stability property of A-stable collocation-based Runge-Kutta methods for neutral delay differential equations

We consider a linear homogeneous system of neutral delay differential equations with a constant delay whose zero solution is asymptotically stable independent of the value of the delay, and discuss the stability of collocation-based Runge-Kutta methods for the system. We show that anA-stable method preserves the asymptotic stability of the analytical solutions of the system whenever a constant step-size of a special form is used.     

The stability of modified Runge-Kutta methods for the pantograph equation

In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the -methods with , the odd stage Gauss-Legendre methods and the even stage Lobatto IIIA and IIIB methods are asymptotically stable. Some experiments are given.

     

Stability Analysis of Runge-Kutta Methods for Non-Linear Delay Differential Equations

This paper is concerned with the numerical solution of delay differential equations(DDEs). We focus on the stability behaviour of Runge-Kutta methods for nonlinear DDEs. The new concepts of GR(l)-stability, GAR(l)-stability and weak GAR(l)-stability are further introduced. We investigate these stability properties for (k, l)-algebraically stable Runge-Kutta methods with a piecewise constant or linear interpolation procedure.     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.     

Parallel iteration across the steps of high-order Runge-Kutta methods for nonstiff initial value problems

For the parallel integration of nonstiff initial value problems (IVPs), three main approaches can be distinguished: approaches based on “parallelism across the problem”, on “parallelism across the method” and on “parallelism across the steps”. The first type of parallelism does not require special integration methods and can be exploited within any available IVP solver. The method-parallelism approach received much attention, particularly within the class of explicit Runge-Kutta methods originating from fixed point iteration of implicit Runge-Kutta methods of Gaussian type. The construction and implementation on a parallel machine of such methods is extremely simple. Since the computational work per processor is modest with respect to the number of data to be exchanged between the various processors, this type of parallelism is most suitable for shared memory systems. The required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration and requires the processors to communicate after each full iteration. If the iterations have sufficient computational volume, then the step-parallel approach may be suitable for implementation on distributed memory systems. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The dynamic step-parallel iteration process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to achieve speed-up factors up to 15.