Local error estimation for multistep collocation methods

Multistep collocation methods for initial value problems in ordinary differential equations are known to be a subclass of multistep Runge-Kutta methods and a generalisation of the well-known class of one-step collocation methods as well as of the one-leg methods of Dahlquist. In this paper we derive an error estimation method of embedded type for multistep collocation methods based on perturbed multistep collocation methods. This parallels and generalizes the results for one-step collocation methods by Nørsett and Wanner. Simple numerical experiments show that this error estimator agrees well with a theoretical error estimate which is a generalisation of an error estimate first derived by Dahlquist for one-leg methods.     

On the stability properties of Brown's multistep multiderivative methods

Summary Brown introducedk-step methods usingl derivatives. We investigate for whichk andl the methods are stable or unstable. It is seen that to anyl the method becomes unstable fork large enough. All methods withk2(l+1) are stable. Fork=1,2,..., 18 there exists a _k such that the methods are stable for anyl _k and unstable for anyl < _k . The _k are given.     

On the order of composite multistep methods for ordinary differential equations

Summary In this paper, a general class ofk-step methods for the numerical solution of ordinary differential equations is discussed. It is shown that methods with order of consistencyq have order of convergence (q+1) if a very simple condition is satisfied. This result gives a new aspect to previous results of Spijker; it also serves as a starting point for a new theory of cyclick-step methods, completing an approach of Donelson and Hansen. It facilitates the practical determination of high-order cyclick-step methods, especially of stiffly stable,k-step methods.     

Contractivity preserving explicit linear multistep methods

Summary We investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on the general test-equation , whereA is a square matrix of arbitrary orders1. The contractivity is analysed with respect to arbitrary norms in thes-dimensional space (which are not necessarily generated by an inner product). For given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize.This research has been supported by the Netherlands organisation for scientific research (NWO)     

On the convergence of multistep methods for nonlinear stiff differential equations

Summary Convergence estimates are given forA()-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturbation arguments with frequency domain techniques.     

On error behaviour of partitioned linearly implicit runge-kutta methods for stiff and differential algebraic systems

This paper studies partitioned linearly implicit Runge-Kutta methods as applied to approximate the smooth solution of a perturbed problem with stepsizes larger than the stiffness parameter. Conditions are supplied for construction of methods of arbitrary order. The local and global error are analyzed and the limiting case 0 considered yielding a partitioned linearly implicit Runge-Kutta method for differential-algebraic equations of index one. Finally, some numerical experiments demonstrate our theoretical results.     

A 0-Stability and stiff stability of Brown's multistep multiderivative methods

Summary Brown [1] introducedk-step methods usingl derivatives. Necessary and sufficient conditions forA ₀-stability and stiff stability of these methods are given. These conditions are used to investigate for whichk andl the methods areA ₀-stable. It is seen that for allk andl withk1.5 (l+1) the methods areA ₀-stable and stiffly stable. This result is conservative and can be improved forl sufficiently large. For smallk andl A ₀-stability has been determined numerically by implementing the necessary and sufficient condition.     

On improving the absolute stability of local extrapolation

Summary A widely used technique for improving the accuracy of solutions of initial value problems in ordinary differential equations is local extrapolation. It is well known, however, that when using methods appropriate for solving stiff systems of ODES, the stability of the method can be seriously degraded if local extrapolation is employed. This is due to the fact that performing local extrapolation on a low order method is equivalent to using a higher order formula and this high order formula may not be suitable for solving stiff systems. In the present paper a general approach is proposed whereby the correction term added on in the process of local extrapolation is in a sense a rational, rather than a polynomial, function. This approach allows high order formulae with bounded growth functions to be developed. As an example we derive anA-stable rational correction algorithm based on the trapezoidal rule. This new algorithm is found to be efficient when low accuracy is requested (say a relative accuracy of about 1%) and its performance is compared with that of the more familiar Richardson extrapolation method on a large set of stiff test problems.     

On projected Runge-Kutta methods for differential-algebraic equations

Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems.     

A class of pseudo Runge-Kutta methods

This paper develops a general theory for a class of Runge-Kutta methods which are based, in addition to the stages of the current step, also on the stages of the previous step. Such methods have been introduced previously for the case of one and two stages. We show that for any numbers of stages methods of orderp withs+1 p 2s can be constructed. The paper terminates with a study of step size change and stability.     

Convergence of multistep methods for Volterra functional differential equations

Summary This paper deals with the convergence of nonstationary quasilinear multistep methods with varying step, used for the numerical integration of Volterra functional differential equations. A Perron type condition (appearing in the differential equations theory) is imposed on the increment function. This gives a generalization of some results of Tavernini ([19–21]).     

Accuracy and stability of multistage multistep formulas

Summary We continue our research concerning relations between accuracy and stability of time discretizations. Here we concentrate, rather than on the size and shape of the stability region, on the obtainable accuracy when the number of parameters in the formulas is large.     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

A note on multistep methods and attracting sets of dynamical systems

Summary We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.This work was supported by the U.S. Department of Energy Contract DE-A503-76 ER72012     

On the D-suitability of implicit Runge-Kutta methods

The concept of suitability means that the nonlinear equations to be solved in an implicit Runga-Kutta method have a unique solution. In this paper, we introduce the concept of D-suitability and show that previous results become special cases of ours. In addition, we also give some examples to illustrate the D-suitability of a matrixA.     

The stability function for multistep collocation methods

Summary C-polynomials for rational approximation to the exponential function was introduced by Nørsett [7] to study stability properties of one-step methods. For one-step collocation methods theC-polynomial has a very simple form. In this paper we studyC-polynomials for multistep collocation methods and obtain results that generalize those in the one-step case, and provide a way to analyze linear stability of such methods.     

Asymptotic expansions for multistep methods applied to nonlinear Volterra integral equations of the second kind

Summary This paper deals with linear multistep methods applied to nonlinear, nonsingular Volterra integral equations of the second kind. Analogously to the theory of W.B. Gragg, the existence of asymptotic expansions in the stepsizeh is proved. Under certain conditions only even powers ofh occur. As a special case, the midpoint rule is treated, a short numerical example for the applicability to extrapolation techniques is given.     

On the order conditions of Runge-Kutta methods with higher derivatives

Summary Runge-Kutta methods have been generalized to procedures with higher derivatives of the right side ofy=f(t,y) e.g. by Fehlberg 1964 and Kastlunger and Wanner 1972. In the present work some sufficient conditions for the order of consistence are derived for these methods using partially the degree of the corresponding numerical integration formulas. In particular, methods of Gauß, Radau, and Lobatto type are generalized to methods with higher derivatives and their maximum order property is proved. The applied technique was developed by Crouzeix 1975 for classical Runge-Kutta methods. Examples of simple explicit and semi-implicit methods are given up to order 7 and 6 respectively.     

On implicit Runge-Kutta methods with higher derivatives

Two families of implicit Runge-Kutta methods with higher derivatives are (re-)considered generalizing classical Runge-Kutta methods of Butcher type and f Ehle type. For generalized Butcher methods the characteristic functionG() is represented by means of the node polynomial directly, thereby showing that in methods of maximum order,G() is connected withs-orthogonal polynomials in exactly the same way as Padé approximations in the classical case.     

Numerical experiments with a multistep Radau method

This paper describes an implementation of multistep collocation methods, which are applicable to stiff differential problems, singular perturbation problems, and D.A.E.s of index 1 and 2.These methods generalize one-step implicit Runge-Kutta methods as well as multistep one-stage BDF methods. We give numerical comparisons of our code with two representative codes for these methods, RADAU5 and LSODE.