The stability function for multistep collocation methods

Summary C-polynomials for rational approximation to the exponential function was introduced by Nørsett [7] to study stability properties of one-step methods. For one-step collocation methods theC-polynomial has a very simple form. In this paper we studyC-polynomials for multistep collocation methods and obtain results that generalize those in the one-step case, and provide a way to analyze linear stability of such methods.     

On a least-squares collocation method for linear differential-algebraic equations

Summary The aim of this note is to extend some results on least-squares collocation methods and to prove the convergence of a least-squares collocation method applied to linear differential-algebraic equations. Some numerical examples are presented.     

B-convergence properties of defect correction methods. I

Summary B-convergence properties of defect correction methods based on the implicit Euler and midpoint schemes are discussed. The property ofB-convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness. It turns out that the orders ofB-convergence of these methods coincide with the conventional orders of convergence of these methods derived under the assumption that.hL is small (whereL is a Lipschitz constant of the right-hand side). In Part I these assertions are reduced to the validity of the so-called Hypothesis A which is discussed in greater detail in Part II. Numerical experiments confirming the theoretical analysis are also given in Part II.     

B-convergence properties of defect correction methods. II

Summary B-convergence properties of defect correction methods based on the implicit Euler and midpoint schemes are discussed. The property ofB-convergence means that there exist global error bounds for nonlinear stiff problems independent of their stiffness. It turns out that the orders ofB-convergence of these methods coincide with the conventional orders of small (whereL is a Lipschitz constant of the right-hand side). In Part I these assertions are reduced to the validity of the so-called HypothesisA which is discussed in greater detail in Part II. Numerical experiments confirming the theoretical analysis are also given in Part II.     

General framework,stability and error analysis for numerical stiff boundary value methods

Summary This paper provides a general framework, called theoretical multiple shooting, within which various numerical methods for stiff boundary value ordinary differential problems can be analyzed. A global stability and error analysis is given, allowing (as much as possible) the specificities of an actual numerical method to come in only locally. We demonstrate the use of our results for both one-sided and symmetric difference schemes. The class of problems treated includes some with internal (e.g. turning point) layers.     

Einige Stabilitätsungleichungen für gewöhnliche Differenzengleichungen in nichtkompakter Form

Summary For finite difference equations in noncompact form approximating ordinary boundary value problems some stability inequalities are proved inl ^p -spaces using Spijker-norms. As applications the convergence of the discrete Green's function and the possibility of lower order approximation near the boundary are treated.

     

On the reduction of Holt's problem to a finite interval

Summary The reduction of Holt's problem to a finite interval is studied. We give the error estimate for the reduction. The solution of the boundary value problem is computed by a variant of the chasing method.This work was partly supported by the Hungarian Academic Research Foundation under project N⁰1-3-86-221     

On some difference schemes for singular singularly-perturbed boundary value problems

Summary Singularly perturbed boundary value ordinary differential problems are considered, where the problem defining the reduced solution is singular. For numerical approximation, families of symmetric difference schemes, which are equivalent to certain collocation schemes based on Gauss and Lobatto points, are used. Convergence results, previously obtained for the regular singularly perturbed case, are extended. While Gauss schemes are extended with no change, Lobatto schemes require a small modification in the mesh selection procedure. With meshes as prescribed in the text, highly accurate solutions can be obtained with these schemes for singular singularly perturbed problems at a very reasonable cost. This is demonstrated by examples.This research was completed while the author was visiting the Department of Applied Mathematics, Weizmann Inst., Rehovot, Israel. The author was supported in part under NSERC grant A4306     

Uniform fourth order difference scheme for a singular perturbation problem

Summary The numerical solution of a nonlinear singularly perturbed two-point boundary value problem is studied. The developed method is based on Hermitian approximation of the second derivative on special discretization mesh. Numerical examples which demonstrate the effectiveness of the method are presented.This research was partly supported by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.-Yugoslav Joint Board on Scientific and Technological Cooperation (grants JF 544, JF 799)     

A class of simple exponential B-splines and their application to numerical solution to singular perturbation problems

Summary We shall consider an application of simple exponential splines to the numerical solution of singular perturbation problem. The computational effort involved in our collocation method is less than that required for the other methods of exponential type.     

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.     

Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

Natural Runge-Kutta and projection methods

Summary Recently the author defined the class of natural Runge-Kutta methods and observed that it includes all the collocation methods. The present paper is devoted to a complete characterization of this class and it is shown that it coincides with the class of the projection methods in some polynomial spaces.This work was supported by the Italian Ministero della Pubblica Istruzione, funds 40%     

On error behaviour of partitioned linearly implicit runge-kutta methods for stiff and differential algebraic systems

This paper studies partitioned linearly implicit Runge-Kutta methods as applied to approximate the smooth solution of a perturbed problem with stepsizes larger than the stiffness parameter. Conditions are supplied for construction of methods of arbitrary order. The local and global error are analyzed and the limiting case 0 considered yielding a partitioned linearly implicit Runge-Kutta method for differential-algebraic equations of index one. Finally, some numerical experiments demonstrate our theoretical results.     

Correction of Numerov's eigenvalue estimates

Summary The error in the estimate of thekth eigenvalue of a regular Sturm-Liouville problem obtained by Numerov's method with mesh lengthh isO(k ⁶ h ⁴). We show that a simple correction technique of Paine, de Hoog and Anderssen reduces the error to one ofO(k ³ h ⁴). Numerical examples demonstrate the usefulness of this correction even for low values ofk.     

Rosenbrock methods for Differential Algebraic Equations

Summary This paper deals with the numerical solution of Differential/Algebraic Equations (DAE) of index one. It begins with the development of a general theory on the Taylor expansion for the exact solutions of these problems, which extends the well-known theory of Butcher for first order ordinary differential equations to DAE's of index one. As an application, we obtain Butcher-type results for Rosenbrock methods applied to DAE's of index one, we characterize numerical methods as applications of certain sets of trees. We derive convergent embedded methods of order 4(3) which require 4 or 5 evaluations of the functions, 1 evaluation of the Jacobian and 1 LU factorization per step.     

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.     

Linearly implicit extrapolation methods for differential-algebraic systems

Summary We consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution-dependent singular matrix. We study extrapolation methods based on two linearly implicit Euler discretizations. Their error behaviour is explained by perturbed asymptotic expansions.     

B-Convergence of Lobatto IIIC formulas

Summary A completion ofB-convergence results of Lobatto IIIC schemes is presented. In particular, it is shown that Lobatto IIIC schemes with more than two stages areB-convergent when applied to IVPs with a negative one-sided Lipschitz constantm; they are notB-convergent, however, for IVPs with a non-negativem.