The equivalence of algebraic stability andAN-stability

It is shown that, under certain restrictions,AN-stability is equivalent to algebraic stability for general linear methods. The restrictions have the purpose of excluding from consideration methods which can be replaced by simpler methods in various specific ways andAN-stability is to be interpreted in the strong sense. This result generalizes known results for Runge-Kutta and for one-leg methods.     

A stability property of implicit Runge-Kutta methods

A class of implicit Runge-Kutta methods is shown to possess a stability property which is a natural extension of the notion ofA-stability for non-linear systems.     

B-Theory of Runge-Kutta methods for stiff volterra functional differential equations

B-stability andB-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra functional differential equations (VFDEs) are established which provide unified theoretical foundation for the study of Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type which appear in practice.     

A necessary condition forB-stability

A necessary condition forB-stability is derived. Then it is shown that the Runge-Kutta methods of Lobatto type III _A and III _B and some other methods are notB-stable.     

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.     

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.     

A- andB-stability for Runge-Kutta methods-characterizations and equivalence

Summary Using a special representation of Runge-Kutta methods (W-transformation), simple characterizations ofA-stability andB-stability have been obtained in [9, 8, 7]. In this article we will make this representation and their conclusions more transparent by considering the exact Runge-Kutta method. Finally we demonstrate by a numerical example that for difficult problemsB-stable methods are superior to methods which are onlyA-stable.Talk, presented at the conference on the occasion of the 25th anniversary of the founding ofNumerische Mathematik, TU Munich, March 19–21, 1984     

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.     

A note onB-stability of Runge-Kutta methods

Summary Burrage and Butcher [1, 2] and Crouzeix [4] introduced for Runge-Kutta methods the concepts ofB-stability,BN-stability and algebraic stability. In this paper we prove that for any irreducible Runge-Kutta method these three stability concepts are equivalent.Chapters 1–3 of this article have been written by the second author, whereas chapter 4 has been written by the first author     

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.     

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.     

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 efficiency of a class of a-stable methods

The numerical solution of systems of differential equations of the formB dx/dt=σ(t)Ax(t)+f(t),x(0) given, whereB andA (withB and —(A+A ^T) positive definite) are supposed to be large sparse matrices, is considered.A-stable methods like the Implicit Runge-Kutta methods based on Radau quadrature are combined with iterative methods for the solution of the algebraic systems of equations.     

An algebraic characterization ofB-convergent Runge-Kutta methods

Summary In the analysis of discretization methods for stiff intial value problems, stability questions have received most part of the attention in the past.B-stability and the equivalent criterion algebraic stability are well known concepts for Runge-Kutta methods applied to dissipative problems. However, for the derivation ofB-convergence results — error bounds which are not affected by stiffness — it is not sufficient in many cases to requireB-stability alone. In this paper, necessary and sufficient conditions forB-convergence are determined.This paper was written while J. Schneid was visiting the Centre for Mathematics and Computer Science with an Erwin-Schrödinger stipend from the Fonds zur Förderung der wissenschaftlichen Forschung     

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.     

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.     

OnA-stable implicit Runge-Kutta methods

An elementary proof is given of theA-stability of implicit Runge-Kutta methods for which the corresponding rational function is on the diagonal or one of the first two subdiagonals of the Padé table for the exponential function. The result is extended to give necessary and sufficient conditions for theA-stability ofn-stage methods of order greater than or equal to 2n–2.     

Additive maps on rings behaving like derivations at idempotent-product elements

For every ring R with the unit I containing a nontrivial idempotent P, we describe the additive maps δ from R into itself which behave like derivations, and show that derivations on such kinds of rings can be determined by the action on the elements A,B∈R with AB=0, AB=P and AB=I respectively. Those results of An and Hou [R. An, J. Hou, Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear Algebra Appl. 431 (2009) 1070-1080], Bres?ar [M. Bres?ar, Characterizing homomorphisms, multipliers and derivations in rings with idempotents, Proc. Roy. Soc. Edinburgh. Sect. A. 137 (2007) 9-21] and Chebotar et al. [M.A. Chebotar, W.-F. Ke, P.-H. Lee, Maps characterized by action on zero products, Pacific J. Math. 216 (2) 2004 217-228] are improved.     

S-Stability properties for generalized Runge-Kutta methods

Summary A class of generalized Runge-Kutta methods is considered for the numerical integration of stiff systems of ordinary differential equations. These methods are characterized by the fact that the coefficients of the integration formulas are matrices depending on the Jacobian, or on an approximation to the Jacobian. Special attention is paid to stability aspects. In particular, theS-stability properties of the method are investigated. The concept of internal stability is discussed. Internal stability imposes conditions on intermediate results in the Runge-Kutta scheme. Some numerical examples are discussed.     

Intervals and tests for weighted sum of percentiles θ(p) and θ(1−p) for symmetrical populations

Available is a random sample from a distribution that is continuous, symmetrical, and reasonably well-behaved. Approximate one-sided and two-sided confidence intervals (provide tests) are developed forAθ(p)+Bθ(1?p), where θ(p) is the population 100p percentile, 1/2≦p<1, andA, B can be any positive or negative numbers. The interquantile and other ranges are special cases. Asymptotically, a confidence coefficient value is precisely determined. The statistics used are weighted sums (with weightsA andB) of two percentage points of the sample. Many comparisons of population percentiles can be made through suitable choice ofp, A, andB.