On a class of multistep methods with strong contractivity properties (Q758139)

Contractivity of second-derivative linear multistep methods for the test equation \(y'=\lambda (t)y\) is investigated. If \(\lambda\) (t) is monotonic then necessary and sufficient conditions for contractivity on intervals on the negative real axis are given. Such a condition requires that \(h^ 2\lambda '(t)\) stays bounded where h denotes the stepsize. If \(\lambda\) (t) is increasing the \(A_ 0\)-contractive schemes involve the first and second derivative on the newest time level only. The maximal order is 2. For these schemes conditions are given when they are A-contractive or only A(\(\alpha\))-contractive for the constant linear test equation. A class of \(A_ 0\)-contractive k-step schemes of order 2 which depends on k-1 free parameters is constructed. If \(\lambda\) (t) is decreasing there is no \(A_ 0\)-contractive scheme.