The author continues several earlier studies of Picard-Lindelöf iteration of linear constant coefficient systems \(\dot x+Ax=f(t),\) \(x(0)=x_ 0\), where the matrix A is assumed to be split as \(A=M-N\) for iteration \(\dot x^ n+Mx^ n=Nx^{n-1}+f(t),\) \(x^ n(0)=x_ 0\) [see e.g. the author, ibid. 57, No.2, 147-156 (1990; Zbl 0697.65058)]. There are two obvious ways to try to speed up the basic iteration: either by accelerating the convergence or by lessening the work done with iterating. In the present paper the author obtains a result which says that if one keeps refining the mesh the iteration errors decay essentially independently of the refinement process. The analysis to prove these results is done among grid functions identifying them as sequences in \(l_ 2\)-spaces. Each sweep is done by a linear multistep method with a fixed time step. As one is forced to prolongate the iterate on the finer mesh each time the step is decreased, an essential point in those results is to make sure that repeated prolongation itself is stable. It is shown that there are arbitrarily high order nonexpansive prolongations in an \(l_ 2\)-setting and that those particular prolongations are power bounded in the \(l_{\infty}\)- norm.