A-stability of Runge-Kutta methods for systems with additive noise

Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of Dahlquist'sA-stability is proposed. It is shown that the discretization of the drift term alone controls theA-stability of the whole scheme. The quantitative effect of implicitness uponA-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.This author was partially supported by the Italian Consiglio Nazionale delle Ricerche.     

On theA-stability of implicit Runge-Kutta processes

A new technique to calculate the characteristic functions and to examine theA-stability of implicit Runge-Kutta processes is presented. This technique is based on a direct algebraic approach and an application of theC-polynomial theory of Nørsett. New processes are suggested. These processes can be exponentially fitted in anA-stable manner.     

A criterion forP-stability properties of Runge-Kutta methods

P-stability is an analogous stability property toA-stability with respect to delay differential equations. It is defined by using a scalar test equation similar to the usual test equation ofA-stability. EveryP-stable method isA-stable, but anA-stable method is not necessarilyP-stable. We considerP-stability of Runge-Kutta (RK) methods and its variation which was originally introduced for multistep methods by Bickart, and derive a sufficient condition for an RK method to have the stability properties on the basis of an algebraic characterization ofA-stable RK methods recently obtained by Schere and Müller. By making use of the condition we clarify stability properties of some SIRK and SDIRK methods, which are easier to implement than fully implicit methods, applied to delay differential equations.     

On the practical value of the notion ofBN-stability

The oldest concept of unconditional stability of numerical integration methods for ordinary differential systems is that ofA-stability. This concept is related to linear systems having constant coefficients and has been introduced by Dahlquist in 1963. More recently, since another contribution of Dahlquist in 1975, there has been much interest in unconditional stability properties of numerical integration methods when applied to non-linear dissipative systems (G-stability,BN-stability,A-contractivity). Various classes of implicit Runge-Kutta methods have already been shown to beBN-stable. However, contrary to the property ofA-stability, when implementing such a method for practical use this unconditional stability property may be lost. The present note clarifies this for a class of diagonally implicit methods and shows at the same time that Rosenbrock's method is notBN-stable.     

Stability analysis of methods employing reducible rules for Volterra integral equations

The concept of (A ₀,S)-stability, for numerical methods approximating solutions of Volterra integral equations, is formally defined. New stability polynomials for the recent multi-lag type methods are obtained. (A ₀, 1)-stability of these and other methods employing reducible quadrature rules are also investigated.     

NP-stability of Runge-Kutta methods based on classical quadrature

Recently Bellen, Jackiewicz and Zennaro have studied stability properties of Runge-Kutta (RK) methods for neutral delay differential equations using a scalar test equation. In particular, they have shown that everyA-stable collocation method isNP-stable, i.e., the method has an analogous stability property toA-stability with respect to the test equation. Consequently, the Gauss, Radau IIA and Lobatto IIIA methods areNP-stable. In this paper, we examine the stability of RK methods based on classical quadrature by a slightly different approach from theirs. As a result, we prove that the Radau IA and Lobatto IIIC methods equipped with suitable continuous extensions are alsoNP-stable by virtue of fundamental notions related to those methods such as simplifying conditions, algebraic stability, and theW-transformation.     

A Class of Multiderivative Hybrid One-step Methods

This paper presents a class of hybrid one-step methods that are obtained by using Cramer's rule and rational approximations to function exp(q). The algorithms fall into the catalogue of implicit formula, which involves sth order derivative and s 1 free parameters. The order of the algorithms satisfies s 1≤p≤2s 2. The stability of the methods is also studied, necessary and sufficient conditions for A-stability and L-stability are given. In addition, some examples are also given to demonstrate the method presented.     

A note on a class of stronglyA-stable methods

Formulae for a class ofA-stable quadrature methods, or equivalently a certain implicit Runge-Kutta scheme, are given. A short proof of the strongA-stability is presented.On leave of absence from Chalmers University of Technology, Göteborg, Sweden.     

The equivalence ofB-stability andA-stability

It is a well-known result of Dahlquist that the linear (A-stability) and non-linear (G-stability) stability concepts are equivalent for multistep methods in their one-leg formulation. We show to what extent this result also holds for Runge-Kutta methods. Dedicated to Germund Dahlquist on the occasion of his 60th birthday.     

Continuous two-step Runge–Kutta methods for ordinary differential equations

New classes of continuous two-step Runge-Kutta methods for the numerical solution of ordinary differential equations are derived. These methods are developed imposing some interpolation and collocation conditions, in order to obtain desirable stability properties such as A-stability and L-stability. Particular structures of the stability polynomial are also investigated.     

Constructive characterization ofA-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methods

Summary All rational approximations to exp(z) of order 2m– (m denotes the maximal degree of nominator and denominator) are given by a closed formula involving real parameters. Using the theory of order stars [9], necessary and sufficient conditions forA-stability (respectivelyI-stability) are given. On the basis of this characterization relations between the concepts ofA-stability and algebraic stability (for implicit Runge-Kutta methods) are investigated. In particular we can partly prove the conjecture that to any irreducibleA-stableR(z) of oderp0 there exist algebraically stable Runge-Kutta methods of the same order withR(z) as stability function.     

An algebraic approach toA-stable linear multistep-multiderivative integration formulas

A general algebraic approach and some new results are given pertaining to the synthesis of linearA-stable multistep-multiderivative formulas used for integrating stiff differential equations. This problem is shown to be considerably simplified by associating to each formula a special two-variable function, termed the canonical polynomial. In particular, the canonical polynomial approach allows to solve the approximation problem in closed form and provides an easy-to-check algebraic criterion forA-stability. A lower bound is established for the maximum order of accuracy compatible withA-stability, which turns out to be identical to the absolute maximum in some particular cases. It is finally conjectured that this property holds true in general.     

G-stability is equivalent toA-stability

In 1975 the author showed that a norm (Liapunov function) can be constructed for the stability and error analysis of a linear multistep method (and the related one-leg method) for the solution of stiff non-linear systems, provided that the system satisfies a monotonicity condition and the method possesses a property calledG-stability.In this paper it is shown thatG-stability is equivalent toA-stability. More generally, a Liapunov function exists if the stability region of the method contains a circle (half-plane), provided that the system satisfies a monotonicity condition related to this circle (half-plane). In the general case this condition depends on the stepsize.     

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.     

A-stable block implicit one-step methods

A class of methods for solving the initial value problem for ordinary differential equations is studied. We developr-block implicit one-step methods which compute a block ofr new values simultaneously with each step of application. These methods are examined for the property ofA-stability. A sub-class of formulas is derived which is related to Newton-Cotes quadrature and it is shown that for block sizesr=1,2,..., 8 these methods areA-stable while those forr=9,10 are not. We constructA-stable formulas having arbitrarily high orders of accuracy, even stiffly (strongly)A-stable formulas.     

A 0-stable linear multistep formulas of theα-type

The-type linear multistep formulas are a generalization of the Adams-type formulas. This paper is concerned with completely characterizing theA ₀-stability of thek-step, orderk -type formulas. Specifically, all such formulas of orders 4 or less are identified and it is shown that no-type formulas of order 5 or more exist. These theorems generalize some previous results.     

Nilpotent injectors in alternating groups

AnN-Injector in an arbitrary finite group is defined as a maximal nilpotent subgroup ofG containing a subgroupA ofG of maximal order, satisfying class (A)≦2. In a previous paper theN-Injectors of Sym(n) were determined. In this paper we determine theN-Injectors of Alt(n), after having determined the set of all nilpotent subgroups,A, of Sym(n) of maximal order satisfying class(A)≦2. It is shown that the set ofN-Injectors of Alt(n) consists of a unique conjugacy class, and ifn≠9, it coincides with the set of the nilpotent subgroups of Alt(n) of maximal order.     

Stability of Numerical Methods for Ordinary Differential Equations

Variable stepsize stability results are found for three representative multivalue methods. For the second order BDF method, a best possible result is found for a maximum stepsize ratio that will still guarantee A(0)-stability behaviour. It is found that under this same restriction, A()-stability holds for 70°. For a new two stage two value first order method, which is L-stable for constant stepsize, A(0)-stability is maintained for stepsize ratios as high as aproximately 2.94. For the third order BDF method, a best possible result of (1/2)(1+ ) is found for a ratio bound that will still guarantee zero-stability.     

High order a-stable methods for the numerical solution of systems of D.E.'s

The following note shows how one can obtain one step methods of arbitrarily high order which satisfy Dahlquist's requirements ofA-stability. Although most of these methods appear at the moment to be largely of theoretical interest the author is working on several practical applications.