On the Convergence of Backward Differentiation Formulas for Stiff Initial Value Problems

The influence of a time-dependent transformation to a numerical method is studied. Thus convergence results of backward differentiation formulas applied to the non-autonomous stiff system y = A(t)y + (t) are given. The approach is based on a special decomposition of the companion matrix.This revised version was published online in October 2005 with corrections to the Cover Date.     

Non-stiff integrators for differential–algebraic systems of index 2

Non-stiff differential-algebraic equations (DAEs) can be solved efficiently by partitioned methods that combine well-known non-stiff integrators from ODE theory with an implicit method to handle the algebraic part of the system. In the present paper we consider partitioned one-step and partitioned multi-step methods for index-2 DAEs in Hessenberg form and the application of these methods to constrained mechanical systems. The methods are presented from a unified point of view. The comparison of various classes of methods is completed by numerical tests for benchmark problems from the literature. This revised version was published online in June 2006 with corrections to the Cover Date.     

An algorithm for the computation of consistent initial values for differential–algebraic equations

We use boundary value methods to compute consistent initial values for fully implicit nonlinear differential-algebraic equations. The obtained algorithm uses variable order formulae and a deferred correction technique to evaluate the error. A rigorous theory is stated for nonlinear index 1, 2 and 3 DAEs of Hessenberg form. Numerical tests on classical index 1, 2 and 3 DAE problems are reported. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Convergence Behaviour of Variable Stepsize Multistep Methods for Singularly Perturbed Problems

In this note, we investigate the convergence behaviour of linear multistep discretizations for singularly perturbed systems, emphasising the features of variable stepsizes. We derive a convergence result for A()-stable linear multistep methods and specify a refined error estimate for backward differentiation formulas. Important ingredients in our convergence analysis are stability bounds for non-autonomous linear problems that are obtained by perturbation techniques.     

Improving the accuracy of BDF methods for index 3 differential-algebraic equations

Methods for solving index 3 DAEs based on BDFs suffer a loss of accuracy when there is a change of step size or a change of order of the method. A layer of nonuniform convergence is observed in these cases, andO(1) errors may appear in the algebraic variables. From the viewpoint of error control, it is beneficial to allow smooth changes of step size, and since most codes based on BDFs are of variable order, it is also of interest to avoid the inaccuracies caused by a change of order of the method. In the case of BDFs applied to index 3 DAEs in semi-explicit form, we present algorithms that correct toO(h) the inaccurate approximations to the algebraic variables when there are changes of step size in the backward Euler method. These algorithms can be included in an existing code at a very small cost. We have also described how to obtain formulas that correct theO(1) errors in the algebraic variables appearing after a change of order.This author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolívar (CESMa) for permitting her free use of its research facilities.     

The uniqueness and existence of solutions for the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation

In this paper, we study the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation, allowing the presence of guided waves. Our assumptions on the perturbed and source terms are too few. On the basis of the Green's function for the 3D homogeneous Helmholtz equation in a step‐index waveguide without perturbation, we introduce a generalized (out‐going) Sommerfeld–Rellich radiation condition, and then we prove the uniqueness and existence of solutions for the studied 3D Helmholtz equation satisfying our radiation condition. Copyright © 2012 John Wiley & Sons, Ltd.