Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems

This paper deals with the analysis of general one-step methods for the numerical solution of initial (-boundary) value problems for stiff ordinary and partial differential equations. Restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size. These restrictions are related to disks contained in the stability region of the method, and the errors are measured with arbitrary norms (not necessarily generated by an inner product). The theory is illustrated in the numerical solution of a diffusion-convection problem where the error growth is measured with the maximum norm.