This paper continues earlier work by the same author concerning the stability and B-convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable, Bor optimally B-convergent ...

Linear multistep methods for the second order differential equation y" = 2 — X y, X real, are said to have an interval of periodicity if for a fixed X. and a stepsize sufficiently small the numerical solution neither explodes nor decays. We give a very simple necessary and sufficient condition under which a linear multistep method has an interval of periodicity. This condition is then applied t...

Fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular Volterra integral equations. The proposed methods are convergent of the order of the underlying multistep method, also in the generic case of solutions which are not smooth at the origin. The stability properties (stability region, A-stability, A(a)-stability) are closely related to those o...

In this paper we consider a large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales, a slow one and a fast one. A small parameter t characterizes the stiffness of these systems. We approximate a system of ODE's in this set by a general class of multistep discretizations which includes both one-leg and linear multistep methods. ...

This paper presents a family of generalized multistep methods that evolves the numerical solution of ordinary di erential equations on con guration spaces formulated as homogeneous manifolds. Any classical multistep method may be employed as an invariant method, and the order of the invariant method is as high as in the classical setting. We present numerical results that re ect some of the pro...

Diagonally split Runge–Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge–Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) timest...

Monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution. We formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations, and propose an efficient algorithm for its solution. This algorithm reliably finds optimal meth...

in this paper, we will present a review of the multistep collocation method for delay volterra integral equations (dvies) from [1] and then, we study the superconvergence analysis of the multistep collocation method for dvies. some numerical examples are given to confirm our theoretical results.