Runge-Kutta methods with a multiple real eigenvalue only

Due to practical reasons one is interested inv-stage Runge-Kutta methods whose defining matrix has just one realv-fold eigenvalue. The purpose of this note is to show that methods of this type can be constructed by the method of collocation using the ratio between the zeros of certain Laguerre polynomials as collocation points.     

Symplectic Runge-Kutta methods of high order based on W-transformation

In this paper , characterizations of symmetric and symplectic Runge-Kutta methods based on the W-transformation of Hairer and Wanner are presented. Using these characterizations, we construct two families symplectic (symmetric and algebraically stable or algebraically stable) Runge-Kutta methods of high order. Methods constructed in this way and presented in this paper include and extend the known classes of high order implicit Runge-Kutta methods.     

Symplectic Partitioned Runge-Kutta Methods

For partitioned Runge-Kutta methods, in the integration of Hamiltonian systems, a condition for symplecticness and its characterization which is based on the W-transformation of Hairer and Wanner are presented. Examples for partitioned Runge-Kutta methods which satisfy the symplecticness condition are given. A special class of symplectic partitioned Runge-Kutta methods is constructed.     

Symplectic matrices with predetermined left eigenvalues

We prove that given four arbitrary quaternion numbers of norm 1 there always exists a 2×2 symplectic matrix for which those numbers are left eigenvalues. The proof is constructive. An application to the LS category of Lie groups is given.     

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.     

Construction of High Order Symplectic Runge-Kutta Methods

Characterizations of symmetric and symplectic Runge-Kutta methods, which are based on the W-transformation of Hairer and Wanner, are presented. Using these characterizations we construct two classes of high order symplectic (symmetric and algebraically stable or algebraically stable) Runge-Kutta methods. They include and extend known classes of high order implicit Runge-Kutta methods.     

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 Generalized W-Transformation for Constructing Symplectic Partitioned Runge-Kutta Methods

This paper deals with the construction of implicit symplectic partitioned Runge–Kutta methods (PRKM) of high order for separable and general partitioned Hamiltonian systems. The main tool is a generalized W-transformation for PRKM based on different quadrature formulas. Methods of high order and special properties can be determined using the transformed coefficient matrices. Examples are given.     

Pseudo-symplectic Runge-Kutta methods

Apart from specific methods amenable to specific problems, symplectic Runge-Kutta methods are necessarily implicit. The aim of this paper is to construct explicit Runge-Kutta methods which mimic symplectic ones as far as the linear growth of the global error is concerned. Such method of orderp have to bepseudo-symplectic of pseudosymplecticness order2p, i.e. to preserve the symplectic form to within ⊗(h ^2p )-terms. Pseudo-symplecticness conditions are then derived and the effective construction of methods discussed. Finally, the performances of the new methods are illustrated on several test problems.     

Reducible Runge-Kutta methods

Reducible Runge-Kutta methods are characterized by means of special matrices. Previous definitions of reducibility are incorporated. This characterization may be useful in the study of algebraic stability and in studies of existence and uniqueness.     

Canonical Runge-Kutta methods

Summary In the present note we provide a complete characterization of all Runge-Kutta methods which generate a canonical transformation if applied to a Hamiltonian system of ordinary differential equations.     

On real eigenvalues of real tridiagonal matrices

Lower bounds for the number of different real eigenvalues as well as for the number of real simple eigenvalues of a class of real irreducible tridiagonal matrices are given. Some numerical implications are discussed.     

Existence of real eigenvalues of real tensors

We use the Brouwer degree to establish the existence of real eigenpairs of higher order real tensors in various settings. Also, we provide some finer criteria for the existence of real eigenpairs of two-dimensional real tensors and give a complete classification of the Brouwer degree zero and ±2 maps induced by general third order two-dimensional real tensors.     

A computational method for eigenvalues and eigenvectors of a matrix with real eigenvalues

This paper describes a new computational procedure for calculating eigenvalues and eigenvectors of a square matrix. The method is based on a matrix function, the sign of a matrix. Eigenvalues and eigenvectors of matrices with distinct eigenvalues and nondefective matrices with repeated roots can be determined in a straightforward manner. Defective matrices require additional calculations.     

Equilibria of Runge-Kutta methods

Summary It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.     

Explicit stabilized Runge-Kutta methods

Explicit Runge-Kutta methods with the stability domains extended along the real axis are examined. For these methods, a simple and efficient procedure for calculating the stability polynomials is proposed. Three techniques for constructing methods with given stability polynomials are considered. Methods of the second and third orders are constructed, and their accuracy as applied to solving the Prothero-Robinson equation is examined. A comparison of the above methods on some test problems is performed.     

Two-derivative Runge-Kutta methods with optimal phase properties

In this work, we consider two-derivative Runge-Kutta methods for the numerical integration of first-order differential equations with oscillatory solution. We construct methods with constant coefficients and special properties as minimum phase-lag and amplification errors with three and four stages. All methods constructed have fifth algebraic order. We also present methods with variable coefficients with zero phase-lag and amplification errors. In order to examine the efficiency of the new methods, we use four well-known oscillatory test problems.     

Generalized singly-implicit Runge-Kutta methods with arbitrary knots

The aim of this paper is to derive Butcher's generalization of singly-implicit methods without restrictions on the knots. Our analysis yields explicit computable expressions for the similarity transformations involved which allow the efficient implementation of the first phase of the method, i.e. the solution of the nonlinear equations. Furthermore, simple formulas for the second phase of the method, i.e. computation of the approximations at the next nodal point, are established. Finally, the matrix which governs the stability of the method is studied.     

On implicit Runge-Kutta methods with higher derivatives

Two families of implicit Runge-Kutta methods with higher derivatives are (re-)considered generalizing classical Runge-Kutta methods of Butcher type and f Ehle type. For generalized Butcher methods the characteristic functionG() is represented by means of the node polynomial directly, thereby showing that in methods of maximum order,G() is connected withs-orthogonal polynomials in exactly the same way as Padé approximations in the classical case.