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.     

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-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.     

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 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 implementation of Runge-Kutta implicit processes

The application ofA-stable, implicit Runge-Kutta processes to the solution of stiff systems of ordinary differential equations is discussed, and an iterative procedure for solving the resulting nonlinear system of equations is suggested.     

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.     

An extension ofA-stability to alternating direction implicit methods

Completely implicit, noniterative, finite-difference schemes have recently been developed by several authors for nonlinear, multidimensional systems of hyperbolic and mixed hyperbolic-parabolic partial differential equations. The method of Douglas and Gunn or the method of approximate factorization can be used to reduce the computational problem to a sequence of one-dimensional or alternating direction implicit (ADI) steps. Since the eigenvalues of partial differential equations (for example, the equations of compressible fluid dynamics) are often widely distributed with large imaginary parts,A-stable integration formulas provide ideal time-differencing approximations. In this paper it is shown that if anA-stable linear multistep method is used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation, then one can always construct an ADI scheme by the method of approximate factorization which is alsoA-stable, i.e., unconditionally stable. A more restrictive result is given for three spatial dimensions. Since necessary and sufficient conditions forA-stability can easily be determined by using the theory of positive real functions, the stability analysis of the factored partial difference equations is reduced to a simple algebraic test.The main results of this paper were presented at the SIAM National Meeting, Madison, Wis., May 24 to 26, 1978, and section 9 was part of a presentation at the 751st Meeting of the American Mathematical Society, San Luis Obispo, California, Nov. 11 to 12, 1977.     

On implicit Runge-Kutta methods with a stability function having distinct real poles

Because of their potential for offering a computational speed-up when used on certain multiprocessor computers, implicit Runge-Kutta methods with a stability function having distinct poles are analyzed. These are calledmultiply implicit (MIRK) methods, and because of the so-calledorder reduction phenomenon, their poles are required to be real, i.e., only real MIRK's are considered. Specifically, it is proved that a necessary condition for aq-stage, real MIRK to beA-stable with maximal orderq+1 is thatq=1, 2, 3 or 5. Nevertheless, it is shown that for every positive integerq, there exists aq-stage, real MIRK which is stronglyA ₀-stable with orderq+1, and for every evenq, there is aq-stage, real MIRK which isI-stable with orderq. Finally, some useful examples of algebraically stable real MIRK's are given.This work was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18107 while the author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665-5225.     

Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations

Summary GeneralizedA()-stable Runge-Kutta methods of order four with stepsize control are studied. The equations of condition for this class of semiimplicit methods are solved taking the truncation error into consideration. For application anA-stable and anA(89.3°)-stable method with small truncation error are proposed and test results for 25 stiff initial value problems for different tolerances are discussed.     

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.     

Note onA-stability of multistep multiderivative methods

Daniel and Moore [4] conjectured that anA-stable multistep method using higher derivatives cannot have an error order exceeding 2l. We confirm partly this conjecture by showing that for a large class ofA-stable methods the error order can not be 2l+1 mod 4. This extends results found in Jeltsch [13].     

Characterization of allA-stable methods of order 2m-4

The characterization ofA-stable methods is often considered as a very difficult task (see e.g. [1]). In recent years, simple proofs have been found for methods of orderp2m-2 (see [2], [3], [7]). In this paper, we characterize theA-acceptable approximations of orderp 2m-4 and apply the result to 12-parameter families of implicit Runge-Kutta methods.     

Two-step numerical methods for parabolic differential equations

A family of two-stepA-stable methods of maximal order for the numerical solution of ordinary differential systems is developed. If these methods are applied to the stiff, large systems which originate from linear parabolic differential equations they yield a large, sparse set of linear algebraic equations of special form. This set is considerably easier to solve than the algebraic equations which are obtained when using diagonal Obrechkoff methods, which are one-step,A-stable and of maximal order     

Multiderivative Runge-Kutta processes for two-point boundary value problems

A class of finite difference schemes for the solution of a nonlinear system of first order differential equations with two point boundary conditions which shares properties with Runge-Kutta processes and gap schemes is discussed. The order conditions for the coefficients of these processes, techniques for reducing these order conditions in number and the symmetry conditions are given. A symmetricA-stable eight order process which has second, fourth and sixth orderA-stable processes embedded in it is given as an example.Research supported in part by the United States Air Force under contract AFOSR-89-0383.     

A Liouville property for spherical averages in the plane

It is shown that any continuous bounded function f on such that , is constant provided r is a strictly positive real function on satisfying The proof is based on a minimum principle exploiting that and on a study of -stable sets, i.e., sets A such that the circle of radius r(x) centered at x is contained in A whenever . The latter reveals that there is no disjoint pair of non-empty closed -stable subsets in unless (taking spheres this holds for any , ). A counterexample is given where . Received November 24, 1999 / Published online December 8, 2000     

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.     

An inverse eigenvalue problem: ComputingB-stable Runge-Kutta methods having real poles

The implementation of implicit Runge-Kutta methods requires the solution of large sets of nonlinear equations. It is known that on serial machines these costs can be reduced if the stability function of ans-stage method has only ans-fold real pole. Here these so-called singly-implicit Runge-Kutta methods (SIRKs) are constructed utilizing a recent result on eigenvalue assignment by state feedback and a new tridiagonalization, which preserves the entries required by theW-transformation. These two algorithms in conjunction with an unconstrained minimization allow the numerical treatment of a difficult inverse eigenvalue problem. In particular we compute an 8-stage SIRK which is of order 8 andB-stable. This solves a problem posed by Hairer and Wanner a decade ago. Furthermore, we finds-stageB-stable SIRKs (s=6,8) of orders, which are evenL-stable.     

A geometric study of the D-stability problem

The set of D-stable matrices is studied from a differentiable viewpoint, and some general properties of the set are obtained. We study also those 4 by 4 D-stable matrices A which have the property that A+tI is D-stable for all t?0. An example of a 4 by 4 D-stable matrix A without the property is given.     

Two-Step Fifth-Order Methods for Evolutionary Problems with Positive Operators

The family of three-level fifth-order time integrators are considered for hyperbolic, parabolic PDEs and stiff ODEs. They are classified into two parts depending on positivity or negativity of the operators corresponding to the second time derivatives. Two options are presented for hyperbolic equations. Stability analysis is performed for linear cases. It is shown that the scheme for ODEs can be nearly A-stable and nearlyL -stable for particular values of its free parameter. Numerical illustration is presented for hyperbolic case.