Summary.We investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations. The d emphasis is on the general test-equation dt U(t)= AU(t), where A is a square matrix of arbitrary order s > 1. The contractivity is analysed with respect to arbitrary norms in the s-dimensional space (which are not necessarily generated by an inner product). For given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize.

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Abstract. This paper deals with the stability analysis of one-step methods in the numerical solution of initial (-boundary) value problems for linear, ordinary, and partial differential equations. Restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size. These restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration.The errors in the one-step methods are measured in arbitrary norms (not necessarily generated by an inner product).The theory is illustrated in the numerical solution of the heat equation and some other differential equations, where the error growth is measured in the maximum norm.

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Abstract.We investigate contractivity properties of implicit linear multistep methods in the numerical solution of ordinary differential equations. The emphasis is on nonlinear and linear systems 4:tJ(t) = f(t, U(t)), where / satisfies a so-called circle condition in an arbitrary norm. The results for the two types of systems turn out to be closely related. We construct optimal multistep methods of given order and stepnumber, which allow the use of a maximal stepsize.