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

Imre Fekete (Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Hungary), Gustaf Söderlind, István Faragó

In this talk we investigate zero stability on compact intervals and smooth nonuniform grids. The grid points \(\{t_n\}_{n=0}^N\) are constructed 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 \(\Phi(1)=1\). We show that for all strongly stable linear multistep methods, there is an \(N^*\) such that a condition of zero stability is always fulfilled for \(N>N^*\), provided that \(\Phi \in C^2[0,1]\). Thus zero stability is maintained whenever adjacent step sizes satisfy \(h_n/h_{n-1} = 1 + \mathrm O(N^{-1})\). This suggests that variable step size should always be implemented using smooth step size changes.
The talk is based on the paper
G. Söderlind, I. Fekete, I. Faragó: On the zero-stability of multistep methods on smooth nonuniform grids, BIT Numer. Math., https://doi.org/10.1007/s10543-018-0716-y, 2018.