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     

On the relation between algebraic stability andB-convergence for Runge-Kutta methods

Summary This paper is concerned with the numerical solution of stiff initial value problems for systems of ordinary differential equations using Runge-Kutta methods. For these and other methods Frank, Schneid and Ueberhuber [7] introduced the important concept ofB-convergence, i.e. convergence with error bounds only depending on the stepsizes, the smoothness of the exact solution and the so-called one-sided Lipschitz constant . Spijker [19] proved for the case <0 thatB-convergence follows from algebraic stability, the well-known criterion for contractivity (cf. [1, 2]). We show that the order ofB-convergence in this case is generally equal to the stage-order, improving by one half the order obtained in [19]. Further it is proved that algebraic stability is not only sufficient but also necessary forB-convergence.This study was completed while this author was visiting the Oxford University Computing Laboratory with a stipend from the Netherlands Organization for Scientific Research (N.W.O.)     

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.     

B-convergence properties of defect correction methods. II

Summary B-convergence properties of defect correction methods based on the implicit Euler and midpoint schemes are discussed. The property ofB-convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness. It turns out that the orders ofB-convergence of these methods coincide with the conventional orders of small (whereL is a Lipschitz constant of the right-hand side). In Part I these assertions are reduced to the validity of the so-called HypothesisA which is discussed in greater detail in Part II. Numerical experiments confirming the theoretical analysis are also given in Part II.     

B-convergence properties of defect correction methods. I

Summary B-convergence properties of defect correction methods based on the implicit Euler and midpoint schemes are discussed. The property ofB-convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness. It turns out that the orders ofB-convergence of these methods coincide with the conventional orders of convergence of these methods derived under the assumption that.hL is small (whereL is a Lipschitz constant of the right-hand side). In Part I these assertions are reduced to the validity of the so-called Hypothesis A which is discussed in greater detail in Part II. Numerical experiments confirming the theoretical analysis are also given in Part II.     

Delay-dependent stability of high order Runge–Kutta methods

This paper is concerned with the study of the delay-dependent stability of Runge–Kutta methods for delay differential equations. First, a new sufficient and necessary condition is given for the asymptotic stability of analytical solution. Then, based on this condition, we establish a relationship between τ(0)-stability and the boundary locus of the stability region of numerical methods for ordinary differential equations. Consequently, a class of high order Runge–Kutta methods are proved to be τ(0)-stable. In particular, the τ(0)-stability of the Radau IIA methods is proved.     

High orderP-stable formulae for the numerical integration of periodic initial value problems

Summary Recently there has been considerable interest in the approximate numerical integration of the special initial value problemy=f(x, y) for cases where it is known in advance that the required solution is periodic. The well known class of Störmer-Cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems. An appropriate stability requirement for the numerical integration of periodic problems is that ofP-stability. However Lambert and Watson have shown that aP-stable linear multistep method cannot have an order of accuracy greater than 2. In the present paper a class of 2-step methods of Runge-Kutta type is discussed for the numerical solution of periodic initial value problems.P-stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods.     

Sur laB-stabilité des méthodes de Runge-Kutta

Résumé Dans cet article, nous modifions légèrement la définition de laB-stabilité donnée par J.C. Butcher [1] afin qu'elle s'applique à une plus large classe d'équations différentielles et nous donnons des caractérisations simples de cette propriété.

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OnB-stability of the methods of Runge Kutta

Summary In this paper, we slightly modify the definition ofB-stability of Butcher [1], so as to cover a wider class of differential equations, and we give simple characterizations of this property.

     

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.     

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     

$L^1$-Stability and error estimates for approximate Hamilton-Jacobi solutions

Summary.   We study the -stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let denote the `small scale' of such approximations (– the viscosity amplitude , the spatial grad-size , etc.), then our -error estimates are of , and are sharper than the classical -results of order one half, . The main building blocks of our theory are the notions of the semi-concave stability condition and -measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the -stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain -bounds on their associated truncation errors; -convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our -theory. Received April 20, 1998 / Revised version received November 8, 1999 / Published online August 24, 2000     

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.     

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.     

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     

The order ofB-convergence of algebraically stable Runge-Kutta methods

In a previous paper it was shown that for a class of semi-linear problems many high order Runge-Kutta methods have order of optimalB-convergence one higher than the stage order. In this paper we show that for the more general class of nonlinear dissipative problems such as result holds only for a small class of Runge-Kutta methods and that such methods have at most classical order 3.     

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.     

B-convergence results for linearly implicit one step methods

B-consistency andB-convergence of linearly implicit one step methods with respect to a class of arbitrarily stiff semi-linear problems are considered. Order conditions are derived. An algorithm for constructing methods of order>1 is shown and examples are given. By suitable modifications of the methods the occurring order reduction is decreased.     

A necessary condition forB-convergence of Runge-Kutta methods

Algebraic stability is a well-known property necessary forB-convergence of a Runge-Kutta method, provided its nodesc _i are such thatc _i – c _j wheneveri j. In this paper a slightly weaker property is established which becomes algebraic stability as soon asc _i – c _j , wheneveri j is excluded.Using this condition it is shown that the Lobatto IIIA methods with more than two stages cannot beB-convergent.This paper was written while the author was visiting the Rijksuniversiteit Leiden in the Netherlands, supported by the Netherlands Organization for Scientific Research (N.W.O.) and by an Erwin Schrödinger scholarship from the Fonds zur Förderung der wissenschaftlichen Forschung.     

Arithmetic tests forA-stability,A[α]-stability,and stiff-stability

Arithmetic tests forA-stability,A[]-stability, and stiff-stability are presented as special cases of a general stability test for numerical integration methods. The test evolves from extracted properties of the characteristic polynomial (in two variables) of the numerical method applied to the prototype scalar ordinary differential equation , Re {q}<0. The several steps of the test impose root clustering conditions — such as being Hurwitz — on a polynomial of one variable.Research sponsored in part by a special grant of the Syracuse University Senate Research Committee to the first author and by the National Science Foundation under Grant ENG76-21816 to the second author.The research reported herein was completed while the first author was on sabbatical leave in the Department of Electrical Engineering and Computer Sciences at the University of California in Berkeley, California.