On circle containment orders

A partially ordered set is called acircle containment order provided one can assign to each element of the poset a circle in the plane so thatxy iff the circle assigned tox is contained in the circle assigned toy. It has been conjectured that every finite three-dimensional partially ordered set is a circle containment order. We show that the infinite three dimensional posetZ ³ isnot a circle containment order.Research supported in part by the Office of Naval Research, contract number N00014-85-K0622.Research supported in part by National Science Foundation, grant number DMS-8403646.     

Bruhat Order for Two Flags and a Line

The classical Ehresmann-Bruhat order describes the possible degenerations of a pair of flags in a linear space V under linear transformations of V; or equivalently, it describes the closure of an orbit of GL(V acting diagonally on the product of two flag varieties.We consider the degenerations of a triple consisting of two flags and a line, or equivalently the closure of an orbit of GL(V) acting diagonally on the product of two flag varieties and a projective space. We give a simple rank criterion to decide whether one triple can degenerate to another. We also classify the minimal degenerations, which involve not only reflections (i.e., transpositions) in the Weyl group S_VSn = dim(V, but also cycles of arbitrary length. Our proofs use only elementary linear algebra and combinatorics.     

A-stable diagonally implicit Runge-Kutta-Nyström methods for parallel computers

In this paper, we study diagonally implicit Runge-Kutta-Nyström methods (DIRKN methods) for use on parallel computers. These methods are obtained by diagonally implicit iteration of fully implicit Runge-Kutta-Nyström methods (corrector methods). The number of iterations is chosen such that the method has the same order of accuracy as the corrector, and the iteration parameters serve to make the method at least A-stable. Since a large number of the stages can be computed in parallel, the methods are very efficient on parallel computers. We derive a number of A-stable, strongly A-stable and L-stable DIRKN methods of orderp withs ^* (p) sequential, singly diagonal-implicit stages wheres ^*(p)=[(p+1)/2] ors ^* (p)=[(p+1)/2]+1,[°] denoting the integer part function.These investigations were supported by the University of Amsterdam with a research grant to enable the author to spend a total of two years in Amsterdam.     

Singly implicit diagonally extended Runge-Kutta methods of fourth order

Singly implicit diagonally extended Runge-Kutta methods make it possible to combine the merits of diagonally implicit methods (namely, the simplicity of implementation) and fully implicit ones (high stage order). Due to this combination, they can be very efficient at solving stiff and differential-algebraic problems. In this paper, fourth-order methods with an explicit first stage are examined. The methods have the third or fourth stage order. Consideration is given to an efficient implementation of these methods. The results of tests in which the proposed methods were compared with the fifth-order RADAU IIA method are presented.     

Two-stage and Three-stage Diagonally Implicit Runge-Kutta Nystrom Methods of Orders Three and Four

We investigate the existence of two-stage and three-stage R-stable,P-stable, RL-stable, and dispersively enhanced diagonally implicitRunge-Kutta Nyström methods of orders three and four. Wefirst show that a one-parameter family of two-stage third-orderR-stable diagonally implicit methods exists, and that theirdispersive order is at most four. From this we show that two-stagefourth-order R-stable, and third-order P-stable and RL-stablediagonally implicit methods do not exist. Next we show a two-parameterfamily of three-stage fourth-order R-stable diagonally implicitmethods exists with dispersive order at most four, and thatthis family contains a one-parameter family of P-stable methodsand a unique RL-stable. We also show that a one-parameter familyof fourth-order diagonally implicit methods with dispersiveorder at least six exists, and that they are not R-stable. Wepresent third- and fourth-order R-stable and P-stable methodswith small principal truncation coefficients and discuss howthese methods might be implemented in an efficient integrator.     

Theorems of ergodic type for ρ-nonexpansive mappings in the Hilbert ball

Summary We present two types of ergodic theorems for contractive iterations in the Hilbert ball. Both explicit and implicit iterations are discussed.     

How to avoid accuracy and order reduction in Runge–Kutta methods as applied to stiff problems

The solution of stiff problems is frequently accompanied by a phenomenon known as order reduction. The reduction in the actual order can be avoided by applying methods with a fairly high stage order, ideally coinciding with the classical order. However, the stage order sometimes fails to be increased; moreover, this is not possible for explicit and diagonally implicit Runge–Kutta methods. An alternative approach is proposed that yields an effect similar to an increase in the stage order. New implicit and stabilized explicit Runge–Kutta methods are constructed that preserve their order when applied to stiff problems.     

The weak* similarity property on dual operator algebras

This paper is devoted to dual operator algebras, that isw ^*-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebrasA with the following weak^* similarity property: for every Hilbert spaceH, anyw ^*-continuous unital homomorphism fromA intoB(H) is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras, in analogy with some recent work of Pisier on the similarity problem on operator algebras.     

A nonlinear optimization approach to the construction of general linear methods of high order

We describe the construction of diagonally implicit multistage integration methods of order and stage order p = q = 7 and p = q = 8 for ordinary differential equations. These methods were obtained using state-of-the-art optimization methods, particularly variable-model trust-region least-squares algorithms.     

An eighth order tridiagonal finite difference method for nonlinear two-point boundary value problems

We present an eighth order finite difference method for the second order nonlinear boundary value problemy=f(x, y), y(a)=A, y(b)=B; the method iseconomical in the sense that each discretization of the differential equation at an interior grid point is based on seven evaluations off. For linear differential equations, the scheme leads to tridiagonal linear systems. We showO(h ⁸)-convergence of the method and demonstrate computationally its eighth order.     

Factorizations of and extensions to J-unitary rational matrix functions on the unit circle

This paper concerns two topics: (1) minimal factorizations in the class ofJ-unitary rational matrix functions on the unit circle and (2) completions of contractive rational matrix functions on the unit circle to two by two block unitary rational matrix functions which do not increase the McMillan degree. The results are given in terms of a special realization which does not require any additional properties at zero and at infinity. The unitary completion result may be viewed as a generalization of Darlington synthesis.     

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.     

Über das Stekloffsche Eigenwertproblem: Isoperimetrische Ungleichungen für symmetrische Gebiete

Summary Isoperimetric inequalities ofPólya [6] for symmetric membranes are extended to the Stekloff problem. The given symmetric domainG _z is mapped conformally onto a circle; some (harmonic) eigenfunctions of the circle are transplanted ontoG _z ; application of Rayleigh's and Poincaré's principles to the transplanted functions gives upper bounds for a number of eigenvalues ofG _z which depends on the order of symmetry of the domain.     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

A character formula for infinite dimensional representations of the type i superalgebras sl(m/n) and c(2)

Infinite dimensional representations of Type I Lie superalgebra sl(m/n) for which the highest weight is singly or multiply and diagonally atypical are shown to have a character formula which is a modified version of the Bernstein-Leites character formula. This formula is also shown to be valid for all infinite dimensional representations of the Lie superalgebra C(2) classified according to the atypicality type of the highest weight A     

The Construction of Practical General Linear Methods

Using the property of inherent Runge—Kutta stability, it is possible to construct diagonally implicit general linear methods with stability regions exactly the same as for Runge—Kutta methods. In addition to A-stable methods found in this way, it is also possible to construct explicit methods with stability regions identical to those of explicit Runge—Kutta methods. The use of doubly companion matrices makes it possible to find all explicit and diagonally-implicit methods possessing the inherent Runge—Kutta stability property.This revised version was published online in October 2005 with corrections to the Cover Date.     

The order of points on the second convex hull of a simple polygon

Given a setV of points on the plane, if {q ₁,...,q _n } is the set of points on the second convex hull ofV, the order in which these points are visited in anyV-gon is characterized. This order must verify two similar conditions to those of Kuratowski's theorem for planar graphs. Moreover, the number of possible orders that verify these conditions is obtained. It isO(5 ⁿ ).     

Runge–Kutta methods and viscous wave equations

We study the numerical time integration of a class of viscous wave equations by means of Runge–Kutta methods. The viscous wave equation is an extension of the standard second-order wave equation including advection–diffusion terms differentiated in time. The viscous wave equation can be very stiff so that for time integration traditional explicit methods are no longer efficient. A-Stable Runge–Kutta methods are then very good candidates for time integration, in particular diagonally implicit ones. Special attention is paid to the question how the A-Stability property can be translated to this non-standard class of viscous wave equations.      

A comparison of Rosenbrock and ESDIRK methods combined with iterative solvers for unsteady compressible flows

In this article, we endeavour to find a fast solver for finite volume discretizations for compressible unsteady viscous flows. Thereby, we concentrate on comparing the efficiency of important classes of time integration schemes, namely time adaptive Rosenbrock, singly diagonally implicit (SDIRK) and explicit first stage singly diagonally implicit Runge-Kutta (ESDIRK) methods. To make the comparison fair, efficient equation system solvers need to be chosen and a smart choice of tolerances is needed. This is determined from the tolerance TOL that steers time adaptivity. For implicit Runge-Kutta methods, the solver is given by preconditioned inexact Jacobian-free Newton-Krylov (JFNK) and for Rosenbrock, it is preconditioned Jacobian-free GMRES. To specify the tolerances in there, we suggest a simple strategy of using TOL/100 that is a good compromise between stability and computational effort. Numerical experiments for different test cases show that the fourth order Rosenbrock method RODASP and the fourth order ESDIRK method ESDIRK4 are best for fine tolerances, with RODASP being the most robust scheme.     

Fehlerabschätzung für die numerische Differentiation analytischer Funktionen durch Quadratur

Summary Numerical treatment of the integral in Cauchy's integral formula produces approximations for the derivatives of an analytic functionf; this fact has already been utilized byLyness andMoler [3, 4]. In the present paper this idea is investigated especially in view of the accuracy of these formulas regarded as quadrature formulas. Since the integration can be reduced to the integration of a periodic analytic function, it is possible to continue the considerations ofDavis [2] in order to find bounds for the error of the differentiation rules. For the application of these bounds one essentially needs estimations of the maximum off on a circle inside of its region of analyticity. Examples show the practical use of the bounds.

---

---

Meinem verehrten LehrerH. Görtler zur Vollendung seines 60. Lebensjahres gewidmet