On the relation between stability and contractivity
The author treats linear initial value problems \[ \frac{d}{dt}U(t)=A(t)U(t),\quad U(0)=u_ 0, \] where, for an arbitrary fixed norm on \({\mathbb{R}}^ s\), \(\| U(t)\|\) is non-increasing. A general type of method is considered which generates a sequence \(\{u_ n\}\) of approximations to \(\{U(t_ n)\}\) where, with arbitrary steplengths, \(t_ n=h_ 1+h_ 2+...+h_ n\). A method is weakly stable if there is a bound \(\| u_ n\| \leq \gamma n^ q\| u_ 0\|,\) \(n=1,2,3,..\).. It is shown that for single step methods where \(u_ n=\phi (A_ n)u_{n-1},\) \(n=1,2,3,...\), with \(\phi\) a rational function, weak stability implies the stronger contractivity bound \(\| u_ n\| \leq \| u_{n-1}\|\), \(n=1,2,3,..\).. It is known that this implies that the order is at most one. It is proved that this is also the case for more general methods under the assumption of weak stability. The order barrier does not hold if attention is restricted to norms generated by inner products.
scientific article; zbMATH DE number 3888824