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 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.     

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.     

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.     

Stability of implicit Runge-Kutta methods for nonlinear stiff differential equations

Under the assumption that an implicit Runge-Kutta method satisfies a certain stability estimate for linear systems with constant coefficientsl ₂-stability for nonlinear systems is proved. This assumption is weaker than algebraic stability since it is satisfied for many methods which are not evenA-stable. Some local smoothness in the right hand side of the differential equation is needed, but it may have a Jacobian and higher derivatives with large norms. The result is applied to a system derived from a strongly nonlinear parabolic equation by the method of lines.     

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 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     

Linear and non-linear stability for general linear methods

We explore the interrelation between a number of linear and non-linear stability properties. The weakest of these,A-stability, is shown by counterexample not to imply any of the various versions ofAN-stability introduced in the paper and two of these properties, weak and strongAN-stability, are also shown not to be equivalent. Finally, another linear stability property defined here, EuclideanAN-stability, is shown to be equivalent to algebraic stability.     

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.     

Stability of the Radau IA and Lobatto IIIC methods for delay differential equations

Recently, we have proved that the Radau IA and Lobatto IIIC methods are P-stable, i.e., they have an analogous stability property to A-stability with respect to scalar delay differential equations (DDEs). In this paper, we study stability of those methods applied to multidimensional DDEs. We show that they have a similar property to P-stability with respect to multidimensional equations which satisfy certain conditions for asymptotic stability of the zero solutions. The conditions are closely related to stability criteria for DDEs considered in systems theory. Received October 8, 1996 / Revised version received February 21, 1997     

A stability property ofA-stable natural Runge-Kutta methods for systems of delay differential equations

A natural Runge-Kutta method is a special type of Runge-Kutta method for delay differential equations (DDEs); it is known that any one-step collocation method is equivalent to one of such methods. In this paper, we consider a linear constant-coefficient system of DDEs with a constant delay, and discuss the application of natural Runge-Kutta methods to the system. We show that anA-stable method preserves the asymptotic stability property of the analytical solutions of the system.     

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.     

On implicit Runge-Kutta methods with high stage order

It is well known that high stage order is a desirable property for implicit Runge-Kutta methods. In this paper it is shown that it is always possible to construct ans-stage IRK method with a given stability function and stage orders−1 if the stability function is an approximation to the exponential function of at least orders. It is further indicated how to construct such methods as well as in which cases the constructed methods will be stiffly accurate.     

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.     

Approximating Runge-Kutta matrices by triangular matrices

The implementation of implicit Runge-Kutta methods requires the solution of large systems of non-linear equations. Normally these equations are solved by a modified Newton process, which can be very expensive for problems of high dimension. The recently proposed triangularly implicit iteration methods for ODE-IVP solvers [5] substitute the Runge-Kutta matrixA in the Newton process for a triangular matrixT that approximatesA, hereby making the method suitable for parallel implementation. The matrixT is constructed according to a simple procedure, such that the stiff error components in the numerical solution are strongly damped. In this paper we prove for a large class of Runge-Kutta methods that this procedure can be carried out and that the diagnoal entries ofT are positive. This means that the linear systems that are to be solved have a non-singular matrix. The research reported in this paper was supported by STW (Dutch Foundation for Technical Sciences).     

A-stable Runge-Kutta processes

The concept of strongA-stability is defined. A class of stronglyA-stable Runge-Kutta processes is introduced. It is also noted that several classes of implicit Runge-Kutta processes defined by Ehle [6] areA-stable.     

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     

Perturbed collocation and Runge-Kutta methods

Summary It is well known thatsome implicit Runge-Kutta methods are equivalent to collocation methods. This fact permits very short and natural proofs of order andA, B, AN, BN-stability properties for this subclass of methods (see [9] and [10]). The present paper answers the natural question, ifall RK methods can be considered as a somewhat perturbed collocation. After having introduced this notion we give a proof on the order of the method and discuss their stability properties. Much of known theory becomes simple and beautiful.     

The NGP-stability of Runge-Kutta methods for systems of neutral delay differential equations

Summary. This paper deals with the stability analysis of implicit Runge-Kutta methods for the numerical solutions of the systems of neutral delay differential equations. We focus on the behavior of such methods with respect to the linear test equations where ,L, M and N are complex matrices. We show that an implicit Runge-Kutta method is NGP-stable if and only if it is A-stable. Received February 10, 1997 / Revised version received January 5, 1998