On the zero-stability of the 3-step BDF-formula on nonuniform grids

In this paper a bound on the quotienth _j+1/h _j of adjacent step-sizes is derived such that the three-stepBDF-formula behaves zero-stable on any sequence of non-uniform grids satisfying this bound.Supported in part by FINEP grant no. 4.3.82.0179.00 and CNPq grant no. 302482-MA.     

On the zero-stability of variable stepsize multistep methods: the spectral radius approach

Summary. In this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients. In particular, we deal with the zero-stability of the 3-step BDF-method on grids with variable stepsize for the numerical solution of IVPs for ODEs. Our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results, which give a slight improvement to the best results already known from the literature. The success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods. Received May 19, 1999 / Revised version received February 21, 2000 / Published online November 8, 2000     

Monotonicity on nonuniform grids

In this paper we study a class of numerical methods used to solve two-point boundary value problems on nonuniform grids. Particular attention is devoted to positive solutions, i.e. conditions under which the solutions of the problem are positive. Applications to steady states of air pollution problems are also referred to.     

Stability of implicit schemes on nonuniform grids

Implicit finite-difference approximations of the quasilinear conservation law are considered. Issues of stability and convergence are discussed, and an accuracy bound is obtained.Translated from Matematicheskie Modeli Estestvoznaniya, Published by Moscow University, Moscow, 1995, pp. 157–161.     

On the convergence of advanced linear multistep methods

A convergence theorem is given showing that zero-stable advanced linear multistep methods with orderp consistency have orderp convergence.     

On the instability of high order backward-difference multistep methods

It is shown that the backward difference multistep method $$\mathop \Sigma \limits_{m = 1}^q \frac{1}{m}\nabla ^m y_p = hf_p$$ for the numerical integration ofy′(x)=f(x, y) is stable in the sense of Dahlquist iff 1≦q≦6.     

On the stability properties of Brown's multistep multiderivative methods

Summary Brown introducedk-step methods usingl derivatives. We investigate for whichk andl the methods are stable or unstable. It is seen that to anyl the method becomes unstable fork large enough. All methods withk2(l+1) are stable. Fork=1,2,..., 18 there exists a _k such that the methods are stable for anyl _k and unstable for anyl < _k . The _k are given.     

On multistep interval methods for solving the initial value problem

In this paper we shortly complete our previous considerations on interval versions of Adams multistep methods [M. Jankowska, A. Marciniak, Implicit interval multistep methods for solving the initial value problem, Comput. Meth. Sci. Technol. 8(1) (2002) 17–30; M. Jankowska, A. Marciniak, On explicit interval methods of Adams–Bashforth type, Comput. Meth. Sci. Technol. 8(2) (2002) 46–57; A. Marciniak, Implicit interval methods for solving the initial value problem, Numerical Algorithms 37 (2004) 241–251]. It appears that there exist two families of implicit interval methods of this kind. More considerations are dealt with two new kinds of interval multistep methods based on conventional well-known Nyström and Milne–Simpson methods. For these new interval methods we prove that the exact solution of the initial value problem belongs to the intervals obtained. Moreover, we present some estimations of the widths of interval solutions. Some conclusions bring this paper to the end.     

On monotonicity and boundedness properties of linear multistep methods

In this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods. Instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with Runge-Kutta starting procedures. This allows many multistep methods of practical interest to be included in the theory. In a related manner, we also consider contractivity and stability in arbitrary norms.

     

On the order of composite multistep methods for ordinary differential equations

Summary In this paper, a general class ofk-step methods for the numerical solution of ordinary differential equations is discussed. It is shown that methods with order of consistencyq have order of convergence (q+1) if a very simple condition is satisfied. This result gives a new aspect to previous results of Spijker; it also serves as a starting point for a new theory of cyclick-step methods, completing an approach of Donelson and Hansen. It facilitates the practical determination of high-order cyclick-step methods, especially of stiffly stable,k-step methods.     

On the convergence of multistep methods for nonlinear stiff differential equations

Summary Convergence estimates are given forA()-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturbation arguments with frequency domain techniques.     

Stiffness characteristic and the applications of the method of lines on nonuniform grids

The method of lines is investigated for the numerical solution of the stream-function-and-vorticity form of the Navier-Stokes equations on nonuniform grids. Stiffness characteristics of a linear one-dimensional model equation are examined to establish the feasibility of applying the method to the vorticity equation in two dimensions. The governing equations are transformed from the physical domain with a highly variable grid to a computational domain with a uniform grid. The method of lines is used to solve only the vorticity equation, and the successive-over relaxation technique is used to solve the stream-function equation. It is observed that the transformed governing equations become stiffer with increased concentration of grid points and also as the number of grid points increases. It is also observed that the differencing technique affects the stiffness characteristics. The use of forward differencing is not feasible, and backward differencing is preferable to central differencing for high Reynolds numbers. The results of specific applications for the solution of flow in curved-wall diffusers and a driven cavity demonstrate that the method of lines under certain circumstances is feasible for the numerical solution of physical problems on domains covered with variable grids.     

Splitting algorithms for the wavelet transform of first-degree splines on nonuniform grids

For the splines of first degree with nonuniform knots, a new type of wavelets with a biased support is proposed. Using splitting with respect to the even and odd knots, a new wavelet decomposition algorithm in the form of the solution of a three-diagonal system of linear algebraic equations with respect to the wavelet coefficients is proposed. The application of the proposed implicit scheme to the point prediction of time series is investigated for the first time. Results of numerical experiments on the prediction accuracy and the compression of spline wavelet decompositions are presented.     

On richardson extrapolation for finite difference methods on regular grids

Summary Difference solutions of partial differential equations can in certain cases be expanded by even powers of a discretization parameterh. If we haven solutions corresponding to different mesh widthsh ₁,...,h _n we can improve the accuracy by Richardson extrapolation and get a solution of order 2n, yet only on the intersection of all grids used, i.e. normally on the coarsest grid. To interpolate this high order solution with the same accuracy in points not belonging to all grids, we need 2n points in an interval of length (2n–1)h ₁.This drawback can be avoided by combining such an interpolation with the extrapolation byh. In this case the approximation depends only on grid points in an interval of length 3/2h ₁. The length of this interval is independent of the desired order.By combining this approach with the method of Kreiss, boundary conditions on curved boundaries can be discretized with a high order even on coarse grids.This paper is based on a lecture held at the 5th Sanmarinian University Session of the International Academy of Sciences San Marino, at San Marino, 1988-08-27-1988-09-05     

Exponential multistep methods of Adams-type

The paper is concerned with the construction, implementation and numerical analysis of exponential multistep methods. These methods are related to explicit Adams methods but, in contrast to the latter, make direct use of the exponential and related matrix functions of a (possibly rough) linearization of the vector field. This feature enables them to integrate stiff problems explicitly in time.     

Stability of linear multistep methods on the imaginary axis

The stability of linear multistep methods of order higher than one is investigated for hyperbolic equations. By means of the Routh array and the Hermite-Biehler theorem, the stability boundary on the imaginary axis is expressed in terms of the error constant of the third order term. As a corollary we state the result that the stability boundary for methods of order higher than two, is at most 3, and this value is attained by the Milne-Simpson method.This work was done during the author's stay at the Mathematical Centre Amsterdam, and the University of Technology, Eindhoven.     

On the local error and the local truncation error of linear multistep methods

Two different measures of the local accuracy of a linear multistep method — the local error and the local trunction error — appear in the literature. It is shown that the principal parts of these errors are not identical for general linear multistep methods, but that they are so for a sub-class which contains all methods of Adams type. It is sometimes argued that local error is the more natural measure; this view is challenged.     

On the convergence of nonnested multigrid methods with nested spaces on coarse grids

Multigrid methods for discretized partial differential problems using nonnested conforming and nonconforming finite elements are here defined in the general setting. The coarse‐grid corrections of these multigrid methods make use of different finite element spaces from those on the finest grid. In general, the finite element spaces on the finest grid are nonnested, while the spaces are nested on the coarse grids. An abstract convergence theory is developed for these multigrid methods for differential problems without full elliptic regularity. This theory applies to multigrid methods of nonnested conforming and nonconforming finite elements with the coarse‐grid corrections established on nested conforming finite element spaces. Uniform convergence rates (independent of the number of grid levels) are obtained for both the V and W‐cycle methods with one smoothing on all coarse grids and with a sufficiently large number of smoothings solely on the finest grid. In some cases, these uniform rates are attained even with one smoothing on all grids. The present theory also applies to multigrid methods for discretized partial differential problems using mixed finite element methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 265–284, 2000     

On some explicit Adams multistep methods for fractional differential equations

In this paper we present a family of explicit formulas for the numerical solution of differential equations of fractional order. The proposed methods are obtained by modifying, in a suitable way, Fractional-Adams–Moulton methods and they represent a way for extending classical Adams–Bashforth multistep methods to the fractional case. The attention is hence focused on the investigation of stability properties. Intervals of stability for k$k$-step methods, k=1,…,5$k=1,\dots ,5$, are computed and plots of stability regions in the complex plane are presented.