On the zero-stability of multistep methods on smooth nonuniform grids

TL;DR

This paper investigates the zero stability of linear multistep methods on smooth, nonuniform grids, establishing conditions under which stability is maintained as the grid becomes increasingly refined.

Contribution

It provides a theoretical framework showing zero stability for multistep methods on smooth nonuniform grids with controlled step size ratios, extending classical stability results.

Findings

Zero stability holds for sufficiently large N on smooth grids

Stability is guaranteed if the grid deformation map is twice continuously differentiable

Results are exemplified for BDF-type methods

Abstract

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid $\{t_n\}_{n=0}^N$ can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., $t_n = \Phi(\tau_n)$, where $\tau_n = n/N$ and the map $\Phi$ is monotonically increasing with $\Phi(0)=0$ and $\Phi(1)=1$. The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines $\Phi$, and a tolerance requirement which determines $N$. Given any strongly stable multistep method, there is an $N^*$ such that the method is zero stable for $N>N^*$, provided that $\Phi \in C^2[0,1]$. Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy $h_n/h_{n-1} = 1 + \mathrm O(N^{-1})$ as $N\rightarrow\infty$. The results are exemplified for BDF-type methods.