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.     

Counterexamples to a stability barrier

Summary We consider multistep difference schemes for the linear, constant coefficient advection equationu _t=cu_x. In the last section of Strang [5], a barrier for the order of stable schemes has been given which was independent of the number of time levels. Here we give two types of counterexamples to this barrier. The first type consists of formulas of highest possible order to a given stencil, which are stable for small positive Courant numbers. Further a formula is given where one does not insist on having the highest possible order for the stencil and uses the gained freedom to ensure stability for small positive and negative Courant numbers. In addition, an explicit formula is derived for the schemes of highest possible order when stability is disregarded.This work has been performed in the project Mehrschritt-Differenzenschemata of the Schwerpunktprogramm Finite Approximationen in der Strömungsmechanik which is supported by the DFG     

Eine Methode zur berechnung sämtlicher Lösungen von Polynomgleichungssystemen

Summary In this paper a numerical method is given to compute all solutions of systemsT ofn polynomial equations inn unknowns on the only premises that the sets of solutions of these systems are finite. The method employed is that of embedding, i.e. the systemT is embedded in a set of systems which are successively solved, starting with one having solutions easily to compute and proceding toT in a finite series of steps. An estimation of the number of steps necessary is given. The practicability of the method is proved for all systemsT. Numerical examples and results are contained.

Diese Arbeit ist eine Zusammenfassung der Ergebnisse der Dissertation, die der Verfasser an der Johannes-Kepler-Universität Linz und der Technischen Universität München unter der Betreuung von Prof. Dr. Hansjörg Wacker angeferetigt hat. 2. Begutachter: Prof. Dr. Manfred Feilmeier     

A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations

This paper deals with adapting Runge-Kutta methods to differential equations with a lagging argument. A new interpolation procedure is introduced which leads to numerical processes that satisfy an important asymptotic stability condition related to the class of testproblemsU(t)=U(t)+U(t–) with , C, Re()<–||, and >0. Ifc _i denotes theith abscissa of a given Runge-Kutta method, then in thenth stept _n–1t _n :=t _n–1+h of the numerical process our interpolation procedure computes an approximation toU(t _n–1+c _i h–) from approximations that have already been generated by the process at pointst _j–1+c _i h(j=1,2,3,...). For two of these new processes and a standard process we shall consider the convergence behaviour in an actual application to a given, stiff problem.     

Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils

Self-adjoint quadratic operator pencilsL()= ² A + B + C with a noninvertible leading operatorA are considered. In particular, a characterization of the spectral points of positive and of negative type ofL is given, and their behavior under a compact perturbation is studied. These results are applied to a pencil arising in magnetohydrodynamics.     

Improvement of numerical solution of boundary layer problems by incorporation of asymptotic approximations

Summary A method for improvement of the numerical solution of differential equations by incorporation of asymptotic approximations is investigated for a class of singular perturbation problems.Uniform error estimates are derived for this method when implemented in known difference schemes and applied to linear second order O.D.E.'s. An improvement by a factor of ⁿ⁺¹ can be obtained (where is the small parameter andn is the order of the asymptotic approximation) for a small amount of extra work. Numerical experiments are presented.     

Numerical application of Euler's series transformation and its generalizations

Summary A nonlinear generalizationÊ _z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE _z and a twoparametric methodE . It is shown how to applyE orE _, to compute the valuef(z_o) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE _, are superior toÊ _z . A compact recursive algorithm is given for computingE andE _,.     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

Adams methods for neutral functional differential equations

Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a non-smooth solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.     

Point systems with extremal properties and conformal mapping

Summary LetD be the unit disk. It is a well-known fact that by use of simply connected domain methods the general conformal mapping problem of doubly connected domains can be reduced to the special case of a regionD bounded by the unit circle and a Jordan curve inD, where . Here we treat this special case and assume to be piecewise analytic without cusps. Let be the conformal mapping of {<|w|<1} onto the doubly connected domainD with (1)=1. We approximate by interpolation with finite Laurent series using point systems with extremal properties. Numerical results for four examples are given.     

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.     

Acceleration of convergence for finite element solutions of the Poisson equation

Summary Letu _h be the finite element solution to–u=f with zero boundary conditions in a convex polyhedral domain . Fromu _h we calculate for eachz and ||1 an approximationu _h ^– (z) toD u(z) with |D u(z)–u _h ^– (z)|=O(h ^2k–2) wherek is the order of the finite elements. The same superconvergence order estimates are obtained also for the boundary flux. We need not work on a regular mesh but we have to compute averages ofu _h where the diameter of the domain of integration must not depend onh.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain

Summary The argument principle is a natural and simple method to determine the number of zeros of an analytic functionf(z) in a bounded domainD. N, the number of zeros (counting multiplicities) off(z), is 1/2 times the change in Argf(z) asz moves along the closed contour D. Since the range of Argf(z) is (–, ] a critical point in the computational procedure is to assure that the discretization of D, {z _i ,i=1, ...,M}, is such that . Discretization control which may violate this inequality can lead to an unreliable algorithm. Mathematical theorems derived for the discretization of D lead to a completely reliable algorithm to computeN. This algorithm also treats in an elementary way the case when a zero is on or near the contour D. Numerical examples are given for the reliable algorithm formulated here and it is pointed out in these examples how inadequate discretization control can lead to failure of other algorithms.Dedicated to Professor Ivo Babuka in commemoration of his sixtieth birthdayThis research is part of the doctoral dissertation of this author     

Small potential corrections for the discrete eigenvalues of the Sturm-Liouville problem

Summary The inverse Sturm-Liouville problem is the problem of finding a good approximation of a potential functionq such that the eigenvalue problem (^*)–y +qy=y holds on (0, ) fory(0)=y()=0 and a set of given eigenvalues . Since this problem has to be solved numerically by discretization and since the higher discrete eigenvalues strongly deviate from the corresponding Sturm-Liouville eigenvalues , asymptotic corrections for the 's serve to get better estimates forq. Let _k (1kn) be the first eigenvalues of (^*), let _k be the corresponding discrete eigenvalues obtained by the finite element method for (^*) and let _{k k} for the special caseq=0. Then, starting from an asymptotic correction technique proposed by Paine, de Hoog and Anderssen, new estimates for the errors of the corrected discrete eigenvalues are obtained and confirm and improve the knownO(kh ²)(h:=/(n+1)) behaviour. The estimates are based on new Sobolev inequalities and on Fourier analysis and it is shown that for 4+c ₂ k(n+1)/2, wherec ₁ andc ₂ are constants depending onq which tend to 0 for vanishingq.     

Computation of the smallest positive eigenvalue of a quadratic λ-matrix

Summary The computation of the smallest positive eigenvalue ^* of a quadratic -matrix is used to determine the stability of structures. In addition to existence results we derive two efficient and reliable methods to calculate ^*. Both methods are based on shift techniques which are discussed thoroughly with respect to convergence.     

Product integration for the linear transport equation in slab geometry

Summary In this paper we present a product quadrature rule for the discretization of the well-known linear transport equation in slab geometry. In particular we give an algorithm for constructing the weights of the rule and prove that the order of convergence isO(n ^–3+ ), >0 small as we like. Numerical examples are given, and our formula is also compared with product Simpson rules. Finally, we examine a Nyström method based on our quadrature.Work sponsored by the Ministero della Pubblica Instruzione of Italy     

Harmonic functions on nilpotent groups

For a probability measure on a locally compact groupG which is not supported on any proper closed subgroup, an elementF ofL (G) is called -harmonic if F(st)d(t)=F(s), for almost alls inG. Constant functions are -harmonic and it is known that for abelianG all -harmonic functions are constant. For other groups it is known that non constant -harmonic functions exist and the question of whether such functions exist on nilpotent groups is open, though a number of partial results are known. We show that for nilpotent groups of class 2 there are no non constant -harmonic functions. Our methods also enable us to give new proofs of results similar to the known partial results.     

Modularity in the lattice of Σ-permutable subgroups

For a Hall system of a finite solvable group G, it is known that the set of -permutable subgroups is a sublattice of the subgroup lattice of G. We investigate the class SPM of groups in which the lattice is modular. We prove that if is modular, then U V for all (an evidently stronger condition). Both of these phenomena—the modularity of and whether two -permutable subgroups U and V permute with each other—are shown to be determined locally, by what happens at each prime. The class SPM is shown to be quotient closed, but not direct product or subgroup closed.This revised electronic version of the Abstract includes the formulas that were missing in the previous electronic version published online in September 2004.     

The average a posteriori error of numerical methods

Summary The definition of the average error of numerical methods (by example of a quadrature formula to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation by an averaging process over the set of possible information, which is used by (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation and compute the resulting average error. The latter is minimal for the knots . (It is well known that the maximal error is minimal for the knotsa _i .) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.