Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.Work supported by the Ministero della Ricerca Scientifica, 40% project, and C.N.R. (contract of research # 92.00535.01).     

Analysis of the dynamics of local error control via a piecewise continuous residual

Positive results are obtained about the effect of local error control in numerical simulations of ordinary differential equations. The results are cast in terms of the local error tolerance. Under theassumption that a local error control strategy is successful, it is shown that a continuous interpolant through the numerical solution exists that satisfies the differential equation to within a small, piecewise continuous, residual. The assumption is known to hold for thematlab ode23 algorithm [10] when applied to a variety of problems. Using the smallness of the residual, it follows that at any finite time the continuous interpolant converges to the true solution as the error tolerance tends to zero. By studying the perturbed differential equation it is also possible to prove discrete analogs of the long-time dynamical properties of the equation—dissipative, contractive and gradient systems are analysed in this way. Supported by the Engineering and Physical Sciences Research Council under grants GR/H94634 and GR/K80228. Supported by the Office of Naval Research under grant N00014-92-J-1876 and by the National Science Foundation under grant DMS-9201727.     

Convergence results for the MATLAB ode23 routine

We prove convergence results on finite time intervals, as the user-defined tolerance τ→0, for a class of adaptive timestepping ODE solvers that includes the ode23 routine supplied in MATLAB Version 4.2. In contrast to existing theories, these convergence results hold with error constants that are uniform in the neighbourhood of equilibria; such uniformity is crucial for the derivation of results concerning the numerical approximation of dynamical systems. For linear problems the error estimates are uniform on compact sets of initial data. The analysis relies upon the identification of explicit embedded Runge-Kutta pairs for which all but the leading order terms of the expansion of the local error estimate areO(∥f(u∥)²). This work was partially supported by NSF Grant DMS-95-04879.     

Error growth analysis via stability regions for discretizations of initial value problems

This paper deals with numerical methods for the solution of linear initial value problems. Two main theorems are presented on the stability of these methods. Both theorems give conditions guaranteeing a mild error growth, for one-step methods characterized by a rational function ϕ(z). The conditions are related to the stability regionS={z:z∈ℂ with |ϕ(z)|≤1}, and can be viewed as variants to the resolvent condition occurring in the reputed Kreiss matrix theorem. Stability estimates are presented in terms of the number of time stepsn and the dimensions of the space. The first theorem gives a stability estimate which implies that errors in the numerical process cannot grow faster than linearly withs orn. It improves previous results in the literature where various restrictions were imposed onS and ϕ(z), including ϕ′(z)≠0 forz∈σS andS be bounded. The new theorem is not subject to any of these restrictions. The second theorem gives a sharper stability result under additional assumptions regarding the differential equation. This result implies that errors cannot grow faster thann ^β, with fixed β<1. The theory is illustrated in the numerical solution of an initial-boundary value problem for a partial differential equation, where the error growth is measured in the maximum norm.     

A note on contractivity in the numerical solution of initial value problems

This paper concerns the stability analysis of numerical methods for approximating the solutions to (stiff) initial value problems. Our analysis includes the case of (nonlinear) systems of differential equations that are essentially more general than the classical test equationU=U, with a complex constant.We explore the relation between two stability concepts, viz. the concepts of contractivity and weak contractivity.General Runge-Kutta methods, one-stage Rosenbrock methods and a notable rational Runge-Kutta method are analysed in some detail.     

A singular initial value problem and self-similar solutions of a nonlinear dissipative wave equation

We present a systematic study of local solutions of the ODE of the form near t=0. Such ODEs occur in the study of self-similar radial solutions of some second order PDEs. A general theorem of existence and uniqueness is established. It is shown that there is a dichotomy between the cases γ>0 and γ<0, where γ=∂f/∂x^′ at t=0. As an application, we study the singular behavior of self-similar radial solutions of a nonlinear wave equation with superlinear damping near an incoming light cone. A smoothing effect is observed as the incoming waves are focused at the tip of the cone.     

Stability radius of polynomials occurring in the numerical solution of initial value problems

This paper deals with polynomial approximations(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by stability requirements, we present a numerical study of the largest diskD()={z C: |z+|} that is contained in the stability regionS()={z C: |(z)|1}. The radius of this largest disk is denoted byr(), the stability radius. On the basis of our numerical study, several conjectures are made concerningr _m,p=sup {r(): _m,p}. Here _{m, p} (1pm; p, m integers) is the class of all polynomials(x) with real coefficients and degree m for which(x)=exp(x)+O(x ^p+1) (forx 0).     

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green’s function approach. O(h⁶) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.     

A posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations

We present constructive a posteriori estimates of inverse operators for initial value problems in linear ordinary differential equations (ODEs) on a bounded interval. Here, “constructive” indicates that we can obtain bounds of the operator norm in which all constants are explicitly given or are represented in a numerically computable form. In general, it is difficult to estimate these inverse operators a priori. We, therefore, propose a technique for obtaining a posteriori estimates by using Galerkin approximation of inverse operators. This type of estimation will play an important role in the numerical verification of solutions for initial value problems in nonlinear ODEs as well as for parabolic initial boundary value problems.     

Convergence of linear multistep and one-leg methods for stiff nonlinear initial value problems

To prove convergence of numerical methods for stiff initial value problems, stability is needed but also estimates for the local errors which are not affected by stiffness. In this paper global error bounds are derived for one-leg and linear multistep methods applied to classes of arbitrarily stiff, nonlinear initial value problems. It will be shown that under suitable stability assumptions the multistep methods are convergent for stiff problems with the same order of convergence as for nonstiff problems, provided that the stepsize variation is sufficiently regular.     

Input-to-state stability of Runge–Kutta methods for nonlinear control systems

In the paper we introduce input-to-state stability (ISS) of Runge–Kutta methods for control systems. The ISS properties of Runge–Kutta methods are studied for linear control systems and nonlinear control systems, respectively. The previously reported results in literature are special cases of ISS of Runge–Kutta methods.     

High-order nonlinear initial-value problems countably determined

We give a numerical approximation of the solution of a high-order nonlinear initial-value problem by making use of certain properties of an adequate Schauder basis.     

Dissipativity of Runge-Kutta methods in Hilbert spaces

This paper concerns the discretization by Runge-Kutta methods of the initial value problemu _t =f(u), under the dissipative structural condition that there exist α≥0, β>0, such thatf:W→H, ℜe, ∀w∈W, for complex Hilbert spacesW⊆H. It is shown that strong A-stability is necessary to ensure the dissipativity of the method, whilst algebraic stability plus |R(∞)|<1 is a sufficient condition in the case of DJ-irreducible methods.     

Relationships among some classes of implicit Runge-Kutta methods and their stability functions

In this paper we apply the theory for implicit Runge-Kutta methods presented by Stetter to a number of subclasses of methods that have recently been discussed in the literature. We first show how each of these classes can be expressed within this theoretical framework and from this we are able to establish a number of relationships among these classes. In addition to improving the current state of understanding of these methods, their expression within this theoretical framework makes it possible for us to obtain results giving general forms for their stability functions.This work was supported by the Natural Sciences and Engineering Research Council of Canada.     

A note on minimization problems and multistep methods

Summary. We use the qualitative properties of the solution flow of the gradient equation to compute a local minimum of a real-valued function . Under the regularity assumption of all equilibria we show a convergence result for bounded trajectories of a consistent, strictly stable linear multistep method applied to the gradient equation. Moreover, we compare the asymptotic features of the numerical and the exact solutions as done by Humphries, Stuart (1994) and Schropp (1995) for one-step methods. In the case of -stable formulae this leads to an efficient solver for stiff minimization problems. Received July 10, 1995 / Revised version received June 27, 1996     

A-stable spline-collocation methods of multivalue type

In this paper the general classV of spline-collocation methods presented by Mülthei is investigated. The methods ofV approximate solutions of first order initial value problems. ClassV contains as subclass the methods of so-called multivalue type, and in particular contains the generalized singly-implicit methods treated by Butcher.Any multivalue type representativeU V yields a matrix valued function corresponding toU, which characterizes the region of absolute stability ofU. If a sequence (U()) of multivalue type representatives ofV tending to some singlevalue type representative V is considered, it can easily be seen by the structure of , that the sequence of the greatest eigenvalues of the (.,) tends to the stability function corresponding to . This fact allows one to construct one-parameter families of A-stable methods of multivalue type.     

Extremal hypergraph problems and convex hulls

Theprofile of a hypergraph onn vertices is (f ₀, f₁, ...,f _n) wheref _i denotes the number ofi-element edges. The extreme points of the set of profiles is determined for certain hypergraph classes. The results contain many old theorems of extremal set theory as particular cases (Sperner. Erdős—Ko—Rado, Daykin—Frankl—Green—Hilton).     

Logical structures and genus of proofs

Any arbitrarily complicated non-oriented graph, that is a graph of arbitrarily large genus, can be encoded in a cut-free proof. This unpublished result of Statman was shown in the early seventies. We provide a proof of it, and of a number of other related facts.