Abstract:
Starting from an A-stable rational approximation to $e^z$ of order $p$, $$r(z) = 1 + z + · · · + z^p/p! + O(z^p+1), $$ families of stable methods are proposed to time discretize abstract initial value problems of the type $u'(t) = Au(t) + f(t)$. These numerical procedures turn out to be of order p, thus overcoming the order reduction phenomenon. A first approach to extend the methods to semilinear problems of the form $u'(t) = Au(t) + f(t,u)$ is also presented.